Abstract
The authors of this paper study a nonlinear viscoelastic equation with variable exponents. By using the FaedoGalerkin 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 exponents1 Introduction
Let () be a bounded Lipschitz domain and . Consider the following nonlinear viscoelastic hyperbolic problem:
where , denotes the lateral boundary of the cylinder .
It will be assumed throughout the paper that the exponents , are continuous in Ω with logarithmic module of continuity:
where
And we also assume that
(H1) is function and satisfies
In the case when m, p are constants, there have been many results about the existence and blowup properties of the solutions, we refer the readers to the bibliography given in [16].
In recent years, much attention has been paid to the study of mathematical models of electrorheological 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 [710] 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 blowup 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 OrliczSobolev 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
Lemma 2.1[14]
For, the following relations hold:
For, ifpsatisfies condition (1.2), thePoincaré inequality
holds, where the positive constantCdepends onpand Ω.
Remark 2.1Note that the following inequality
does not in general hold.
Lemma 2.3[17]
Let Ω be an open domain (that may be unbounded) inwith cone property. Ifis a Lipschitz continuous function satisfyingandis measurable and satisfies
then there is a continuous embedding.
The main theorem in this section is the following.
Theorem 2.1Let, the exponents, satisfy conditions (1.2)(1.4). Then Problem (1.1) has at least one weak solutionin the class
And one of the following conditions holds:
Proof Let be an orthogonal basis of with
is the subspace generated by the first k vectors of the basis . By normalization, we have . Let us define the operator
For any given integer k, we consider the approximate solution
which satisfies
Here we denote by the inner product in .
Problem (1.1) generates the system of k ordinary differential equations
By the standard theory of the ODE system, we infer that problem (2.2) admits a unique solution in , where . Then we can obtain an approximate solution for (1.1), in , over . And the solution can be extended to , for any given , by the estimate below. Multiplying (2.1) and summing with respect to i, we conclude that
By simple calculation, we have
here
Combining (2.3)(2.4) and (H1)(H2), we get
Integrating (2.5) over , and using assumptions (1.2)(1.4), we have
where C1 is a positive constant depending only on , .
Hence, by Lemma 2.1, we also have
In view of (H1)(H2) and (A1)(A2), we also have
where C2 is a positive constant depending only on , , l, , . It follows from (2.7) that
Next, multiplying (1.1) by and then summing with respect to i, we get that the following holds:
Note that
From Lemma 2.2, we have
where C, are embedding constants. From (2.10)(2.14), we obtain that
Integrating (2.15) over and using (2.7), Lemma 2.3, we get
where C3 is a positive constant depending only on .
Taking α, ε small enough in (2.16), we obtain the estimate
Hence, by Lemma 2.1, we have
where C4 is a positive constant depending only on , , l, , T.
From estimate (2.17), we get
By (2.7)(2.9) and (2.18), we infer that there exist a subsequence of and a function u such that
Next, we will deal with the nonlinear term. From the AubinLions theorem, see Lions [[18], pp.5758], it follows from (2.21) and (2.22) that there exists a subsequence of , still represented by the same notation, such that
which implies almost everywhere in . Hence, by (2.19)(2.22),
Multiplying (2.2) by (which is the space of function with compact support in ) and integrating the obtained result over , we obtain that
Note that is a basis of . Convergence (2.19)(2.24) is sufficient to pass to the limit in (2.25) in order to get
for arbitrary . In view of (2.19)(2.22) and Lemma 3.3.17 in [19], we obtain
Hence, we get , . 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

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

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)

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

Messaoudi, SA: Blowup of positiveinitialenergy solutions of a nonlinear viscoelastic hyperbolic equation. J. Math. Anal. Appl.. 320, 902–915 (2006). Publisher Full Text

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

Messaoudi, SA, SaidHouari, 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

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

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

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

Gao, Y, Guo, B, Gao, W: Weak solutions for a highorder pseudoparabolic equation with variable exponents. Appl. Anal. (2013). Publisher Full Text

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

AndreuVaillo, F, Caselles, V, Mazón, JM: Parabolic Quasilinear Equations Minimizing Linear Growth Functions, Birkhäuser, Basel (2004)

Antontsev, SN: Wave equation with Laplacian and damping term: blowup of solutions. C. R., Méc.. 339, 751–755 (2011). PubMed Abstract  Publisher Full Text

Fan, X, Zhao, D: On the spaces and . J. Math. Anal. Appl.. 263, 424–446 (2001). Publisher Full Text

Kováčik, O, Rákosník, J: On spaces and . Czechoslov. Math. J.. 41(116), 592–618 (1991)

Zhao, JN: Existence and nonexistence of solutions for . J. Math. Anal. Appl.. 172, 130–146 (1993). Publisher Full Text

Fan, X, Shen, J, Zhao, D: Sobolev embedding theorems for spaces . J. Math. Anal. Appl.. 262, 749–760 (2001). Publisher Full Text

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

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