Open Access Research

Global solutions to a class of nonlinear damped wave operator equations

Zhigang Pan1*, Zhilin Pu2 and Tian Ma1

Author Affiliations

1 Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, P. R. China

2 College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan 610066, P. R. China

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Boundary Value Problems 2012, 2012:42  doi:10.1186/1687-2770-2012-42


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/42


Received:6 October 2011
Accepted:13 April 2012
Published:13 April 2012

© 2012 Pan et al; 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

This study investigates the existence of global solutions to a class of nonlinear damped wave operator equations. Dividing the differential operator into two parts, variational and non-variational structure, we obtain the existence, uniformly bounded and regularity of solutions.

Mathematics Subject Classification 2000: 35L05; 35A01; 35L35.

Keywords:
nonlinear damped wave operator equations; global solutions; uniformly bounded; regularity

1 Introduction

In recent years, there have been extensive studies on well-posedness of the following nonlinear variational wave equation with general data:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M1">View MathML</a>

(1.1)

where c(·) is given smooth, bounded, and positive function with c'(·) ≥ 0 and c'(u0) > 0,u0 H1(R),u1(x) ∈ L2(R). Equation (1.1) appears naturally in the study for liquid crystals [1-4]. In addition, Chang et al. [5], Su [6] and Kian [7] discussed globally Lipschitz continuous solutions to a class one dimension quasilinear wave equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M2">View MathML</a>

(1.2)

where (x,t) ∈ R × R+, u0(x),ω0(x) ∈ R. Furthermore, Nishihara [8] and Hayashi [9] obtained the global solution to one dimension semilinear damped wave equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M3">View MathML</a>

(1.3)

Ikehata [10] and Vitillaro [11] proved global existence of solutions for semilinear damped wave equations in RN with noncompactly supported initial data or in the energy space, in where the nonlinear term f(u) = |u|p or f(u) = 0 is too special; some authors [12-14] discussed the regularity of invariant sets in semilinear wave equation, but they didn't refer to any the initial value condition of it. Unfortunately, it is difficulty to classify a class wave operator equations, since the differential operator structure is too complex to identify whether have variational property. Our aim is to classify a class of nonlinear damped wave operator equations in order to research them more extensively and go beyond the results of [12].

In this article, we are interested in the existence of global solutions of the following nonlinear damped wave operator equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M4">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M5">View MathML</a> is a mapping, X2 X1, X1, X2 are Banach spaces and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M6">View MathML</a> is the dual spaces of X1, R+ = [0, ∞), u = u(x,t). If k > 0, (1.4) is called damped wave equation. We obtain the existence, uniformly bounded and regularity of solutions by dividing the differential operator G(u) into two parts, variational and non-variational structure.

2 Preliminaries

First we introduce a sequence of function spaces:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M7">View MathML</a>

(2.1)

where H, H1, H2 are Hilbert spaces, X is a linear space, X1, X2 are Banach spaces and all inclusions are dense embeddings. Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M8">View MathML</a>

(2.2)

In addition, the operator L has an eigenvalue sequence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M9">View MathML</a>

(2.3)

such that {ek} ⊂ X is the common orthogonal basis of H and H2. We investigate the existence of global solutions of the Equation (1.4), so we need define its solution. Firstly, in Banach space X, introduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M10">View MathML</a>

where p = (p1, p2,..., pm),pi ≥ 1(1 ≤ i m),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M11">View MathML</a>

where | · |k is semi-norm in X, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M12">View MathML</a>. Similarily, we can define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M13">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M14">View MathML</a>.

Definition 2.1. Set (φ, ψ) ∈ X2 × H1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M15">View MathML</a> is called a globally weak solution of (1.4), if for ∀v X1, it has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M16">View MathML</a>

(2.4)

Definition 2.2. Let Y1,Y2 be Banach spaces, the solution u(t, φ, ψ) of (1.4) is called uniformly bounded in Y1 × Y2, if for any bounded domain Ω1 × Ω2Y1 × Y2, there exists a constant C which only depends the domain Ω1 × Ω2, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M17">View MathML</a>

Definition 2.3. A mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M18">View MathML</a> is called weakly continuous, if for any sequence {un} ⊂ X2, un u0 in X2,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M19">View MathML</a>

Lemma 2.1. [15]Let H2, H be Hilbert spaces, and H2 H be a continuous embedding. Then there exists a orthonormal basis {ek} of H, and also is one orthogonal basis of H2.

Proof. Let I : H2 H be imbedded. According to assume I is a linear compact operator, we define the mapping A : H2 H as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M20">View MathML</a>

obviously, A : H2 H2 is linear symmetrical compact operator and positive definite. Therefore, A has a complete eigenvalue sequence {λk} and eigenvector sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M21">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M22">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M23">View MathML</a> is orthogonal basis of H2. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M24">View MathML</a>

it implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M25">View MathML</a> is also orthogonal sequence of H. Since H2 H is dense, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M25">View MathML</a> is also orthogonal sequence of H, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M26">View MathML</a> is norm orthogonal basis of H. The proof is completed.

