Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents

Gao and Gao Boundary Value Problems 2013, 2013:208 http://www.boundaryvalueproblems.com/content/2013/1/208 RESEARCH Open Access Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents Yunzhu Gao1* and Wenjie Gao2 * Correspondence: 1 Department of Mathematics and Statistics, Beihua University, Jilin, P.R. China Full list of author information is available at the end of the article 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  ⊂ RN (N ≥ ) be a bounded Lipschitz domain and  < T < ∞. Consider the following nonlinear viscoelastic hyperbolic problem: ⎧ t m(x)– ⎪ ut = |u|p(x)– u, ⎪ ⎨utt – u – utt +  g(t – τ )u(τ ) dτ + |ut | ⎪ ⎪ ⎩ u(x, t) = , u(x, ) = u (x), ut (x, ) = u (x), (x, t) ∈ QT , (x, t) ∈ ST , (.) x ∈ , where QT =  × (, T], ST denotes the lateral boundary of the cylinder QT . It will be assumed throughout the paper that the exponents m(x), p(x) are continuous in  with logarithmic module of continuity:  < m– = inf m(x) ≤ m(x) ≤ m+ = sup m(x) < ∞, x∈ (.) x∈  < p– = inf p(x) ≤ p(x) ≤ p+ = sup p(x) < ∞, x∈ (.) x∈ ∀z, ξ ∈ , |z – ξ | < ,      m(z) – m(ξ ) + p(z) – p(ξ ) ≤ ω |z – ξ | , (.) where lim sup ω(τ ) ln τ →+  = C < +∞. τ And we also assume that (H) g : R+ → R+ is C  function and satisfies ∞ g() > , – g(s) ds = l > ;  © 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. Gao and Gao Boundary Value Problems 2013, 2013:208 http://www.boundaryvalueproblems.com/content/2013/1/208 Page 2 of 8 (H) there exists η >  such that g (t) < –ηg(t), t ≥ . 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 [–]. 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 [–] 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 [, ]. 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 [, ], we shall study the existence and energy decay of the solutions to Problem (.) and state some properties to the solutions. The outline of this paper is the following. In Section , 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 (.) 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 |u|p(x) dx < ∞ , Lp(x) () = u(x) : u is measurable in , Ap(·) (u) =  u p(·) = inf λ > , Ap(·) (u/λ) ≤  .  Lemma . [] For u ∈ Lp(x) (), the following relations hold: () u p(·) <  (= ; > ) ⇔ Ap(·) (u) <  (= ; > ); p+ p– p+ p– () u p(·) <  ⇒ u p(·) ≤ Ap(·) (u) ≤ u p(·) ; u p(·) >  ⇒ u p(·) ≥ Ap(·) (u) ≥ u p(·) ; () u p(·) →  ⇔ Ap(·) (u) → ; u p(·) → ∞ ⇔ Ap(·) (u) → ∞. ,p(·) Lemma . [, ] For u ∈ W equality (), if p satisfies condition (.), the p(·)-Poincaré in- u p(x) ≤ C ∇u p(x) holds, where the positive constant C depends on p and . Remark . Note that the following inequality |u|p(x) dx ≤ C  |∇u|p(x) dx  does not in general hold. Gao and Gao Boundary Value Problems 2013, 2013:208 http://www.boundaryvalueproblems.com/content/2013/1/208 Page 3 of 8 Lemma . [] Let  be an open domain (that may be unbounded) in RN with cone property. If p(x) :  → R is a Lipschitz continuous function satisfying  < p– ≤ p+ < Nk and r(x) :  → R is measurable and satisfies p(x) ≤ r(x) ≤ p∗ (x) = Np(x) N – kp(x) a.e. x ∈ , then there is a continuous embedding W k,p(x) () → Lr(x) (). The main theorem in this section is the following. Theorem . Let u , u ∈ H (), the exponents m(x), p(x) satisfy conditions (.)-(.). Then Problem (.) has at least one weak solution u :  × (, ∞) → R in the class  u ∈ L∞ , ∞; H () ,  u ∈ L∞ , ∞; H () ,  u ∈ L , ∞; H () . And one of the following conditions holds: Np– – + – (A)  < p– < p+ < max{N, N–p – },  < m < m < p ; p– – N – + – + (A) max{, N+ } < p < p < ,  < m < m < p– < .  Proof Let {wj }∞ j= be an orthogonal basis of H () with wj x ∈ , –wj = λj wj , wj = , x ∈ ∂. Vk = span{wi , . . . , wk } is the subspace generated by the first k vectors of the basis {wj }∞ j= . By normalization, we have wj  = . Let us define the operator  Lu,  = utt + ∇u∇  t – dτ + |ut |m(x)– ut   – α|u|p(x)– u g(t – τ )∇u∇ dx, ∈ Vk . For any given integer k, we consider the approximate solution uk = k  cki (t)wi , i= which satisfies ⎧ ⎨Lu , w  = , i = , , . . . k, ⎩uk () = uk , ukt () = uk , k i   here uk = ki= (u , wi )wi , uk = ki= (u , wi )wi and uk → u , uk → u in H (). Here we denote by (·, ·) the inner product in L (). (.) Gao and Gao Boundary Value Problems 2013, 2013:208 http://www.boundaryvalueproblems.com/content/2013/1/208 Page 4 of 8 Problem (.) generates the system of k ordinary differential equations ⎧ t ⎪ (cki (t)) = –λi cki (t) + λi  g(t – τ )cki (τ ) dτ ⎪ ⎪ ⎪   ⎪ ⎨ + |( ki= (cki (t)) , wi )|m(x)– ( ki= (cki (t)) , wi )   ⎪ ⎪ – α|( ki= cki (t), wi )|p(x)– ( ki= cki (t), wi ), ⎪ ⎪ ⎪ ⎩ k ci () = (u , wi ), (cki ()) = (u , wi ), i = , , . . . , k. (.) By the standard theory of the ODE system, we infer that problem (.) admits a unique solution cki (t) in [, tk ], where tk > . Then we can obtain an approximate solution uk (t) for (.), in Vk , over [, tk ). And the solution can be extended to [, T], for any given T > , by the estimate below. Multiplying (.) (cki (t)) and summing with respect to i, we conclude that   t     d  u  (...truncated)