Arbitrary decays for a viscoelastic equation
© Wu; licensee Springer. 2011
Received: 16 February 2011
Accepted: 6 October 2011
Published: 6 October 2011
In this paper, we consider the nonlinear viscoelastic equation , in a bounded domain with initial conditions and Dirichlet boundary conditions. We prove an arbitrary decay result for a class of kernel function g without setting the function g itself to be of exponential (polynomial) type, which is a necessary condition for the exponential (polynomial) decay of the solution energy for the viscoelastic problem. The key ingredient in the proof is based on the idea of Pata (Q Appl Math 64:499-513, 2006) and the work of Tatar (J Math Phys 52:013502, 2010), with necessary modification imposed by our problem.
Mathematical Subject Classification (2010): 35B35, 35B40, 35B60
KeywordsViscoelastic equation Kernel function Exponential decay Polynomial decay
It is well known that viscoelastic materials have memory effects. These properties are due to the mechanical response influenced by the history of the materials themselves. As these materials have a wide application in the natural sciences, their dynamics are of great importance and interest. From the mathematical point of view, their memory effects are modeled by an integro-differential equations. Hence, questions related to the behavior of the solutions for the PDE system have attracted considerable attention in recent years. Many authors have focused on this problem for the last two decades and several results concerning existence, decay and blow-up have been obtained, see [1–28] and the reference therein.
for all t ≥ 0 and some positive constants ξ1 and ξ2. Later, this result was extended by Messaoudi and Tatar  to a situation where a nonlinear source term is competing with the dissipation terms induced by both the viscoelasticity and the viscosity. Recently Messaoudi and Tatar  studied problem (1.1) for the case of γ = 0, they improved the result in  by showing that the solution goes to zero with an exponential or polynomial rate, depending on the decay rate of the relaxation function g.
The assumptions (1.2) and (1.3), on g, are frequently encountered in the linear case (ρ = 0), see [1, 2, 4–6, 13, 22, 23, 29–31]. Lately, these conditions have been weakened by some researchers. For instance, instead of (1.3) Furati and Tatar  required the functions e αt g(t) and e αt g'(t) to have sufficiently small L1-norm on (0, ∞) for some α > 0 and they can also have an exponential decay of solutions. In particular, they do not impose a rate of decreasingness for g. Later on Messaoudi and Tatar  improved this result further by removing the condition on g'. They established an exponential decay under the conditions g'(t) ≤ 0 and e αt g(t) ∈ L1(0, ∞) for some large α > 0. This last condition was shown to be necessary condition for exponential decay . More recently Tatar  investigated the asymptotic behavior to problem (1.1) with ρ = γ = 0 when h(t)g(t) ∈ L1(0, ∞) for some nonnegative function h(t). He generalized earlier works to an arbitrary decay not necessary of exponential or polynomial rate.
where Ω ⊂ R N , N ≥ 1, is a bounded domain with a smooth boundary ∂Ω. Here ρ, p > 0 and g represents the kernel of the memory term, with conditions to be stated later [see assumption (A1)-(A3)].
We intend to study the arbitrary decay result for problem (1.4)-(1.6) under the weaker assumption on g, which is not necessarily decaying in an exponential or polynomial fashion. Indeed, our result will be established under the conditions g'(t) ≤ 0 and for some nonnegative function ξ(t). Therefore, our result allows a larger class of relaxation functions and improves some earlier results concerning the exponential decay or polynomial decay.
The content of this paper is organized as follows. In Section 2, we give some lemmas and assumptions which will be used later, and we mention the local existence result in Theorem 2.2. In Section 3, we establish the statement and proof of our result related to the arbitrary decay.
2 Preliminary results
In this section, we give some assumptions and lemmas which will be used throughout this work. We use the standard Lebesgue space L p (Ω) and Sobolev space with their usual inner products and norms.
holds with the optimal positive constant c s , where || · || p denotes the norm of L p (Ω).
With regards to the relaxation function g(t), we assume that it verifies
(A2) g'(t) ≤ 0 for almost all t > 0.
Proof. Multiplying Eq. (1.4) by u t and integrating it over Ω, then using integration by parts and the assumption (A1)-(A2), we obtain (2.6).
3 Decay of the solution energy
Remark. This functional was first introduced by Tatar  for the case of ρ = 0 and without imposing the dispersion term and forcing term as far as (1.4) is concerned.
The following Lemma tells us that L(t) and E(t) + Φ3(t) are equivalent.
holds for all t ≥ 0 and λ i small, i = 1, 2.
To obtain a better estimate for , we need the following Lemma which repeats Lemma 2 in .
Proof. Straightforward computations yield this identity.
since g is nonnegative and continuous.
where μ and K are positive constants.
We now estimate the first two terms on the right-hand side of (3.11) as in .
for some positive constants K and μ.
Similar to those remarks as in , we have the following remark.
Remark. Note that there is a wide class of relaxation functions satisfying (A3). More precisely, if ξ(t) = e αt , α > 0, then η(t) = α, this gives the exponential decay estimate , for some positive constants c1 and c2. Similarly, if ξ(t) = (1 + t) α , α > 0, then we obtain the polynomial decay estimate E (t) ≤ c3 (1 + t)-μ, for some positive constants c3 and μ.
The authors would like to thank very much the anonymous referees for their valuable comments on this work.
