Open Access

Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents

Boundary Value Problems20132013:208

DOI: 10.1186/1687-2770-2013-208

Received: 23 June 2013

Accepted: 21 August 2013

Published: 11 September 2013

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 Ω R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq1_HTML.gif ( N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq2_HTML.gif) be a bounded Lipschitz domain and 0 < T < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq3_HTML.gif. Consider the following nonlinear viscoelastic hyperbolic problem:
{ u t t Δ u Δ u t t + 0 t g ( t τ ) Δ u ( τ ) d τ + | u t | m ( x ) 2 u t = | u | p ( x ) 2 u , ( x , t ) Q T , u ( x , t ) = 0 , ( x , t ) S T , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ1_HTML.gif
(1.1)

where Q T = Ω × ( 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq4_HTML.gif, S T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq5_HTML.gif denotes the lateral boundary of the cylinder Q T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq6_HTML.gif.

It will be assumed throughout the paper that the exponents m ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq7_HTML.gif, p ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq8_HTML.gif are continuous in Ω with logarithmic module of continuity:
1 < m = inf x Ω m ( x ) m ( x ) m + = sup x Ω m ( x ) < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ2_HTML.gif
(1.2)
1 < p = inf x Ω p ( x ) p ( x ) p + = sup x Ω p ( x ) < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ3_HTML.gif
(1.3)
z , ξ Ω , | z ξ | < 1 , | m ( z ) m ( ξ ) | + | p ( z ) p ( ξ ) | ω ( | z ξ | ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ4_HTML.gif
(1.4)
where
lim sup τ 0 + ω ( τ ) ln 1 τ = C < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equa_HTML.gif

And we also assume that

(H1) g : R + R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq9_HTML.gif is C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq10_HTML.gif function and satisfies
g ( 0 ) > 0 , 1 0 g ( s ) d s = l > 0 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equb_HTML.gif
(H2) there exists η > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq11_HTML.gif such that
g ( t ) < η g ( t ) , t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equc_HTML.gif

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 [16].

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 [710] 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
L p ( x ) ( Ω ) = { u ( x ) : u  is measurable in  Ω , A p ( ) ( u ) = Ω | u | p ( x ) d x < } , u p ( ) = inf { λ > 0 , A p ( ) ( u / λ ) 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equd_HTML.gif

Lemma 2.1 [14]

For u L p ( x ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq12_HTML.gif, the following relations hold:
  1. (1)

    u p ( ) < 1 ( = 1 ; > 1 ) A p ( ) ( u ) < 1 ( = 1 ; > 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq13_HTML.gif;

     
  2. (2)

    u p ( ) < 1 u p ( ) p + A p ( ) ( u ) u p ( ) p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq14_HTML.gif; u p ( ) > 1 u p ( ) p + A p ( ) ( u ) u p ( ) p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq15_HTML.gif;

     
  3. (3)

    u p ( ) 0 A p ( ) ( u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq16_HTML.gif; u p ( ) A p ( ) ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq17_HTML.gif.

     

Lemma 2.2 [15, 16]

For u W 0 1 , p ( ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq18_HTML.gif, if p satisfies condition (1.2), the p ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq19_HTML.gif-Poincaré inequality
u p ( x ) C u p ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Eque_HTML.gif

holds, where the positive constant C depends on p and Ω.

Remark 2.1 Note that the following inequality
Ω | u | p ( x ) d x C Ω | u | p ( x ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equf_HTML.gif

does not in general hold.

Lemma 2.3 [17]

Let Ω be an open domain (that may be unbounded) in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq20_HTML.gif with cone property. If p ( x ) : Ω ¯ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq21_HTML.gif is a Lipschitz continuous function satisfying 1 < p p + < N k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq22_HTML.gif and r ( x ) : Ω ¯ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq23_HTML.gif is measurable and satisfies
p ( x ) r ( x ) p ( x ) = N p ( x ) N k p ( x ) a.e.  x Ω ¯ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equg_HTML.gif

then there is a continuous embedding W k , p ( x ) ( Ω ) L r ( x ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq24_HTML.gif.

The main theorem in this section is the following.