Now, we introduce an important inequality

Lemma 2.2. [16] (Gronwall inequality) Let x(t), y(t), z(t) be real function on [a, b], where x(t) ≥ 0,∀a t b, z(t) ∈ C[a, b], y(t) is differentiable on [a, b]. If the inequality as follows is hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M27">View MathML</a>

(2.5)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M28">View MathML</a>

(2.6)

3 Main results

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M29">View MathML</a>. Throughout of this article, we assume that

(i) There exists a function F C1 : X2 R1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M30">View MathML</a>

(3.1)

(ii) Function F is coercive, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M31">View MathML</a>

(3.2)

(iii) B as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M32">View MathML</a>

(3.3)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M33">View MathML</a>.

Theorem 3.1. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M5">View MathML</a>is weakly continuous, (φ, ψ) ∈ X2 × H1, then we obtain the results as follows:

(1) If G = A satisfies the assumption (i) and (ii), then there exists a globally weak solution of (1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M34">View MathML</a>

and u is uniformly bounded in X2 × H1;

(2) If G = A + B satisfies the assumption (i), (ii) and (iii), then there exists a globally weak solution of (1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M35">View MathML</a>

(3) Furthermore, if G = A + B satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M36">View MathML</a>

(3.4)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M33">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M37">View MathML</a>.

Proof. Let {ek} ⊂ X be the public orthogonal basis of H and H2, satisfies (2.3).

Note

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M38">View MathML</a>

(3.5)

From the assumption, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M39">View MathML</a>, apply the Galerkin method to make truncate in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M40">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M41">View MathML</a>

(3.6)

there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M42">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M43">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M44">View MathML</a>

(3.7)

for any v Xn, it yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M45">View MathML</a>

(3.8)

(1) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M46">View MathML</a> substitute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M47">View MathML</a> into (3.7), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M48">View MathML</a>

combine condition (2.2) with (3.1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M49">View MathML</a>

consequently, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M50">View MathML</a>

(3.9)

Assume φ H2, combine(2.2)with(2.3), we know {en} is also the orthogonal basis of H1, then φn φ in H2, ψn ψ in H1, owing to H2 X2 is embedded, so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M51">View MathML</a>

(3.10)

due to the condition (3.6), from (3.9)and (3.10) we easily know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M52">View MathML</a>

consequently, assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M53">View MathML</a>

i.e. un u0 in X2 a.e. t > 0, and G is weakly continuous, so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M54">View MathML</a>

By (3.8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M55">View MathML</a>

it indicates for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M56">View MathML</a>, it holds. Hence, for any v X2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M57">View MathML</a>

(3.11)

Consequently, u0 is a globally weak solution of (1.4).

Furthermore, by (3.9) and (3.10), for any R > 0, there exists a constant C such that if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M58">View MathML</a>

(3.12)

then the weak solution u(t, φ, ψ) of (1.4) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M59">View MathML</a>

(3.13)

Assume (φ,ψ) ∈ X2 × H1 satisfies (3.12), by H2 X2 is dense. May fix φn H2 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M60">View MathML</a>

by (3.13), the solution {u(t, φn, ψ)} of (1.4) is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M61">View MathML</a> a.e. t > 0.

Therefore, assume u(t, φn, ψ) ⇀ u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M61">View MathML</a> then u(t) is a weak solution of (1.4), it satisfies uniformly bounded of (3.13). So the conclusion (1) is proved.

(2) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M62">View MathML</a>, substitute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M47">View MathML</a> into (3.7), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M63">View MathML</a>

combine the condition (2.2) and (3.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M64">View MathML</a>

consequently, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M65">View MathML</a>

(3.14)

by the condition (3.3),(3.14)implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M66">View MathML</a>

(3.15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M67">View MathML</a>.

by Gronwall inequality [Lemma(2.2)], from (3.15) we easily know:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M68">View MathML</a>

(3.16)

it implies that, for any 0 < T < ∞

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M69">View MathML</a>

now, use the same way as (1), we can obtain the result (2).

(3) If the condition (3.4) is hold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M70">View MathML</a>, substitute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M71">View MathML</a> into (3.7), we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M72">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M73">View MathML</a>

by (3.16), it implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M74">View MathML</a>

consequently, for any 0 < T < ∞

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/42/mathml/M75">View MathML</a>

it implies that u W2,2((0,T), H), the main theorem (3.1) has been proved.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors typed, read and approved the final manuscript.

Acknowledgements

The author is very grateful to the anonymous referees whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript. Foundation item: the National Natural Science Foundation of China (No. 10971148).

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