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3-D flow of a compressible viscous micropolar fluid with spherical symmetry: a local existence theorem

Ivan Dražić1* and Nermina Mujaković2

Author Affiliations

1 Faculty of Engineering, University of Rijeka, Rijeka, Croatia

2 Department of Mathematics, University of Rijeka, Rijeka, Croatia

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Boundary Value Problems 2012, 2012:69  doi:10.1186/1687-2770-2012-69

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/69


Received:26 December 2011
Accepted:12 June 2012
Published:2 July 2012

© 2012 Dražić and Mujaković; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider nonstationary 3-D flow of a compressible viscous heat-conducting micropolar fluid in the domain to be the subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M1">View MathML</a> bounded with two concentric spheres that present solid thermoinsulated walls. In thermodynamical sense fluid is perfect and polytropic. Assuming that the initial density and temperature are strictly positive we will prove that for smooth enough spherically symmetric initial data there exists a spherically symmetric generalized solution locally in time.

Keywords:
micropolar fluid; generalized solution; spherical symmetry; weak and strong convergence

1 Introduction

The theory of micropolar fluids is introduced in [1] by Eringen. Various problems with different initial and boundary conditions for incompressible micropolar fluid are presented in [2], but the theory of compressible micropolar fluid is still in the beginning. N. Mujakovic in [3] developed the model for one-dimensional isotropic, viscous, compressible micropolar fluid which is in thermodynamical sense perfect and polytropic. In the same work, the local existence of the solution for homogeneous boundary conditions is proved. N. Mujakovic in [4] and in the references cited therein proved the local and global existence of inhomogeneous boundary conditions for velocity and microrotation as well as stabilization and regularity. In [5] the Cauchy problem for the described problem was also considered. In the last years we find some interesting works with different kind of problems concerning micropolar fluid, e.g., [6,7], but till now the described model of compressible micropolar fluid in three-dimensional case has not been considered.

In this work we consider the three-dimensional model with spherical symmetry. The first article in which the problem of spherical symmetry was described is [8], but for classical fluid. The spherical symmetry for classical fluid is also considered in articles [9-12].

In the setting of the field equations we use the Eulerian description.

In what follows we use the notation:

ρ - mass density

v - velocity

p - pressure

T - stress tensor

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M2">View MathML</a> - an axial vector with the Cartesian components <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M3">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M4">View MathML</a> is Levi-Civita alternating tensora

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M6">View MathML</a>

ω - microrotation velocity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M7">View MathML</a> - skew tensor with Cartesian components <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M8">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M9">View MathML</a> - microinertia density (a positive scalar field)

C - couple stress tensor

θ - absolute temperature

E - internal energy density

q - heat flux density vector

f - body force density

g - body couple density

δ - body heat density

The problem we consider here is based on local forms of the conservation laws for mass, momentum, momentum moment and energy, which are stated respectively as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M10">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M11">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M12">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M13">View MathML</a>

(4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M14">View MathML</a> denotes material derivative of a field a:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M15">View MathML</a>

The scalar product of tensors A and B is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M16">View MathML</a>

The linear constitutive equations for stress tensor, couple stress tensor and heat flux density vector are respectively in the forms:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M17">View MathML</a>

(5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M18">View MathML</a>

(6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M19">View MathML</a>

(7)

where

λ, μ - coefficients of viscosity,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M23">View MathML</a> - coefficients of microviscosity,

k - heat conduction coefficient

are constants with the properties

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M24">View MathML</a>

(8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M25">View MathML</a>

(9)

Assuming that the fluid is perfect and polytropic, for pressure and internal energy we have the equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M26">View MathML</a>

(10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M27">View MathML</a>

(11)

where R and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M28">View MathML</a> are positive constants.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M30">View MathML</a>, denote the domain bounded by two concentric spheres with radii a and b. The boundary of the described domain is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M31">View MathML</a>. We shall consider the problem (1)-(11) in the region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M32">View MathML</a> (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M33">View MathML</a> is arbitrary) with the following initial conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M34">View MathML</a>

(12)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M35">View MathML</a> and boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M36">View MathML</a>

(13)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M37">View MathML</a>; the vector ν is an exterior unit normal vector.

