Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents

Boundary Value Problems, Sep 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.

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Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents

Boundary Value Problems Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents Yunzhu Gao 0 1 3 Wenjie Gao 0 2 0 Statistics, Beihua University , Jilin, P.R 1 Department of Mathematics 2 Institute of Mathematics, Jilin University , Changchun , P.R. China 3 Department of Mathematics and Statistics, Beihua University , Jilin , P.R. China 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. 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: ⎧ utt – ⎪⎪⎨ ⎪ ⎪⎩ u(x, ) = u(x), ut(x, ) = u(x), u – t utt +  g(t – τ ) u(τ ) dτ + |ut|m(x)–ut = |u|p(x)–u, (x, t) ∈ QT , (.) 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 τ→+ lim sup ω(τ ) ln  τ = C < +∞. 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. (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 Lp(x)( ) = u(x) : u is measurable in , Ap(·)(u) = |u|p(x) dx < ∞ , u p(·) = inf λ > , Ap(·)(u/λ) ≤  . Lemma . [] For u ∈ Lp(x)( ), the following relations hold: () u p(·) <  (= ; > )+⇔ Ap(·)(u) <  (= ;–> ); + – () u p(·) <  ⇒ u pp(·) ≤ Ap(·)(u) ≤ u pp(·); u p(·) >  ⇒ u pp(·) ≥ Ap(·)(u) ≥ u pp(·); () u p(·) →  ⇔ Ap(·)(u) → ; u p(·) → ∞ ⇔ Ap(·)(u) → ∞. Lemma . [, ] For u ∈ W,p(·)( ), if p satisfies condition (.), the p(·)-Poincaré inequality 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. 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) = NN–pk(px()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: (A)  < p– < p+ < max{N , NN–pp–– },  < m– < m+ < p–; (A) max{, NN+ } < p– < p+ < ,  < m– < m+ < pp––– < . Proof Let {wj}j∞= be an orthogonal basis of H( ) with wj – wj = λjwj, x ∈ , 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  t Lu, = utt + ∇u∇ – g(t – τ )∇u∇ dτ + |ut|m(x)–ut – α|u|p(x)–u dx, ∈ Vk. For any given integer k, we consider the approximate solution k i= uk = cik(t)wi, which satisfies ⎧⎨ Luk, wi = , i = , , . . . k, ⎩ uk() = uk, ukt() = uk, here uk = ik=(u, wi)wi, uk = ik=(u, wi)wi and uk → u, uk → u in H( ). Here we denote by (·, ·) the inner product in L( ). (.) Problem (.) generates the system of k ordinary differential equations ⎧⎪ (cik(t)) = –λicik(t) + λi t g(t – τ )cik(τ ) dτ ⎪⎪⎨⎪⎪ ⎪⎪⎪⎪⎪⎩ cik() = (u, wi), + |( ik=(cik(t)) , wi)|m(x)–( ik=(cik(t)) , wi) – α|( ik= cik(t), wi)|p(x)–( ik= cik(t), wi), (cik()) = (u, wi), i = , , . . . , k. By the standard theory of the ODE system, we infer that problem (.) admits a unique solution cik(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 (.) (cik(t)) and summing with respect to i, we conclude that ddt  uk  +  ∇uk  – t – α ddt p(x) |uk|p(x) dx = . By simple calculation, we have  t – ∇uk)(t) –  g ∇uk(τ )∇uk(t) dx dτ + uk m(x) dx  d  dt  t ∇uk (t) – g(s) ds ∇uk   p(x) |uk|p(x) dx (.) (.) (.) (.)  t  t (ϕ ∇ψ )(t) ϕ(t – τ ) ∇ψ (t) – ∇ψ (τ )  dτ . Combining (.)-(.) and (H)-(H), we get ddt  uk  +  ∇uk  +   – =  g  ∇uk (t) –  g(t) ∇uk  – ∇uk)(t) – α uk m(x) dx.  uk  +  ∇uk  +   –  t +  (g ∇uk)(t) – α p(x) |uk|p(x) ≤ C, Integrating (.) over (, t), and using assumptions (.)-(.), we have where C is a positive constant depending only on u H , u H .  Hence, by Lemma ., we also have      uk  +  ∇uk  +    t  – Next, multiplying (.) by (cik(t)) and then summing with respect to i, we get that the following holds: (.) Note that –  t From Lemma ., we have  uk  ≤ C   ∇uk , d dt  t   uk  dx + ∇uk  +  m(x) uk m(x) = – ∇uk∇uk dx + α |uk|p(x)–ukuk ≤ αε uk  + αε |uk|p(x)–uk   ≤ αε uk  + |uk|p(x)–uk  dx.  t  t where C, C∗ are embedding constants. From (.)-(.), we obtain that uk  dx + ( – ε – αεC) ∇uk  + ddt m(x) uk m(x)  ≤ ε ∇uk  + ( – l)g() ε ∇uk(τ )  dτ + max C∗ (p––) ∇uk p–– , C∗ (p+–) ∇uk p+– . Integrating (.) over (, t) and using (.), Lemma ., we get uk  dτ + ( – ε – αεC) ∇uk  dτ +  ≤ ε C + ( – l)g()T + C, where C is a positive constant depending only on u H .  Taking α, ε small enough in (.), we obtain the estimate m(x) uk m(x) dx |uk|p(x)–uk  dx = |uk|(p(x)–) dx ≤ max |uk|(p––) dx, |uk|(p+–) dx Hence, by Lemma ., we have uk  dτ + m(x) uk m(x) dx ≤ C. uk  dτ + min   m+ m+ uk mm(–x), m+ uk m(x) ≤ C, where C is a positive constant depending only on u H , u H , l, g(), T .  From estimate (.), we get uk is uniformly bounded in L , T ; H( ) . By (.)-(.) and (.), we infer that there exist a subsequence {ui} of {uk} and a function u such that ui ui ui ui u u u u weakly star in L∞ , T ; H( ) , weakly in Lp– , T ; W ,p(x)( ) , weakly star in L∞ , T ; H( ) , weakly in L , T ; H( ) . Next, we will deal with the nonlinear term. From the Aubin-Lions theorem, see Lions [, pp.-], it follows from (.) and (.) that there exists a subsequence of {ui}, still (.) (.) (.) (.) (.) (.) (.) (.) represented by the same notation, such that ui → u strongly in L , T ; L( ) , which implies ui → u almost everywhere in × (, T ). Hence, by (.)-(.), |ui|p(x)–ui |u|p(x)–u for arbitrary T > . In view of (.)-(.) and Lemma .. in [], we obtain uk () u()  weakly in H( ), uk () u ()  weakly in H( ). Hence, we get u() = u, u() = u. 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. 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Yunzhu Gao, Wenjie Gao. Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents, Boundary Value Problems, 2013, 208,