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.
Keywordsexistence weak solutions viscoelastic variable exponents
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).
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