For simplicity we also assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M39">View MathML</a>.

The initial boundary problems for the system (1)-(13) so far have not been considered in three-dimensional case. The same and similar models in one-dimensional case were considered in [3,5,13] and [4]. In [2] the three-dimensional model was considered but for an incompressible micropolar fluid.

In this paper we prove the local existence of generalized spherically symmetric solution to the problem (1)-(13) in the domain Ω, assuming that the initial functions are also spherically symmetric. In the proof we use the Faedo-Galerkin method. We follow some ideas of [14] where this method was applied to a classical fluid (where microrotation is equal to zero) in one-dimensional case as well as the ideas from [3] and [13] where the same result as here was provided for one-dimensional case.

The paper is organized as follows. In the second section, we derive a spherically symmetric form of (1)-(4), introduce Lagrangian description, and present the main result. In the third section, we consider an approximate problem and get an approximate solution for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M40">View MathML</a>. In the forth section, we prove uniform a priori estimates for the approximate solutions. The proof of the main result is given in the fifth section.

2 Spherically symmetric form and the main result

We first derive the spherically symmetric form of (1)-(7) and (10)-(11). A spherically symmetric solution of (1)-(7) has the form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M41">View MathML</a>

(14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M45">View MathML</a>. We assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M46">View MathML</a>

(15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M49">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M50">View MathML</a> are known real functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M51">View MathML</a>, and thus we reduce system (1)-(7) and conditions (10)-(13) to the following equations for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M55">View MathML</a> of the form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M56">View MathML</a>

(16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M57">View MathML</a>

(17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M58">View MathML</a>

(18)

(19)

with the following initial and boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M60">View MathML</a>

(20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M61">View MathML</a>

(21)

To investigate the local existence, it is convenient to transform the system (16)-(19) to that in Lagrangian coordinates. The Eulerian coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M62">View MathML</a> are connected to the Lagrangian coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M63">View MathML</a> by the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M64">View MathML</a>

(22)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M65">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M66">View MathML</a>

(23)

We introduce the new function η by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M67">View MathML</a>

(24)

Note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M68">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M69">View MathML</a> (which is assumed in Theorem 2.1 later), then there exists an inverse function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M70">View MathML</a>. Let the constant L be defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M71">View MathML</a>

(25)

From (16) we can easily get the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M72">View MathML</a>

(26)

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M73">View MathML</a>

(27)

It is useful to introduce the next coordinate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M74">View MathML</a>

(28)

and the following functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M75">View MathML</a>

Similarly as in [15], for a new coordinate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M76">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M77">View MathML</a>

(29)

Taking into account (26) and (24), we obtain that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M81">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M82">View MathML</a> satisfy the system that we write omitting the primes for simplicity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M83">View MathML</a>

(30)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M84">View MathML</a>

(31)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M85">View MathML</a>

(32)

(33)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M87">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M33">View MathML</a> is arbitrary. Now we have the following boundary and initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M89">View MathML</a>

(34)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M90">View MathML</a>

(35)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M91">View MathML</a>

(36)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M92">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M93">View MathML</a>

(37)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M94">View MathML</a>

(38)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M95">View MathML</a>. We also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M96">View MathML</a>

(39)

From

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M97">View MathML</a>

putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M98">View MathML</a> and integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M99">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M100">View MathML</a>

(40)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M30">View MathML</a> is a radius of the smaller boundary sphere.

We assume the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M102">View MathML</a>

(41)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M103">View MathML</a>.

Before stating the main result, we introduce the following definition.

Definition 2.1 A generalized solution of the problem (30)-(38) in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M104">View MathML</a> is a function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M105">View MathML</a>

(42)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M106">View MathML</a>

(43)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M107">View MathML</a>

(44)

that satisfies Equations (30)-(33) a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M108">View MathML</a> and conditions (34)-(38) in the sense of traces.

Remark 2.1 From the embedding and interpolation theorems (e.g., [16] and [17]) one can conclude that from (43) and (44) it follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M109">View MathML</a>

(45)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M110">View MathML</a>

(46)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M111">View MathML</a>

(47)

It is easy to check that the solution (42) with properties (43)-(44) satisfies the condition for a strong solution of the described problem.