Theorem 2.1 Let u 0 , u 1 H 0 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq25_HTML.gif, the exponents m ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq7_HTML.gif, p ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq8_HTML.gif satisfy conditions (1.2)-(1.4). Then Problem (1.1) has at least one weak solution u : Ω × ( 0 , ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq26_HTML.gif in the class
u L ( 0 , ; H 0 1 ( Ω ) ) , u L ( 0 , ; H 0 1 ( Ω ) ) , u L 2 ( 0 , ; H 0 1 ( Ω ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equh_HTML.gif

And one of the following conditions holds:

(A1) 2 < p < p + < max { N , N p N p } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq27_HTML.gif, 2 < m < m + < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq28_HTML.gif;

(A2) max { 1 , 2 N N + 2 } < p < p + < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq29_HTML.gif, 1 < m < m + < 3 p 2 p < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq30_HTML.gif.

Proof Let { w j } j = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq31_HTML.gif be an orthogonal basis of H 0 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq32_HTML.gif with w j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq33_HTML.gif
Δ w j = λ j w j , x Ω , w j = 0 , x Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equi_HTML.gif
V k = span { w i , , w k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq34_HTML.gif is the subspace generated by the first k vectors of the basis { w j } j = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq31_HTML.gif. By normalization, we have w j 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq35_HTML.gif. Let us define the operator
L u , Φ = Ω [ u t t Φ + u Φ 0 t g ( t τ ) u Φ d τ + | u t | m ( x ) 2 u t Φ α | u | p ( x ) 2 u Φ ] d x , Φ V k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equj_HTML.gif
For any given integer k, we consider the approximate solution
u k = i = 1 k c i k ( t ) w i , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equk_HTML.gif
which satisfies
{ L u k , w i = 0 , i = 1 , 2 , k , u k ( 0 ) = u 0 k , u k t ( 0 ) = u 1 k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ5_HTML.gif
(2.1)

here u 0 k = i = 1 k ( u 0 , w i ) w i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq36_HTML.gif, u 1 k = i = 1 k ( u 1 , w i ) w i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq37_HTML.gif and u 0 k u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq38_HTML.gif, u 1 k u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq39_HTML.gif in H 0 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq32_HTML.gif.

Here we denote by ( , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq40_HTML.gif the inner product in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq41_HTML.gif.

Problem (1.1) generates the system of k ordinary differential equations
{ ( c i k ( t ) ) = λ i c i k ( t ) + λ i 0 t g ( t τ ) c i k ( τ ) d τ ( c i k ( t ) ) = + | ( i = 1 k ( c i k ( t ) ) , w i ) | m ( x ) 2 ( i = 1 k ( c i k ( t ) ) , w i ) ( c i k ( t ) ) = α | ( i = 1 k c i k ( t ) , w i ) | p ( x ) 2 ( i = 1 k c i k ( t ) , w i ) , c i k ( 0 ) = ( u 0 , w i ) , ( c i k ( 0 ) ) = ( u 1 , w i ) , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ6_HTML.gif
(2.2)
By the standard theory of the ODE system, we infer that problem (2.2) admits a unique solution c i k ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq42_HTML.gif in [ 0 , t k ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq43_HTML.gif, where t k > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq44_HTML.gif. Then we can obtain an approximate solution u k ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq45_HTML.gif for (1.1), in V k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq46_HTML.gif, over [ 0 , t k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq47_HTML.gif. And the solution can be extended to [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq48_HTML.gif, for any given T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq49_HTML.gif, by the estimate below. Multiplying (2.1) ( c i k ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq50_HTML.gif and summing with respect to i, we conclude that
d d t ( 1 2 u k 2 2 + 1 2 u k 2 2 0 t g ( t τ ) Ω ( u k ( τ ) u k ( t ) ) d x d τ ) + Ω | u k | m ( x ) d x α d d t ( Ω 1 p ( x ) | u k | p ( x ) d x ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ7_HTML.gif
(2.3)
By simple calculation, we have
0 t g ( t τ ) Ω ( u k ( τ ) , u k ( t ) ) d x d τ = 1 2 d d t ( g u k ) ( t ) 1 2 ( g u k ) ( t ) 1 2 d d t 0 t g ( s ) d s u k 2 2 + 1 2 g ( t ) u k 2 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ8_HTML.gif
(2.4)
here
( φ ψ ) ( t ) 0 t φ ( t τ ) ψ ( t ) ψ ( τ ) 2 2 d τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equl_HTML.gif
Combining (2.3)-(2.4) and (H1)-(H2), we get
d d t ( 1 2 u k 2 2 + 1 2 u k 2 2 + 1 2 ( 1 0 t g ( s ) d s ) u k 2 2 + 1 2 ( g u k ) ( t ) α Ω 1 p ( x ) | u k | p ( x ) d x ) = 1 2 ( g u k ) ( t ) 1 2 g ( t ) u k 2 2 Ω | u k | m ( x ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ9_HTML.gif
(2.5)
Integrating (2.5) over ( 0 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq51_HTML.gif, and using assumptions (1.2)-(1.4), we have
1 2 u k 2 2 + 1 2 u k 2 2 + 1 2 ( 1 0 t g ( s ) d s ) u k 2 2 + 1 2 ( g u k ) ( t ) α 1 p ( x ) | u k | p ( x ) C 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equm_HTML.gif

