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Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents

Yunzhu Gao1* and Wenjie Gao2

Author Affiliations

1 Department of Mathematics and Statistics, Beihua University, Jilin, P.R. China

2 Institute of Mathematics, Jilin University, Changchun, P.R. China

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Boundary Value Problems 2013, 2013:208  doi:10.1186/1687-2770-2013-208

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/208


Received:23 June 2013
Accepted:21 August 2013
Published:11 September 2013

© 2013 Gao and Gao; 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

The authors of this paper study a nonlinear viscoelastic equation with variable exponents. By using the Faedo-Galerkin method and embedding theory, the existence of weak solutions is given to the initial and boundary value problem under suitable assumptions.

Keywords:
existence; weak solutions; viscoelastic; variable exponents

1 Introduction

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M1">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M2">View MathML</a>) be a bounded Lipschitz domain and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M3">View MathML</a>. Consider the following nonlinear viscoelastic hyperbolic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M4">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M6">View MathML</a> denotes the lateral boundary of the cylinder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M7">View MathML</a>.

It will be assumed throughout the paper that the exponents <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M9">View MathML</a> are continuous in Ω with logarithmic module of continuity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M10">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M11">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M12">View MathML</a>

(1.4)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M13">View MathML</a>

And we also assume that

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M14">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M15">View MathML</a> function and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M16">View MathML</a>

(H2) there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M17">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M18">View MathML</a>

In the case when m, p are constants, there have been many results about the existence and blow-up properties of the solutions, we refer the readers to the bibliography given in [1-6].

In recent years, much attention has been paid to the study of mathematical models of electro-rheological fluids. These models include hyperbolic, parabolic or elliptic equations which are nonlinear with respect to gradient of the thought solution and with variable exponents of nonlinearity; see [7-10] and references therein. Besides, another important application is the image processing where the anisotropy and nonlinearity of the diffusion operator and convection terms are used to underline the borders of the distorted image and to eliminate the noise [11,12].

To the best of our knowledge, there are only a few works about viscoelastic hyperbolic equations with variable exponents of nonlinearity. In [13] the authors studied the finite time blow-up of solutions for viscoelastic hyperbolic equations, and in [1] the authors discussed only the viscoelastic hyperbolic problem with constant exponents. Motivated by the works of [1,13], we shall study the existence and energy decay of the solutions to Problem (1.1) and state some properties to the solutions.

The outline of this paper is the following. In Section 2, we introduce the function spaces of Orlicz-Sobolev type, give the definition of the weak solution to the problem and prove the existence of weak solutions for Problem (1.1) with Galerkin’s method.

2 Existence of weak solutions

In this section, the existence of weak solutions is studied. Firstly, we introduce some Banach spaces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M19">View MathML</a>

Lemma 2.1[14]

For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M20">View MathML</a>, the following relations hold:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M21">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M22">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M23">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M24">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M25">View MathML</a>.

Lemma 2.2[15,16]

For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M26">View MathML</a>, ifpsatisfies condition (1.2), the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M27">View MathML</a>-Poincaré inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M28">View MathML</a>

holds, where the positive constantCdepends onpand Ω.

Remark 2.1Note that the following inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M29">View MathML</a>

does not in general hold.

Lemma 2.3[17]

Let Ω be an open domain (that may be unbounded) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M30">View MathML</a>with cone property. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M31">View MathML</a>is a Lipschitz continuous function satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M32">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M33">View MathML</a>is measurable and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M34">View MathML</a>

then there is a continuous embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M35">View MathML</a>.

The main theorem in this section is the following.

Theorem 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M36">View MathML</a>, the exponents<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M9">View MathML</a>satisfy conditions (1.2)-(1.4). Then Problem (1.1) has at least one weak solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M39">View MathML</a>in the class

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M40">View MathML</a>

And one of the following conditions holds:

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M42">View MathML</a>;

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M44">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M45">View MathML</a> be an orthogonal basis of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M46">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M47">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M48">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M49">View MathML</a> is the subspace generated by the first k vectors of the basis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M45">View MathML</a>. By normalization, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M51">View MathML</a>. Let us define the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M52">View MathML</a>

For any given integer k, we consider the approximate solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M53">View MathML</a>

which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M54">View MathML</a>

(2.1)

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M58">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M46">View MathML</a>.