The aim of this paper is to prove the following statements.

Theorem 2.1Let the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M112">View MathML</a>

(48)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M113">View MathML</a>

(49)

satisfy conditions (41). Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M115">View MathML</a>, such that the problem (30)-(38) has a generalized solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M116">View MathML</a>, having the property

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M117">View MathML</a>

(50)

For the functionr, it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M118">View MathML</a>

(51)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M119">View MathML</a>

(52)

Remark 2.2 Notice that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M120">View MathML</a> introduced by (40) belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M121">View MathML</a>. Because of the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M122">View MathML</a> we can conclude that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M123">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M124">View MathML</a>

(53)

From (40) and (41) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M125">View MathML</a>

(54)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M126">View MathML</a>

(55)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M127">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M128">View MathML</a>.

The proof of Theorem 2.1 is essentially based on a careful examination of a priori estimates and a limit procedure. We first study, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M40">View MathML</a>, an approximate problem and derive the a priori estimates for approximate solutions independent of n by utilizing a technique of Kazhikov [14,18] and Mujakovic [3,13] for one-dimensional case. Using the obtained a priori estimates and results of weak compactness, we extract the subsequence of approximate solutions, which, when n tends to infinity, has limit in the same weak sense on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M130">View MathML</a> for sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M115">View MathML</a>. Finally, we show that this limit is the solution to our problem.

3 Approximate solutions

We shall find a local generalized solution to the problem (30)-(38) as a limit of approximate solutions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M133">View MathML</a>

(56)

obtained in what follows. First, we introduce the approximations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M135">View MathML</a> of the functions v and r by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M136">View MathML</a>

(57)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M137">View MathML</a>

(58)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M138">View MathML</a> is defined by (40) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M140">View MathML</a> are unknown smooth functions defined on an interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M142">View MathML</a>.

Then, we can write the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M143">View MathML</a> to the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M144">View MathML</a>

(59)

in the similar way as in [3] and [13] in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M145">View MathML</a>

(60)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M135">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134">View MathML</a> are sufficiently smooth functions, we can conclude that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M143">View MathML</a> is continuous on the rectangle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M149">View MathML</a> with the property <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M150">View MathML</a>. Because of aforementioned, we can conclude that there exists such <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M152">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M153">View MathML</a>

(61)

We also introduce the approximations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M154">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M155">View MathML</a> of the functions ω and θ respectively by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M156">View MathML</a>

(62)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M157">View MathML</a>

(63)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M158">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M159">View MathML</a> are again unknown smooth functions defined on an interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M142">View MathML</a>.

Evidently, the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M162">View MathML</a>

(64)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M163">View MathML</a> are satisfied.

According to the Faedo-Galerkin method, we take the following approximate conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M164">View MathML</a>

(65)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M165">View MathML</a>

(66)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M166">View MathML</a>

(67)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M168">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M170">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M171">View MathML</a> be the Fourier coefficients of the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M49">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M50">View MathML</a> respectively:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M175">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M177">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M178">View MathML</a> be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M179">View MathML</a>

(68)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M180">View MathML</a>

(69)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M181">View MathML</a>

(70)

We take the initial conditions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M154">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M155">View MathML</a> in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M185">View MathML</a>

(71)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M186">View MathML</a>

(72)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M187">View MathML</a>

(73)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M189">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M190">View MathML</a> be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M191">View MathML</a>

(74)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M192">View MathML</a>

(75)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M193">View MathML</a>

(76)

then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M194">View MathML</a>

(77)

(78)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M138">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M197">View MathML</a> are known functions. Taking into account (57), (62), (63), (74)-(78), from (65)-(67) we obtain for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M198">View MathML</a>, the following Cauchy problem:

(79)

(80)

(81)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M202">View MathML</a>

(82)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M203">View MathML</a>

(83)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M204">View MathML</a>

(84)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M205">View MathML</a>

(85)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M206">View MathML</a>

(86)

Here we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M208">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M209">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M210">View MathML</a>

(87)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M211">View MathML</a>

(88)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M212">View MathML</a>

(89)

Notice that the functions on the right-hand side of (79)-(84) satisfy the conditions of the Cauchy-Picard theorem [19,20] and we can easily conclude that the following statements are valid.

