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.
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