- Berrimi S, Messaoudi SA: Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal Theory Methods Appl 2006, 64: 2314-2331. 10.1016/j.na.2005.08.015View ArticleMathSciNetGoogle Scholar
- Berrimi S, Messaoudi SA: Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping. Electron J Diff Equ 2004, 88: 1-10.MathSciNetGoogle Scholar
- Cavalcanti MM, Domingos Cavalcanti VN, Ferreira J: Existence and uniform decay of nonlinear viscoelastic equation with strong damping. Math Methods Appl Sci 2001, 24: 1043-1053. 10.1002/mma.250View ArticleMathSciNetGoogle Scholar
- Cavalcanti MM, Domingos Cavalcanti VN, Soriano JA: Exponential decay for the solution of semilinear viscoelastic wave equation with localized damping. Electron J Diff Equ 2002, 44: 1-14.MathSciNetGoogle Scholar
- Cavalcanti MM, Domingos Cavalcanti VN, Prates Filho JS, Soriano JA: Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Diff Integr Equ 2001, 14(1):85-116.MathSciNetGoogle 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/S0363012902408010View ArticleMathSciNetGoogle Scholar
- Fabrizo M, Polidoro S: Asymptotic decay for some differential systems with fading memory. Appl Anal 2002, 81: 1245-1264. 10.1080/0003681021000035588View ArticleMathSciNetGoogle Scholar
- Furati K, Tatar N-e: Uniform boundedness and stability for a viscoelastic problem. Appl Math Comput 2005, 167: 1211-1220. 10.1016/j.amc.2004.08.036View ArticleMathSciNetGoogle Scholar
- Han X, Wang M: Global existence and uniform decay for a nonlinear viscoelastic equation with damping. Nonlinear Anal Theory Methods Appl 2009, 70: 3090-3098. 10.1016/j.na.2008.04.011View ArticleGoogle Scholar
- Kawashima S, Shibata Y: Global existence and exponential stability of small solutions to nonlinear viscoelasticity. Commun Math Phys 1992, 148: 189-208. 10.1007/BF02102372View ArticleMathSciNetGoogle Scholar
- Kirane M, Tatar N-e: A memory type boundary stabilization of a mildy damped wave equation. Electron J Qual Theory Diff Equ 1999, 6: 1-7.View ArticleMathSciNetGoogle Scholar
- Liu WJ: General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms. J Math Phys 2009, 50: 113506. 10.1063/1.3254323View ArticleMathSciNetGoogle Scholar
- Medjden M, Tatar N-e: Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel. Appl Math Comput 2005, 167: 1221-1235. 10.1016/j.amc.2004.08.035View ArticleMathSciNetGoogle Scholar
- Messaoudi SA, Tatar N-e: Exponential and polynomial decay for quasilinear viscoelastic equation. Nonlinear Anal Theory Methods Appl 2007, 68: 785-793.View ArticleMathSciNetGoogle Scholar
- Messaoudi SA, Tatar N-e: Global existence and asymptotic behavior for a nonlinear viscoelastic problem. Math Sci Res J 2003, 7(4):136-149.MathSciNetGoogle Scholar
- Messaoudi SA, Tatar N-e: Global existence and uniform stability of solutions for a quasilinear viscoelastic problem. Math Methods Appl Sci 2007, 30: 665-680. 10.1002/mma.804View ArticleMathSciNetGoogle Scholar
- Messaoudi SA: Blow-up and global existence in a nonlinear viscoelastic wave equation. Math Nachr 2003, 260: 58-66. 10.1002/mana.200310104View ArticleMathSciNetGoogle 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.2005.07.022View ArticleMathSciNetGoogle Scholar
- Messaoudi SA: General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Anal Theory Methods Appl 2008, 69: 2589-2598. 10.1016/j.na.2007.08.035View ArticleMathSciNetGoogle Scholar
- Messaoudi SA: General decay of solutions of a viscoelastic equation. J Math Anal Appl 2008, 341: 1457-1467. 10.1016/j.jmaa.2007.11.048View ArticleMathSciNetGoogle Scholar
- Messaoudi SA, Tatar N-e: Exponential decay for a quasilinear viscoelastic equation. Math Nachr 2009, 282: 1443-1450. 10.1002/mana.200610800View ArticleMathSciNetGoogle Scholar
- Munoz Rivera JE, Lapa EC, Baretto R: Decay rates for viscoelastic plates with memory. J Elast 1996, 44: 61-87. 10.1007/BF00042192View ArticleGoogle Scholar
- Nečas MJ, Šverák V: On weak solutions to a viscoelasticity model. Comment Math Univ Carolin 1990, 31(3):557-565.MathSciNetGoogle Scholar
- Pata V: Exponential stability in linear viscoelasticity. Q Appl Math 2006, 64: 499-513.View ArticleMathSciNetGoogle Scholar
- Tatar N-e: Arbitrary decay in linear viscoelasticity. J Math Phys 2010, 52: 013502.View ArticleMathSciNetGoogle Scholar
- Wu ST: Blow-up of solutions for an integro-differential equation with a nonlinear source. Electron J Diff Equ 2006, 45: 1-9.Google Scholar
- Wu ST: General decay of solutions for a viscoelastic equation with nonlinear damping and source terms. Acta Math Sci 2011, 31(4):1436-1448.View ArticleMathSciNetGoogle Scholar
- Wu ST: General decay of energy for a viscoelastic equation with linear damping and source term. Taiwan J Math, in press.
- Hrusa WJ: Global existence and asymptotic stability for a nonlinear hyperbolic Volterra equation with large initial data. SIAM J Math Anal 1985, 16: 110-134. 10.1137/0516007View ArticleMathSciNetGoogle Scholar
- Medjden M, Tatar N-e: On the wave equation with a temporal nonlocal term. Dyn Syst Appl 2007, 16: 665-672.MathSciNetGoogle Scholar
- Tiehu Q: Asymptotic behavior of a class of abstract integrodifferential equations and applications. J Math Anal Appl 1999, 233: 130-147. 10.1006/jmaa.1999.6271View ArticleMathSciNetGoogle Scholar