Lemma 3.1For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M213">View MathML</a>there exists such<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M152">View MathML</a>that the Cauchy problem (79)-(86) has a unique solution defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M141">View MathML</a>. The functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M154">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M155">View MathML</a>defined by the formulas (57), (62) and (63) belong to the class<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M221">View MathML</a>and satisfy conditions (71)-(73).

From (77) and (78) we can also easily conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M222">View MathML</a>

(90)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M223">View MathML</a>

(91)

Lemma 3.2There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M152">View MathML</a>, such that the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M135">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M228">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M229">View MathML</a>

(92)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M230">View MathML</a>

(93)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M231">View MathML</a>

(94)

on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M232">View MathML</a>. The constantsm, a, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M233">View MathML</a>, Mand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M234">View MathML</a>are introduced by (40), (41), (53) and (55).

Proof The statements follow from (90)-(91), (41), (53) and (55). □

4 A priori estimates

Our purpose is to find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M115">View MathML</a> such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M213">View MathML</a> there exists a solution to the problem (79)-(86), defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M238">View MathML</a>. It will be sufficient to find uniform (in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M213">View MathML</a>) a priori estimates for the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M240">View MathML</a> defined through Lemmas 3.1 and 3.2.

In what follows we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M241">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M242">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M243">View MathML</a>) a generic constant, not depending on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M40">View MathML</a>, which may have different values at different places.

We also use the notation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M245">View MathML</a>

Some of our considerations are very similar or identical to that of [3] or [13]. In these cases we omit proofs or details of proofs making references to corresponding pages of the articles [3] or [13].

Lemma 4.1For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M246">View MathML</a>it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M247">View MathML</a>

(95)

Proof From (58) follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M248">View MathML</a>

and using Remark 2.2 we get (95) immediately. □

Lemma 4.2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M246">View MathML</a>, the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M250">View MathML</a>

(96)

Proof Multiplying (66) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M251">View MathML</a> and summing over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M252">View MathML</a>, after integration by parts, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M253">View MathML</a>

Integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M255">View MathML</a>, and taking into account (72), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M256">View MathML</a>

Using (92), we get (96). □

In what follows, we use the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M257">View MathML</a>

(97)

(for a function f vanishing at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M258">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M259">View MathML</a> and with the first derivative vanishing at some point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M260">View MathML</a>) that satisfy the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M154">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M263">View MathML</a>.

Lemma 4.3For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M264">View MathML</a>, the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M265">View MathML</a>

(98)

Proof Multiplying (65) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M266">View MathML</a> and summing over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M267">View MathML</a>, after integration by parts and adding to (67) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M268">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M269">View MathML</a>

Integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M255">View MathML</a> and using (92) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M272">View MathML</a>

Taking into account (96), (71), (73), and (97) we obtain (98). □

Lemma 4.4 ([3], Lemma 5.3)

For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M273">View MathML</a>, the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M274">View MathML</a>

(99)

Lemma 4.5For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M246">View MathML</a>, the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M276">View MathML</a>

(100)

Proof Taking the derivative of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M143">View MathML</a> with respect to x and using the estimates (92)-(94), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M278">View MathML</a>

With the help of (97) applied to the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134">View MathML</a>, the Hoelder and Young inequalities as well as (95), we get (100). □

Lemma 4.6For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M246">View MathML</a>it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M281">View MathML</a>

(101)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M282">View MathML</a>

Proof As in [3] Lemma 5.5, [14] pp.63-66 and in [13] Lemma 5.6, multiplying (65), (66) and (67) respectively by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M283">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M284">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M285">View MathML</a> and taking into account (57), (62) and (63), after summation over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M286">View MathML</a> and addition of the obtained equations, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M287">View MathML</a>

(102)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M288">View MathML</a>

Taking into account (92)-(94) and (95)-(100), we estimate the terms on the right-hand side of (102). For instance,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M289">View MathML</a>

Applying the Young inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M290">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M291">View MathML</a> is arbitrary. In the analogous way, we obtain the following inequalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M292">View MathML</a>

Using these inequalities with sufficiently small ε and estimates (92)-(94), from (102) we get (101). □

Lemma 4.7There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M115">View MathML</a>) such that for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M40">View MathML</a>the Cauchy problem (79)-(86) has a unique solution defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M238">View MathML</a>. Moreover, the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M143">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M135">View MathML</a>satisfy the inequalities

(103)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M303">View MathML</a>

(104)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M304">View MathML</a>

(105)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M305">View MathML</a>

(106)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M306">View MathML</a>

(107)

(a, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M233">View MathML</a>, mandMare defined by (41) and (53)-(55)).

