General stability result for a viscoelastic plate equation with past history and general kernel

Abstract

In this paper, we consider the following plate problem:utt−σΔutt+Δ2u−∫0+∞k(s)Δ2u(t−s)ds=0, and we show that the stability of this problem holds for a much larger class of kernels. More precisely, we consider the kernel k:[0,+∞)→(0,+∞) satisfiesk′(t)≤−ξ(t)Ψ(k(t)),t≥0, where ξ and Ψ are functions satisfying some specific properties. Under this very general assumption on the behavior of k at infinity, we establish a relation between the decay rate of the solution and the growth of k at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumption on the history data, usually made in the literature.