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Global existence and uniform decay for the one-dimensional model of thermodiffusion with second sound

Ming Zhang1* and Yuming Qin2

Author Affiliations

1 College of Information Science and Technology, Donghua University, Shanghai, 201620, P.R. China

2 Department of Applied Mathematics, Donghua University, Shanghai, 201620, P.R. China

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Boundary Value Problems 2013, 2013:222  doi:10.1186/1687-2770-2013-222

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/222


Received:19 June 2013
Accepted:4 September 2013
Published:7 November 2013

© 2013 Zhang and Qin; 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

In this paper, we investigate an initial boundary value problem for the one-dimensional linear model of thermodiffusion with second sound in a bounded region. Using the semigroup approach, boundary control and the multiplier method, we obtain the existence of global solutions and the uniform decay estimates for the energy.

MSC: 35B40, 35M13, 35Q79.

Keywords:
thermodiffusion; second sound; global existence; exponential decay

1 Introduction

In this paper, we investigate the global existence and uniform decay rate of the energy for solutions for the one-dimensional model of thermodiffusion with second sound:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M1">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M2">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M3">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M4">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M5">View MathML</a>

(1.5)

together with the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M6">View MathML</a>

(1.6)

and the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M7">View MathML</a>

(1.7)

where u, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M8">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M9">View MathML</a> are the displacement, temperature, and heat flux, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M10">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M11">View MathML</a> are the chemical potentials and the associated flux. The boundary conditions (1.7) model a rigidly clamped medium with temperature and chemical potentials held constant on the boundary.

Here, we denote by λ, μ the material constants, ρ the density, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M13">View MathML</a> the coefficients of thermal and diffusion dilatation, k, D the coefficients of thermal conductivity, n, c, d the coefficients of thermodiffusion, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M15">View MathML</a> the (in general very small) relaxation time. All the coefficients above are positive constants and satisfy the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M16">View MathML</a>

(1.8)

The classical thermodiffusion equations were first given by Nowacki [1,2] in 1971. The equations describe the process of thermodiffusion in a solid body (see, e.g., [1-5]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M17">View MathML</a>

(1.9)

There are many results about the classical thermodiffusion equations. By the method of integral transformations and integral equations, Nowacki [2], Podstrigach [6] and Fichera [7] investigated the initial boundary value problem for the linear homogeneous system. Gawinecki [8] proved the existence, uniqueness and regularity of solutions to an initial boundary value problem for the linear system of thermodiffusion in a solid body. Szymaniec [5] proved the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M18">View MathML</a>-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M19">View MathML</a> time decay estimates along the conjugate line for the solutions of the linear thermodiffusion system. Using the results from [5], Szymaniec [9] obtained the global existence and uniqueness of small data solutions to the Cauchy problem of nonlinear thermodiffusion equations in a solid body. Using the semigroup approach and the multiplier method, Qin et al.[4] obtained the global existence and exponential stability of solutions for homogeneous, nonhomogeneous and semilinear thermodiffusion equations subject to various boundary conditions. Liu and Reissig [3] studied the Cauchy problem for one-dimensional models of thermodiffusion and explained qualitative properties of solutions and showed which part of the model has a dominant influence on wellposedness, propagation of singularities, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M18">View MathML</a>-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M19">View MathML</a> decay estimates on the conjugate line and the diffusion phenomenon.

If we neglect the diffusion in (1.9), then we obtain the classical thermo-elasticity equations. Today models of type I (classical model of thermo-elasticity), of type II (thermal wave), of type III (visco-elastic damping) or second sound present some classification of models of thermo-elasticity (see, e.g., [3,10,11]). By considerations of the total energy equation and comparisons with the models of classical thermo-elasticity and thermodiffusion, we shall propose the linear one-dimensional model of thermodiffusion with second sound as mentioned above. Due to our knowledge, there exist no results for thermodiffusion models with second sound.

Our paper is organized as follows. In Section 2, we present some notations and the main result. Section 3 is devoted to the proof of the main result.

2 Notations and main result

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M22">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M23">View MathML</a>

(2.1)

The associated first-order and second-order energy is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M24">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M25">View MathML</a>

(2.3)

The energy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M26">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M27">View MathML</a>

(2.4)

Our main result reads as follows.

Theorem 2.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M31">View MathML</a>. Then problem (1.1)-(1.7) has a unique global solution such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M32">View MathML</a>

Moreover, the associated energy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M26">View MathML</a>defined by (2.4) decays exponentially, i.e., there exist positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M34">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M35">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M36">View MathML</a>

(2.5)

Remark 2.1 If the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M38">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M39">View MathML</a> will be defined later), then the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M40">View MathML</a>, and problem (1.1)-(1.7) yields higher regularity in t.

3 Proof of the main result

We shall divide the proof into two steps: in Step 1, we shall use the semigroup approach to prove the existence of global solutions and the Remark 2.1; Step 2 is devoted to proving the uniform decay of the energy by the boundary control and the multiplier method.

Step 1. Existence of global solutions.

