Note on intrinsic decay rates for abstract wave equations with memory

Abstract

In this paper we consider a viscoelastic abstract wave equation with memory kernel satisfying the inequality g′ + H(g) ⩽ 0, s ⩾ 0 where H(s) is a given continuous, positive, increasing, and convex function such that H(0) = 0. We shall develop an intrinsic method, based on the main idea introduced by Lasiecka and Tataru [“Uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation,” Differential and Integral Equations 6, 507–533 (1993)], for determining decay rates of the energy given in terms of the function H(s). This will be accomplished by expressing the decay rates as a solution to a given nonlinear dissipative ODE. We shall show that the obtained result, while generalizing previous results obtained in the literature, is also capable of proving optimal decay rates for polynomially decaying memory kernels (H(s) ∼ sp) and for the full range of admissible parameters p ∈ [1, 2). While such result has been known for certain restrictive ranges of the parameters p ∈ [1, 3/2), the methods introduced previously break down when p ⩾ 3/2. The present paper develops a new and general tool that is applicable to all admissible parameters.

Intrinsic Decay Rate Estimates for Semilinear Abstract Second Order Equations with Memory

“…This unnatural restriction is removed by using the methodology presented in this paper. In fact, sub-optimality of the results in the polynomial case was a motivation for introducing new method in [20], which not only generalizes previous theories, but it also provides results in the cases which were explicitly ruled out in previous treatments. However, the proof in [20] is inductive and it requires verification of a "dynamic" hypothesis (we note that the inductive hypothesis introduced in [20] is very different from "successive energy decay estimates or "boot strap" arguments used in [1,13].…”

“…Non-optimality is already manifested in the case of algebraically decaying kernels where the method used forces the restricted range of parameters, as explained in Remark 14.3 above. Lasiecka et al [20] achieves sharp decay rates for the case of inequality, however the most general result requires an inductive-dynamic hypothesis to be satisfied. The contribution of the present work is that it removes the limitation of [20] and there is no need for the dynamic hypothesis.…”

“…Lasiecka et al [20] achieves sharp decay rates for the case of inequality, however the most general result requires an inductive-dynamic hypothesis to be satisfied. The contribution of the present work is that it removes the limitation of [20] and there is no need for the dynamic hypothesis.…”

“…While vast majority of the work deals with well quantized relaxation kernels or with kernels specifying "linear type" of differential inequalities characterized by g 0 .t/ C .t/g.t/ Ä 0 with suitable assumptions imposed on time dependent function .t/ [7-9, 12, 18, 19, 25, 26, 32], there have been few attempts to reach the level of generality which does not require such specifications. In this context we should mention [2,20,32]. All these works follow a general idea of employing convex analysis (as in [22]) in order to obtain estimates independent on specific characterization of the kernel.…”

“…It should also be mentioned that from the point of view of methodology, the methods used in [2,27,32] rely on weighted energy inequalities, while the method used now and in [20] pursues the approach of [22] which is ODE based. It appears that ODE based method is more effective when formulating the results for relaxation kernels satisfying differential inequalities-rather than equalities.…”