where C 1 is a positive constant depending only on u 0 H 0 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq52_HTML.gif, u 1 H 0 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq53_HTML.gif.

Hence, by Lemma 2.1, we also have
1 2 u k 2 2 + 1 2 u k 2 2 + 1 2 ( 1 0 t g ( s ) d s ) u k 2 2 + 1 2 ( g u k ) ( t ) max { α 1 p u k p ( x ) p , α 1 p u k p ( x ) p + } C 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ10_HTML.gif
(2.6)
In view of (H1)-(H2) and (A1)-(A2), we also have
u k 2 2 + u k 2 2 + u k 2 2 + ( g u k ) ( t ) C 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ11_HTML.gif
(2.7)
where C 2 is a positive constant depending only on u 0 H 0 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq52_HTML.gif, u 1 H 0 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq53_HTML.gif, l, p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq54_HTML.gif, p + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq55_HTML.gif. It follows from (2.7) that
u k is uniformly bounded in  L ( 0 , T ; H 0 1 ( Ω ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ12_HTML.gif
(2.8)
u k is uniformly bounded in  L ( 0 , T ; H 0 1 ( Ω ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ13_HTML.gif
(2.9)
Next, multiplying (1.1) by ( c i k ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq56_HTML.gif and then summing with respect to i, we get that the following holds:
Ω | u k | 2 2 d x + u k 2 2 + d d t ( 1 m ( x ) | u k | m ( x ) ) = Ω u k u k d x + 0 t g ( t τ ) Ω u k ( τ ) u k ( t ) d x d τ + α Ω | u k | p ( x ) 2 u k u k d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ14_HTML.gif
(2.10)
Note that
| Ω u k u k d x | ε u k 2 2 + 1 4 ε u k 2 2 , ε > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ15_HTML.gif
(2.11)
| 0 t g ( t τ ) Ω u k ( τ ) u k ( t ) d x d τ | ε u k 2 2 + 1 4 ε Ω ( 0 t g ( t τ ) u k ( τ ) d τ ) 2 d x ε u k 2 2 + 1 4 ε 0 t g ( s ) d s 0 t g ( t τ ) Ω | u k ( τ ) | 2 d x d τ ε u k 2 2 + ( 1 l ) g ( 0 ) 4 ε 0 t u k ( τ ) 2 2 d τ , α | u k | p ( x ) 2 u k u k α ε u k 2 2 + α 4 ε | u k | p ( x ) 2 u k 2 2 α ε u k 2 2 + α 4 ε Ω ( | u k | p ( x ) 2 u k ) 2 d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ16_HTML.gif
(2.12)
From Lemma 2.2, we have
u k 2 2 C 2 u k 2 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ17_HTML.gif
(2.13)
Ω ( | u k | p ( x ) 2 u k ) 2 d x = Ω | u k | 2 ( p ( x ) 1 ) d x max { Ω | u k | 2 ( p 1 ) d x , Ω | u k | 2 ( p + 1 ) d x } max { C 1 2 ( p 1 ) u k 2 2 ( p 1 ) , C 1 2 ( p + 1 ) u k 2 2 ( p + 1 ) } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ18_HTML.gif
(2.14)
where C, C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq57_HTML.gif are embedding constants. From (2.10)-(2.14), we obtain that
Ω | u k | 2 d x + ( 1 2 ε α ε C ) u k 2 2 + d d t ( 1 m ( x ) | u k | m ( x ) ) 1 4 ε u k 2 2 + ( 1 l ) g ( 0 ) 4 ε 0 t u k ( τ ) 2 2 d τ + max { C 1 2 ( p 1 ) u k 1 p 1 , C 1 2 ( p + 1 ) u k 1 p + 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ19_HTML.gif
(2.15)
Integrating (2.15) over ( 0 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq51_HTML.gif and using (2.7), Lemma 2.3, we get
0 t u k 2 d τ + ( 1 2 ε α ε C ) 0 t u k 2 2 d τ + Ω 1 m ( x ) | u k | m ( x ) d x 1 4 ε ( C 2 + ( 1 l ) g ( 0 ) T ) + C 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ20_HTML.gif
(2.16)