The proof is based on the semigroup approach (see [4,12]) that can be used to reduce problem (1.1)-(1.7) to an abstract initial value problem for a first-order evolution equation. In order to choose proper space for (1.1)-(1.7), we shall consider the static system associated with them (see [4]). Considering the energy and the property of operator A, we can choose the following state space and the domain of operator A for problem (1.1)-(1.7):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M41">View MathML</a>

(3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M42">View MathML</a>

(3.2)

Using the same method as in [4,12], we can prove that the operator A generates a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M35">View MathML</a>-semigroup of contractions on the Hilbert space H. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M44">View MathML</a>

Then by Theorem 2.3.1 of [13] about the existence and regularities of solutions, we can complete the proof.

Step 2. Uniform decay of the energy.

In this section, we shall assume the existence of solutions in the Sobolev spaces that we need for our computations. The proof of uniform decay is difficult. It is necessary to construct a suitable Lyapunov function and to combine various techniques from energy method, multiplier approaches and boundary control (see [10,11]). We mainly refer to Racke [11] for the approaches of thermo-elastic models with second sound.

Multiplying (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M45">View MathML</a>, (1.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M8">View MathML</a>, (1.3) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M10">View MathML</a>, (1.4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M9">View MathML</a>, and (1.5) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M11">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M50">View MathML</a>, respectively, and summing up the results, yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M51">View MathML</a>

(3.3)

Similarly, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M52">View MathML</a>

(3.4)

Multiplying (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M53">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M50">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M55">View MathML</a>

(3.5)

Multiplying (1.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M56">View MathML</a>, (1.3) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M57">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M50">View MathML</a> and summing them up, yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M59">View MathML</a>

(3.6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M60">View MathML</a>

Combining (3.5) with (3.6), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M61">View MathML</a>

(3.7)

Now, we conclude from (1.1), (1.4), (1.5) and Poincaré inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M62">View MathML</a>

(3.8)

Multiplying (1.1) by u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M50">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M64">View MathML</a>

(3.9)

From (1.2) and (1.3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M65">View MathML</a>

(3.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M66">View MathML</a>

(3.11)

Multiplying (3.10) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M67">View MathML</a>, (3.11) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M68">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M50">View MathML</a>, and summing up the results, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M70">View MathML</a>

(3.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M71">View MathML</a>.

The boundary terms are estimated as follows.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M72">View MathML</a>

(3.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M73">View MathML</a>

(3.14)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M74">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M75">View MathML</a>

(3.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M76">View MathML</a>

(3.16)

Combining (3.13)-(3.16), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M77">View MathML</a>

(3.17)

Differentiating (1.1) with respect to t and multiplying by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M78">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M79">View MathML</a>

(3.18)

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M80">View MathML</a>

(3.19)

Multiplications of (1.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M81">View MathML</a> and (1.3) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M82">View MathML</a> yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M83">View MathML</a>

which implies, using (1.4) and (1.5),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M84">View MathML</a>

(3.20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M85">View MathML</a>

(3.21)

Combining (3.19)-(3.21), we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M86">View MathML</a>

(3.22)

Using (1.2) and (1.3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M87">View MathML</a>

(3.23)

With (3.17) and (3.23), we can estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M88">View MathML</a>

(3.24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M89">View MathML</a>.

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M90">View MathML</a>. Multiplying both sides of (3.12) by ξ and combining the result with (3.7) and (3.24), we obtain for sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M91">View MathML</a> the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M92">View MathML</a>

(3.25)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M93">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M94">View MathML</a>. Now, we can define the desired Lyapunov functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M95">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M96">View MathML</a>, to be determined later on, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M97">View MathML</a>

(3.26)

Then we conclude from (3.3), (3.4), and (3.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M98">View MathML</a>

(3.27)

By using (3.8), we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M99">View MathML</a>

(3.28)

while (3.9) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M100">View MathML</a>

(3.29)

Combining (3.27)-(3.29), we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M101">View MathML</a>

(3.30)

We choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M102">View MathML</a> such that all terms on the right-hand side of (3.30) become negative,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M103">View MathML</a>

(3.31)

Choosing ε as in (3.31), we obtain from (3.30)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M104">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M105">View MathML</a>

(3.32)

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M106">View MathML</a>

(3.33)

There exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M108">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M109">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M110">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M111">View MathML</a>, it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M112">View MathML</a>

(3.34)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M108">View MathML</a> are determined later on. In fact,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M115">View MathML</a>

(3.35)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M116">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M117">View MathML</a>

At this point, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M119">View MathML</a>. Finally, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M120">View MathML</a>. Thus, we have the validity of (3.34). Combining (3.33) with (3.34), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M121">View MathML</a>

(3.36)

Hence, it follows from (3.36), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M122">View MathML</a>. Applying (3.34) again, we can conclude (2.5) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/222/mathml/M123">View MathML</a>. The proof is complete.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

This paper was in part supported by the NNSF of China with contract numbers 11031003, 11271066 and a grant from Shanghai Education Commission 13ZZ048.

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