Open Access Research

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

For all author emails, please log on.

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 Ω R N ( N 2 ) be a bounded Lipschitz domain and 0 < T < . 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 , (1.1)

where Q T = Ω × ( 0 , T ] , S T denotes the lateral boundary of the cylinder Q T .

It will be assumed throughout the paper that the exponents m ( x ) , p ( x ) are continuous in Ω with logarithmic module of continuity:

1 < m = inf x Ω m ( x ) m ( x ) m + = sup x Ω m ( x ) < , (1.2)

1 < p = inf x Ω p ( x ) p ( x ) p + = sup x Ω p ( x ) < , (1.3)

z , ξ Ω , | z ξ | < 1 , | m ( z ) m ( ξ ) | + | p ( z ) p ( ξ ) | ω ( | z ξ | ) , (1.4)

where

lim sup τ 0 + ω ( τ ) ln 1 τ = C < + .

And we also assume that

(H1) g : R + R + is C 1 function and satisfies

g ( 0 ) > 0 , 1 0 g ( s ) d s = l > 0 ;

(H2) there exists η > 0 such that

g ( t ) < η g ( t ) , t 0 .

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

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 } .

Lemma 2.1[14]

For u L p ( x ) ( Ω ) , the following relations hold:

(1) u p ( ) < 1 ( = 1 ; > 1 ) A p ( ) ( u ) < 1 ( = 1 ; > 1 ) ;

(2) u p ( ) < 1 u p ( ) p + A p ( ) ( u ) u p ( ) p ; u p ( ) > 1 u p ( ) p + A p ( ) ( u ) u p ( ) p ;

(3) u p ( ) 0 A p ( ) ( u ) 0 ; u p ( ) A p ( ) ( u ) .

Lemma 2.2[15,16]

For u W 0 1 , p ( ) ( Ω ) , ifpsatisfies condition (1.2), the p ( ) -Poincaré inequality

u p ( x ) C u p ( x )

holds, where the positive constantCdepends onpand Ω.

Remark 2.1Note that the following inequality

Ω | u | p ( x ) d x C Ω | u | p ( x ) d x

does not in general hold.

Lemma 2.3[17]

Let Ω be an open domain (that may be unbounded) in R N with cone property. If p ( x ) : Ω ¯ R is a Lipschitz continuous function satisfying 1 < p p + < N k and r ( x ) : Ω ¯ R is measurable and satisfies

p ( x ) r ( x ) p ( x ) = N p ( x ) N k p ( x ) a.e.  x Ω ¯ ,

then there is a continuous embedding W k , p ( x ) ( Ω ) L r ( x ) ( Ω ) .

The main theorem in this section is the following.

Theorem 2.1Let u 0 , u 1 H 0 1 ( Ω ) , the exponents m ( x ) , p ( x ) satisfy conditions (1.2)-(1.4). Then Problem (1.1) has at least one weak solution u : Ω × ( 0 , ) R 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 ) ) .

And one of the following conditions holds:

(A1) 2 < p < p + < max { N , N p N p } , 2 < m < m + < p ;

(A2) max { 1 , 2 N N + 2 } < p < p + < 2 , 1 < m < m + < 3 p 2 p < 2 .

Proof Let { w j } j = 1 be an orthogonal basis of H 0 1 ( Ω ) with w j

Δ w j = λ j w j , x Ω , w j = 0 , x Ω .

V k = span { w i , , w k } is the subspace generated by the first k vectors of the basis { w j } j = 1 . By normalization, we have w j 2 = 1 . 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 .

For any given integer k, we consider the approximate solution

u k = i = 1 k c i k ( t ) w i ,

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 , (2.1)

here u 0 k = i = 1 k ( u 0 , w i ) w i , u 1 k = i = 1 k ( u 1 , w i ) w i and u 0 k u 0 , u 1 k u 1 in H 0 1 ( Ω ) .

Here we denote by ( , ) the inner product in L 2 ( Ω ) .

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 . (2.2)

By the standard theory of the ODE system, we infer that problem (2.2) admits a unique solution c i k ( t ) in [ 0 , t k ] , where t k > 0 . Then we can obtain an approximate solution u k ( t ) for (1.1), in V k , over [ 0 , t k ) . And the solution can be extended to [ 0 , T ] , for any given T > 0 , by the estimate below. Multiplying (2.1) ( c i k ( t ) ) 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 . (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 , (2.4)

here

( φ ψ ) ( t ) 0 t φ ( t τ ) ψ ( t ) ψ ( τ ) 2 2 d τ .

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 . (2.5)

Integrating (2.5) over ( 0 , t ) , 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 ,

where C1 is a positive constant depending only on u 0 H 0 1 , u 1 H 0 1 .

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 . (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 , (2.7)

where C2 is a positive constant depending only on u 0 H 0 1 , u 1 H 0 1 , l, p , p + . It follows from (2.7) that

u k is uniformly bounded in  L ( 0 , T ; H 0 1 ( Ω ) ) , (2.8)

u k is uniformly bounded in  L ( 0 , T ; H 0 1 ( Ω ) ) . (2.9)

Next, multiplying (1.1) by ( c i k ( t ) ) 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 . (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 , (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 . (2.12)

From Lemma 2.2, we have

u k 2 2 C 2 u k 2 2 , (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 ) } , (2.14)

where C, C 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 } . (2.15)

Integrating (2.15) over ( 0 , t ) 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 , (2.16)

where C3 is a positive constant depending only on u 1 H 0 1 .