where C 3 is a positive constant depending only on u 1 H 0 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq53_HTML.gif.

Taking α, ε small enough in (2.16), we obtain the estimate
0 t u k 2 d τ + Ω 1 m ( x ) | u k | m ( x ) d x C 4 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equn_HTML.gif
Hence, by Lemma 2.1, we have
0 t u k 2 d τ + min { 1 m + u k m ( x ) m , 1 m + u k m ( x ) m + } C 4 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ21_HTML.gif
(2.17)

where C 4 is a positive constant depending only on u 0 H 0 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq52_HTML.gif, u 1 H 0 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq53_HTML.gif, l, g ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq58_HTML.gif, T.

From estimate (2.17), we get
u k is uniformly bounded in  L 2 ( 0 , T ; H 0 1 ( Ω ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ22_HTML.gif
(2.18)
By (2.7)-(2.9) and (2.18), we infer that there exist a subsequence { u i } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq59_HTML.gif of { u k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq60_HTML.gif and a function u such that
u i u weakly star in  L ( 0 , T ; H 0 1 ( Ω ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ23_HTML.gif
(2.19)
u i u weakly in  L p ( 0 , T ; W 1 , p ( x ) ( Ω ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ24_HTML.gif
(2.20)
u i u weakly star in  L ( 0 , T ; H 0 1 ( Ω ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ25_HTML.gif
(2.21)
u i u weakly in  L 2 ( 0 , T ; H 0 1 ( Ω ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ26_HTML.gif
(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 { u i } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq59_HTML.gif, still represented by the same notation, such that
u i u strongly in  L 2 ( 0 , T ; L 2 ( Ω ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equo_HTML.gif
which implies u i u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq61_HTML.gif almost everywhere in Ω × ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq62_HTML.gif. Hence, by (2.19)-(2.22),
| u i | p ( x ) 2 u i | u | p ( x ) 2 u weakly in  Ω × ( 0 , T ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ27_HTML.gif
(2.23)
| u i | m ( x ) 2 u i | u | m ( x ) 2 u almost everywhere in  Ω × ( 0 , T ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ28_HTML.gif
(2.24)
Multiplying (2.2) by ϕ ( t ) C ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq63_HTML.gif (which C ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq64_HTML.gif is the space of C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq65_HTML.gif function with compact support in ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq66_HTML.gif) and integrating the obtained result over ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq66_HTML.gif, we obtain that
L u k , w i ϕ ( t ) = 0 , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equ29_HTML.gif
(2.25)
Note that { w i } i = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq67_HTML.gif is a basis of H 0 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq32_HTML.gif. Convergence (2.19)-(2.24) is sufficient to pass to the limit in (2.25) in order to get
u t t Δ u Δ u t t + 0 t g ( t τ ) Δ u ( τ ) d τ + | u t | m ( x ) 2 u t = | u | p ( x ) 2 u ,  in  L 2 ( 0 , T ; H 1 ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equp_HTML.gif
for arbitrary T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq49_HTML.gif. In view of (2.19)-(2.22) and Lemma 3.3.17 in [19], we obtain
u k ( 0 ) u ( 0 ) weakly in  H 0 1 ( Ω ) , u k ( 0 ) u ( 0 ) weakly in  H 0 1 ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_Equq_HTML.gif

Hence, we get u ( 0 ) = u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq68_HTML.gif, u 1 ( 0 ) = u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq69_HTML.gif. Then, the existence of weak solutions is established. □

