Optimal decay rates for the viscoelastic wave equation

Abstract

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.

Optimal Decay Rate Estimates of a Nonlinear Viscoelastic Kirchhoff Plate

“…In this paper, we continue to study (1)-(3), in which we consider σ � 1 for simplicity, with minimal conditions on the L 1 (0, ∞) relaxation function g (see (12)). We establish explicit and general energy decay results of systems (1)-(3) by using the idea of Mustafa [22,23] and some properties of convex functions developed in [18,39]. We point out that the decay results established here are optimal exponential and polynomial rates for 1 ≤ p < 2 when G(t) � t p , which improved the previous known results for 1 ≤ p < (3/2).…”

“…We refer the reader to Cavalcanti et al [15,16], Lasiecka et al [17,18], Mustafa [19], Mustafa and Messaoudi [20], and Xiao and Liang [21]. Very recently, in [22,23], Mustafa considered two classes of wave equations and proved general and explicit decay results of energy by using a new general assumption on g: g ′ (t) ≤ − ξ(t)H(g(t)). For viscoelastic plate equation, Rivera et al [24] studied the following equation:…”

Optimal decay result for Kirchhoff plate equations with nonlinear damping and very general type of relaxation functions

“…As in [44], we can deduce that α i C α i → 0 when α i → 0. Then there exists 0 < α 0i < 1 such that if α i < α 0i , then…”

“…(1) by u t , integrating by parts over Ω, and using (28) and the boundary conditions (10) and (11). With the help of the ideas in [44], one can establish the following two helpful lemmas. satisfies, along the solution of (1)-(5), the estimate…”

Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay

“…It is well-known that the memory term generates stabilizing mechanism in viscoelastic wave systems (see [1,3,9,24])…”