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 .

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 , (2.17)

where C4 is a positive constant depending only on u 0 H 0 1 , u 1 H 0 1 , l, g ( 0 ) , T.

From estimate (2.17), we get

u k is uniformly bounded in  L 2 ( 0 , T ; H 0 1 ( Ω ) ) . (2.18)

By (2.7)-(2.9) and (2.18), we infer that there exist a subsequence { u i } of { u k } and a function u such that

u i u weakly star in  L ( 0 , T ; H 0 1 ( Ω ) ) , (2.19)

u i u weakly in  L p ( 0 , T ; W 1 , p ( x ) ( Ω ) ) , (2.20)

u i u weakly star in  L ( 0 , T ; H 0 1 ( Ω ) ) , (2.21)

u i u weakly in  L 2 ( 0 , T ; H 0 1 ( Ω ) ) . (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 } , still represented by the same notation, such that

u i u strongly in  L 2 ( 0 , T ; L 2 ( Ω ) ) ,

which implies u i u almost everywhere in Ω × ( 0 , T ) . Hence, by (2.19)-(2.22),

| u i | p ( x ) 2 u i | u | p ( x ) 2 u weakly in  Ω × ( 0 , T ) , (2.23)

| u i | m ( x ) 2 u i | u | m ( x ) 2 u almost everywhere in  Ω × ( 0 , T ) . (2.24)

Multiplying (2.2) by ϕ ( t ) C ( 0 , T ) (which C ( 0 , T ) is the space of C function with compact support in ( 0 , T ) ) and integrating the obtained result over ( 0 , T ) , we obtain that

L u k , w i ϕ ( t ) = 0 , i = 1 , 2 , , k . (2.25)

Note that { w i } i = 1 is a basis of H 0 1 ( Ω ) . 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 ) )

for arbitrary T > 0 . 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 ) .

Hence, we get u ( 0 ) = u 0 , u 1 ( 0 ) = u 1 . 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).

References

  1. Cavalcanti, MM, Domingos Cavalcanti, VN, Ferreira, J: Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci.. 24, 1043–1053 (2001). Publisher Full Text OpenURL

  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(44), 1–14 (2002)

  3. Cavalcanti, MM, Oquendo, HP: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim.. 42(4), 1310–1324 (2003). Publisher Full Text OpenURL

  4. Messaoudi, SA: Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation. J. Math. Anal. Appl.. 320, 902–915 (2006). Publisher Full Text OpenURL

  5. Messaoudi, SA: General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Anal.. 69, 2589–2598 (2008). Publisher Full Text OpenURL

  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.. 27, 1687–1696 (2004). Publisher Full Text OpenURL

  7. Antontsev, SN, Zhikov, V: Higher integrability for parabolic equations of p ( x , t ) -Laplacian type. Adv. Differ. Equ.. 10, 1053–1080 (2005)

  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.. 342, 27–38 (2008). Publisher Full Text OpenURL

  9. Chen, Y, Levine, S, Rao, M: Variable exponent, linear growth functions in image restoration. SIAM J. Appl. Math.. 66, 1383–1406 (2006). Publisher Full Text OpenURL

  10. Gao, Y, Guo, B, Gao, W: Weak solutions for a high-order pseudo-parabolic equation with variable exponents. Appl. Anal. (2013). Publisher Full Text OpenURL

  11. Aboulaicha, R, Meskinea, D, Souissia, A: New diffusion models in image processing. Comput. Math. Appl.. 56, 874–882 (2008). Publisher Full Text OpenURL

  12. Andreu-Vaillo, F, Caselles, V, Mazón, JM: Parabolic Quasilinear Equations Minimizing Linear Growth Functions, Birkhäuser, Basel (2004)

  13. Antontsev, SN: Wave equation with p ( x , t ) -Laplacian and damping term: blow-up of solutions. C. R., Méc.. 339, 751–755 (2011). PubMed Abstract | Publisher Full Text OpenURL

  14. Fan, X, Zhao, D: On the spaces L p ( x ) ( Ω ) and L m , p ( x ) ( Ω ) . J. Math. Anal. Appl.. 263, 424–446 (2001). Publisher Full Text OpenURL

  15. Kováčik, O, Rákosník, J: On spaces L p ( x ) and W 1 , p ( x ) . Czechoslov. Math. J.. 41(116), 592–618 (1991)

  16. Zhao, JN: Existence and nonexistence of solutions for u t = div ( | u | p 2 u ) + f ( u , u , x , t ) . J. Math. Anal. Appl.. 172, 130–146 (1993). Publisher Full Text OpenURL

  17. Fan, X, Shen, J, Zhao, D: Sobolev embedding theorems for spaces W k , p ( x ) ( Ω ) . J. Math. Anal. Appl.. 262, 749–760 (2001). Publisher Full Text OpenURL

  18. Lions, JL: Quelques Metodes De Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris (1969)

  19. Zheng, SM: Nonlinear Evolution Equation, CRC Press, Boca Raton (2004)