Proof To get the estimate (103) we use an approach similar to that in [3] (Lemma 5.6) and [14] (pp.64-67). First, we introduce the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M308">View MathML</a>

(108)

Using Lemma 4.6, we find that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M309">View MathML</a> satisfies the differential inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M310">View MathML</a>

(109)

There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M311">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M312">View MathML</a>

and we can conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M313">View MathML</a>

(110)

We compare the solution of the problem (109)-(110) with the solution of the Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M314">View MathML</a>

(111)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M315">View MathML</a>

(112)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M316">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M317">View MathML</a> be an existence interval of the solution of the problem (111)-(112). From (109)-(112) we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M318">View MathML</a>

(113)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M320">View MathML</a>. From (113) and (108) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M321">View MathML</a>

(114)

and from (101) it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M322">View MathML</a>

(115)

Integrating (101) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M324">View MathML</a> and using estimates (110) and (115), we immediately get (103).

Now, using the inequalities (97) for the function v, we easily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M325">View MathML</a>

(116)

Using (116), we derive the following estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M326">View MathML</a>

(117)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M327">View MathML</a>

(118)

where C and B are from (103). Assuming that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M328">View MathML</a>

and using (117) and (118) from (58) and (60), we get (104)-(105).

Because of (57), (62) and (63), from (103) and (98), we easily get that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M329">View MathML</a> hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M330">View MathML</a>

(119)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M331">View MathML</a>

(120)

and we can conclude that the solution of the problem (79)-(86) is defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M238">View MathML</a>. □

From (119) and (120), we can easily conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M333">View MathML</a>

(121)

and from (95), (100) and (99) it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M334">View MathML</a>

(122)

Lemma 4.8Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114">View MathML</a>be defined by Lemma 4.7. Then for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M336">View MathML</a>the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M337">View MathML</a>

(123)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M338">View MathML</a>

(124)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M339">View MathML</a>

(125)

hold true.

Proof Multiplying (65) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M340">View MathML</a>, summing over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M140">View MathML</a> and using (104)-(105), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M342">View MathML</a>

(126)

Using (121), (122), (103), (116) and applying the Young inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M343">View MathML</a>

(127)

Taking into account (103) for sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M291">View MathML</a> from (127), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M345">View MathML</a>

(128)

In the same way, from (66) and (67) we obtain the estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M346">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M347">View MathML</a> respectively. The estimates (124) and (125) follow directly from (59) and (58). □

From Lemmas 4.7 and 4.8 we easily derive the following statements.

Proposition 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114">View MathML</a>be defined by Lemma 4.7. Then for the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M349">View MathML</a>the following properties are satisfied:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M350">View MathML</a>is bounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M351">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M352">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M353">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M354">View MathML</a>is bounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M351">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M356">View MathML</a>is bounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M351">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M358">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M359">View MathML</a>;

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M360">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M361">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M362">View MathML</a>are bounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M358">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M359">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M365">View MathML</a>.

5 The proof of Theorem 2.1

In the proofs we use some well-known facts of functional analysis (e.g., [21]). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M366">View MathML</a> be defined by Lemma 4.7. Theorem 2.1 is a consequence of the following lemmas.

Lemma 5.1There exists a function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M367">View MathML</a>

(129)

and the subsequence of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M350">View MathML</a> (for simplicity reasons denoted again as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M350">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M370">View MathML</a>

(130)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M371">View MathML</a>

(131)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M372">View MathML</a>

(132)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M373">View MathML</a>

(133)

The functionrsatisfies the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M374">View MathML</a>

(134)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M375">View MathML</a>

(135)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M120">View MathML</a>is defined by (40).