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics and Statistics, Beihua University
(2)
Institute of Mathematics, Jilin University

References

  1. Cavalcanti MM, Domingos Cavalcanti VN, Ferreira J: Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci. 2001, 24: 1043-1053. 10.1002/mma.250MathSciNetView ArticleMATH
  2. Cavalcanti MM, Domingos Cavalcanti VN, Soriano JA: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. 2002, 2002(44):1-14.MathSciNetMATH
  3. Cavalcanti MM, Oquendo HP: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 2003, 42(4):1310-1324. 10.1137/S0363012902408010MathSciNetView ArticleMATH
  4. Messaoudi SA: Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation. J. Math. Anal. Appl. 2006, 320: 902-915. 10.1016/j.jmaa.2005.07.022MathSciNetView ArticleMATH
  5. Messaoudi SA: General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Anal. 2008, 69: 2589-2598. 10.1016/j.na.2007.08.035MathSciNetView ArticleMATH
  6. Messaoudi SA, Said-Houari B: Blow up of solutions of a class of wave equations with nonlinear damping and source terms. Math. Methods Appl. Sci. 2004, 27: 1687-1696. 10.1002/mma.522MathSciNetView ArticleMATH
  7. Antontsev SN, Zhikov V:Higher integrability for parabolic equations of p ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq70_HTML.gif-Laplacian type. Adv. Differ. Equ. 2005, 10: 1053-1080.MathSciNetMATH
  8. Lian SZ, Gao WJ, Cao CL, Yuan HJ: Study of the solutions to a model porous medium equation with variable exponents of nonlinearity. J. Math. Anal. Appl. 2008, 342: 27-38. 10.1016/j.jmaa.2007.11.046MathSciNetView ArticleMATH
  9. Chen Y, Levine S, Rao M: Variable exponent, linear growth functions in image restoration. SIAM J. Appl. Math. 2006, 66: 1383-1406. 10.1137/050624522MathSciNetView ArticleMATH
  10. Gao Y, Guo B, Gao W: Weak solutions for a high-order pseudo-parabolic equation with variable exponents. Appl. Anal. 2013. 10.1080/00036811.2013.772138
  11. Aboulaicha R, Meskinea D, Souissia A: New diffusion models in image processing. Comput. Math. Appl. 2008, 56: 874-882. 10.1016/j.camwa.2008.01.017MathSciNetView Article
  12. Andreu-Vaillo F, Caselles V, Mazón JM Progress in Mathematics 223. In Parabolic Quasilinear Equations Minimizing Linear Growth Functions. Birkhäuser, Basel; 2004.View Article
  13. Antontsev SN:Wave equation with p ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq71_HTML.gif-Laplacian and damping term: blow-up of solutions. C. R., Méc. 2011, 339: 751-755.View ArticleMATH
  14. Fan X, Zhao D:On the spaces L p ( x ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq72_HTML.gif and L m , p ( x ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq73_HTML.gif. J. Math. Anal. Appl. 2001, 263: 424-446. 10.1006/jmaa.2000.7617MathSciNetView ArticleMATH
  15. Kováčik O, Rákosník J:On spaces L p ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq74_HTML.gif and W 1 , p ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq75_HTML.gif. Czechoslov. Math. J. 1991, 41(116):592-618.MATH
  16. Zhao JN:Existence and nonexistence of solutions for u t = div ( | u | p 2 u ) + f ( u , u , x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq76_HTML.gif. J. Math. Anal. Appl. 1993, 172: 130-146. 10.1006/jmaa.1993.1012MathSciNetView ArticleMATH
  17. Fan X, Shen J, Zhao D:Sobolev embedding theorems for spaces W k , p ( x ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-208/MediaObjects/13661_2013_Article_456_IEq77_HTML.gif. J. Math. Anal. Appl. 2001, 262: 749-760. 10.1006/jmaa.2001.7618MathSciNetView ArticleMATH
  18. Lions JL: Quelques Metodes De Resolution des Problemes aux Limites Non Lineaires. Dunod, Paris; 1969.MATH
  19. Zheng SM: Nonlinear Evolution Equation. CRC Press, Boca Raton; 2004.View Article

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