Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents
© Gao and Gao; licensee Springer 2013
Received: 23 June 2013
Accepted: 21 August 2013
Published: 11 September 2013
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.
where , denotes the lateral boundary of the cylinder .
And we also assume that
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  the authors studied the finite time blow-up of solutions for viscoelastic hyperbolic equations, and in  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
Lemma 2.1 
holds, where the positive constant C depends on p and Ω.
does not in general hold.
Lemma 2.3 
then there is a continuous embedding .
The main theorem in this section is the following.
And one of the following conditions holds:
(A1) , ;
(A2) , .
here , and , in .
Here we denote by the inner product in .
where C 1 is a positive constant depending only on , .
where C 3 is a positive constant depending only on .
where C 4 is a positive constant depending only on , , l, , T.
Hence, we get , . Then, the existence of weak solutions is established. □
Supported by NSFC (11271154) and by Department of Education for Jilin Province (2013439).
- 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 ArticleMATHGoogle Scholar
- 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.MathSciNetMATHGoogle Scholar
- 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 ArticleMATHGoogle Scholar
- 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 ArticleMATHGoogle Scholar
- 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 ArticleMATHGoogle Scholar
- 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 ArticleMATHGoogle Scholar
- Antontsev SN, Zhikov V:Higher integrability for parabolic equations of -Laplacian type. Adv. Differ. Equ. 2005, 10: 1053-1080.MathSciNetMATHGoogle Scholar
- 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 ArticleMATHGoogle Scholar
- 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 ArticleMATHGoogle Scholar
- Gao Y, Guo B, Gao W: Weak solutions for a high-order pseudo-parabolic equation with variable exponents. Appl. Anal. 2013. 10.1080/00036811.2013.772138Google Scholar
- 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 ArticleGoogle Scholar
- Andreu-Vaillo F, Caselles V, Mazón JM Progress in Mathematics 223. In Parabolic Quasilinear Equations Minimizing Linear Growth Functions. Birkhäuser, Basel; 2004.View ArticleGoogle Scholar
- Antontsev SN:Wave equation with -Laplacian and damping term: blow-up of solutions. C. R., Méc. 2011, 339: 751-755.View ArticleMATHGoogle Scholar
- Fan X, Zhao D:On the spaces and . J. Math. Anal. Appl. 2001, 263: 424-446. 10.1006/jmaa.2000.7617MathSciNetView ArticleMATHGoogle Scholar
- Kováčik O, Rákosník J:On spaces and . Czechoslov. Math. J. 1991, 41(116):592-618.MATHGoogle Scholar
- Zhao JN:Existence and nonexistence of solutions for . J. Math. Anal. Appl. 1993, 172: 130-146. 10.1006/jmaa.1993.1012MathSciNetView ArticleMATHGoogle Scholar
- Fan X, Shen J, Zhao D:Sobolev embedding theorems for spaces . J. Math. Anal. Appl. 2001, 262: 749-760. 10.1006/jmaa.2001.7618MathSciNetView ArticleMATHGoogle Scholar
- Lions JL: Quelques Metodes De Resolution des Problemes aux Limites Non Lineaires. Dunod, Paris; 1969.MATHGoogle Scholar
- Zheng SM: Nonlinear Evolution Equation. CRC Press, Boca Raton; 2004.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.