ASYMPTOTIC BEHAVIOR FOR A NONLINEAR VISCOELASTIC PROBLEM WITH A VELOCITY-DEPENDENT MATERIAL DENSITY

We consider a nonlinear viscoelastic problem and prove that the solutions are uniformly bounded and decay exponentially to zero as time goes to infinity. This is established under weaker conditions on the relaxation function than the usually used ones. In particular, we remove the assumptions on the derivative of the kernel. In fact, our kernels are not necessarily differentiable.


Introduction
The problem we would like to investigate is the following: where Ω is a bounded domain in R n , n ≥ 1, with a smooth boundary Γ.The real number ρ is assumed to satisfy 0 < ρ ≤ 2/(n − 2) if n ≥ 3 or ρ > 0 if n = 1,2.The function g(t) is positive and will be specified further below.This model appears in viscoelasticity.We are in the case where the material density depends on u t (see [5,11]).In [1], Cavalcanti et al. studied this nonlinear problem (ρ > 0) and proved well posedness as well as a uniform decay result.It has been shown that solutions go to zero in an exponential manner provided that the kernel g(t) is also exponentially decaying to zero.Namely, the following assumptions were assumed: (H1) g : R + → R + is a bounded C 1 -function such that (H2) there exist positive constants ξ 1 , ξ 2 such that These two assumptions are in fact frequently used also in the linear case (ρ = 0) (see [3,4,5,6,7,8] and also [10,13]).In [9], the present author with Messaoudi have improved the result in [1] by showing that the same asymptotic behavior occurs also for the case γ = 0.This means that the convolution term produces a weak dissipation which is able to drive solutions to the equilibrium state in an exponential manner.We do not need the strong damping.In [6], the present author with Furati proved that for "sufficiently small" g and g , we also have exponential decay (in case ρ > 0).Namely, we need e αt g(t) and e αt g (t) to have "small" L 1 -norms for some α > 0. The conditions in (H2) are not imposed.In particular, g is not necessarily always negative.
Here in this work, we intend to improve further this latter result by removing the condition on g .To this end, we combine the multiplier technique with some appropriate estimations and some new "Lyapunov-type" functionals.These functionals are somewhat similar in spirit to the one introduced by the author in [12].
The plan of the paper is as follows.In the next section, we state an existence theorem, introduce our functionals, and prove some useful propositions for our result.Section 3 is devoted to the exponential decay theorem.

Preliminaries
We start by stating an existence result due to Cavalcanti et al. [1] (see also [2]).
Theorem 2.1.Assume that the kernel g : We point out here that the differentiability of g is not needed to prove local existence.In this paper, we consider γ = 0. We may assume that γ = 1.
The (classical) energy associated to problem (1.1) is defined by If we differentiate E(t) with respect to t along solutions of (1.1), we get This expression is of an undefined sign, and therefore the boundedness (and the dissipativity) of the energy functional E(t) is not clear.In the prior works, the authors defined (2.5) Then, considering the modified energy functional (2.6) it appears that At this point, they use the fact that g (t) ≤ 0 to obtain uniform boundedness.In our case, we do not have this assumption.To overcome this, a new functional has been proposed in [6].An exponential decay result has been obtained under some "smallness" condition on g(t) and g (t).It is our objective here to remove the smallness condition on g (t).In fact, even the differentiability of g is not required.We will need the assumptions (G1) g : R + → R + is a bounded continuous function such that (G2) g(t)e αt ∈ L 1 (0,∞) for some α > 0. We will use repeatedly the following inequality.Lemma 2.2.For any a,b ∈ R and δ > 0, (2.10) Next, we prove the uniform boundedness of the classical energy.
Proof.We have (2.12) From (2.12) and Lemma 2.2 with δ = 1/4, we find for some α > 0. A differentiation of (2.17) yields By Lemma 2.2 with δ = 1/8λ, for some λ > 0 to be determined, we have (2.20) Nasser-Eddine Tatar 1501 Notice that We define Clearly, by (2.16) and (2.22), we have If ḡα ≤ α/2, then it is possible to choose λ so that λ ≥ ḡα (notice that ḡα > ḡ) and λ ≤ α 2 /4 ḡα .Hence, V (t) ≤ 0. Consequently, e(t) and thereafter E(t) are uniformly bounded for all t ≥ 0 by e(0).This proposition will be used in a crucial manner in our main result.However, the functional V (t) is still not suitable to work with.We introduce (2.25) Then, we form the expression for some ε > 0 to be determined later.
The next proposition will show, in particular, that the result we will derive for W(t) will also hold for the classical energy.

Long-time behavior
In this section, we state and prove our main result.Observe that assuming the hypotheses in Proposition 2.3, we have uniqueness of the weak solution.The solution corresponding to E(0) = 0 is the trivial one and is included in our next result.
We also have Therefore, By Lemma 2.2 again with δ = l/4, we find Nasser-Eddine Tatar 1505 Finally, by virtue of the embedding stated at the beginning of the proof of Proposition 2.4 (see (2.29)), we can deduce that (3.10) We must point out here that, to avoid a contradiction, the term in Ω |u t | ρ+2 dx which appears in the derivative of Ψ(t) (see (3.2)) has been estimated by From (3.10), it is clear that for sufficiently small ε and ḡα ≤ α 2 /8λ, there exists The right-hand side inequality in Proposition 2.4 implies that From this, we infer that Then, the left-hand side inequality in Proposition 2.4 allows us to conclude that This completes the proof of the theorem.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation