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.
In this paper, we consider a viscoelastic equation with minimal conditions on, where H is an increasing and convex function near the origin and is a nonincreasing function.With only these very general assumptions on the behavior of gat infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when H(s) = s p and p covers the full admissible range [1, 2). We get the best decay rates expected under this level of generality, and our new results substantially improve several earlier related results in the literature.
In this paper, we consider a viscoelastic equation and establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity. Our result allows a wider class of relaxation functions and improves earlier results in the literature.
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