Here we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M60">View MathML</a> the inner product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M61">View MathML</a>.

Problem (1.1) generates the system of k ordinary differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M62">View MathML</a>

(2.2)

By the standard theory of the ODE system, we infer that problem (2.2) admits a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M63">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M64">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M65">View MathML</a>. Then we can obtain an approximate solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M66">View MathML</a> for (1.1), in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M67">View MathML</a>, over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M68">View MathML</a>. And the solution can be extended to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M69">View MathML</a>, for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M70">View MathML</a>, by the estimate below. Multiplying (2.1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M71">View MathML</a> and summing with respect to i, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M72">View MathML</a>

(2.3)

By simple calculation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M73">View MathML</a>

(2.4)

here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M74">View MathML</a>

Combining (2.3)-(2.4) and (H1)-(H2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M75">View MathML</a>

(2.5)

Integrating (2.5) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M76">View MathML</a>, and using assumptions (1.2)-(1.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M77">View MathML</a>

where C1 is a positive constant depending only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M79">View MathML</a>.

Hence, by Lemma 2.1, we also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M80">View MathML</a>

(2.6)

In view of (H1)-(H2) and (A1)-(A2), we also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M81">View MathML</a>

(2.7)

where C2 is a positive constant depending only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M79">View MathML</a>, l, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M85">View MathML</a>. It follows from (2.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M86">View MathML</a>

(2.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M87">View MathML</a>

(2.9)

Next, multiplying (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M88">View MathML</a> and then summing with respect to i, we get that the following holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M89">View MathML</a>

(2.10)

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M90">View MathML</a>

(2.11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M91">View MathML</a>

(2.12)

From Lemma 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M92">View MathML</a>

(2.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M93">View MathML</a>

(2.14)

where C, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M94">View MathML</a> are embedding constants. From (2.10)-(2.14), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M95">View MathML</a>

(2.15)

Integrating (2.15) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M76">View MathML</a> and using (2.7), Lemma 2.3, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M97">View MathML</a>

(2.16)

where C3 is a positive constant depending only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M79">View MathML</a>.

Taking α, ε small enough in (2.16), we obtain the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M99">View MathML</a>

Hence, by Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M100">View MathML</a>

(2.17)

where C4 is a positive constant depending only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M79">View MathML</a>, l, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M103">View MathML</a>, T.

From estimate (2.17), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M104">View MathML</a>

(2.18)

By (2.7)-(2.9) and (2.18), we infer that there exist a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M105">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M106">View MathML</a> and a function u such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M107">View MathML</a>

(2.19)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M108">View MathML</a>

(2.20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M109">View MathML</a>

(2.21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M110">View MathML</a>

(2.22)

Next, we will deal with the nonlinear term. From the Aubin-Lions theorem, see Lions [[18], pp.57-58], it follows from (2.21) and (2.22) that there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M105">View MathML</a>, still represented by the same notation, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M112">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M113">View MathML</a> almost everywhere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M114">View MathML</a>. Hence, by (2.19)-(2.22),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M115">View MathML</a>

(2.23)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M116">View MathML</a>

(2.24)

Multiplying (2.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M117">View MathML</a> (which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M118">View MathML</a> is the space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M119">View MathML</a> function with compact support in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M120">View MathML</a>) and integrating the obtained result over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M120">View MathML</a>, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M122">View MathML</a>

(2.25)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M123">View MathML</a> is a basis of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M46">View MathML</a>. Convergence (2.19)-(2.24) is sufficient to pass to the limit in (2.25) in order to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M125">View MathML</a>

for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M70">View MathML</a>. In view of (2.19)-(2.22) and Lemma 3.3.17 in [19], we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M127">View MathML</a>

Hence, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/208/mathml/M129">View MathML</a>. Then, the existence of weak solutions is established. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YG carried out the study of existence of weak solutions for viscoelastic hyperbolic equations with variable exponents and drafted the manuscript. WG participated in the discussion of existence of weak solutions for viscoelastic hyperbolic equations with variable exponents.

Acknowledgements

Supported by NSFC (11271154) and by Department of Education for Jilin Province (2013439).

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