Proof The conclusions (130) and (131) follow immediately from Proposition 4.1. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M377">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M378">View MathML</a> belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M379">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M380">View MathML</a>

Using (104), (58), (116), (103) and (107), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M381">View MathML</a>

(136)

(137)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M383">View MathML</a>

(138)

(139)

and we can conclude that the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M385">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M386">View MathML</a> satisfy the conditions of Arzelà-Ascoli theorem. Applying that theorem, we get the strong convergence in (132) and (133). Because of (132) and (104) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M387">View MathML</a>

(140)

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M291">View MathML</a> and sufficiently big <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M40">View MathML</a>. From (140) we can easily conclude that (134) is satisfied. From (132) it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M390">View MathML</a>

(141)

and because of that we have (135). □

Lemma 5.2There exists a function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M391">View MathML</a>

(142)

and the subsequence of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M356">View MathML</a> (denoted again as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M356">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M394">View MathML</a>

(143)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M395">View MathML</a>

(144)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M396">View MathML</a>

(145)

The functionρsatisfies the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M397">View MathML</a>

(146)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M398">View MathML</a>

(147)

Proof Taking into account Proposition 4.1, estimates (103)-(106) and the Arzelà-Ascoli theorem, we prove in the same way as in the previous lemma the properties (143)-(147). □

Lemma 5.3There exist functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M399">View MathML</a>

and the subsequence of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M400">View MathML</a> (denoted again as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M400">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M402">View MathML</a>

(148)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M403">View MathML</a>

(149)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M404">View MathML</a>

(150)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M405">View MathML</a>

(151)

The functionsv, ωandθsatisfy the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M406">View MathML</a>

(152)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M407">View MathML</a>

(153)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M408">View MathML</a>

(154)

Proof The conclusions (148)-(151) follow from Proposition 4.1 and embedding properties (see Remark 2.1). For verification of the boundary and initial conditions (152), (153) and (154), we use the Green formula as follows.

Let φ be a function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M409">View MathML</a> equal to zero in a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M411">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M412">View MathML</a>. Using the integration by parts we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M134">View MathML</a> and v (e.g.) the following equalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M414">View MathML</a>

(155)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M415">View MathML</a>

(156)

Passing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M416">View MathML</a> in (155) and comparing (155) and (156), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M417">View MathML</a>

(157)

and conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M418">View MathML</a>

(158)

In the similar way, we get all the remaining properties in (152)-(154). □

Lemma 5.4The functionsr, ρ, v, ω, θ, defined by Lemmas 5.1, 5.2 and 5.3 satisfy the Equations (30)-(33) a.e. in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M419">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M420">View MathML</a> be subsequence defined by Lemmas 5.1, 5.2 and 5.3. Taking into account (144), (149) and strong convergencies (132), (133), (145) and (151) we get that (30) follows immediately from (59). We can write Equation (65) in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M421">View MathML</a>

(159)

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M422">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M423">View MathML</a> denotes the space of test functions. Now we consider the convergence of one integrand. For example, we will prove the convergence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M424">View MathML</a>

(160)

when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M416">View MathML</a>. Taking into account

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M426">View MathML</a>

(161)

and using already mentioned convergences, we can easily conclude that (160) is satisfied. In the same way, we can derive the convergences of other integrals in (159). Analogously, we get that (32) and (33) follow from (66) and (67). □

Remark 5.1 Taking into account (150) and (58), we can easily prove that the function r defined by Lemma 5.1 has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M427">View MathML</a>

(162)

where v is from Lemma 5.3.

Lemma 5.5There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M115">View MathML</a>such that the functionθdefined by Lemma 5.3 satisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M430">View MathML</a>

(163)

Proof Because of the inclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M431">View MathML</a> (see Remark 2.1), in the same way as in [3], we conclude that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M291">View MathML</a> there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M434">View MathML</a>, such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M435">View MathML</a> holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/69/mathml/M436">View MathML</a>

The conclusions of Theorem 2.1 are an immediate consequence of the above lemmas. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The paper is the result of joint work of both authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.

Acknowledgement

The paper was made with financial support of the scientific project Mathematical analysis of composite and thin structures (037-0693014-2765), Ministry of Science, Education and Sports, Republic of Croatia.

End notes

  1. We assume Einstein notation for summation.

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