General stability result for viscoelastic wave equations

Abstract

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.

Decay rates for memory‐type plate system with delay and source term

“…Here, a new theorem was announced, which was applied to some new examples giving optimal decay rates. Later on, the preceding conditions were weakened by Mustafa and Messaoudi and Lasiecka et al . for the finite memory case and Guesmia for the infinite memory case, and explicit and general decay rate formulas were obtained.…”

General stability of solutions for a viscoelastic wave equations of Kirchhoff type with acoustic boundary conditions

“…Wu and Tsai discussed the energy decay for an integro‐differential equation with strong damping under the usual conditions for some r > 0. Later, Wu proved the exponential energy decay of solutions for viscoelastic equation with strong damping under condition on the nonnegative kernel g such as Besides, the problem of stability of the solutions to a viscoelastic system with memory has been investigated by many authors and the references cited therein . For example, Cavalcanti et al , proved exponential and polynomial decay for a viscoelastic wave equation under the usual conditions for some c i , i = 1,2,3.…”

“…Cavalcanti et al , obtained general stability for a viscoelastic dissipative systems under very weak conditions on the kernel function. Recently, Messaoudi and Mustafa considered the general stability result for a viscoelastic equations under weaker condition on g such as where H is a non‐negative function, with H (0) = 0, and H is a linear or strictly increasing and strictly convex on (0, r ], for some r > 0. Motivated by these results, in this paper, we show the general decay of the solutions to a viscoelastic wave equations of Kirchhoff type with acoustic boundary condition.…”

Energy decay of dissipative plate equations with memory-type boundary conditions

“…Also, by taking D(t) = t α , (2.8) is satisfied for any α > 1. Therefore, if h satisfies (A3) with this function H , then we can use Theorem 2.1 and do some calculations (see [38]) to deduce that the energy decays at the rate…”

“…Recently, the above conditions were strongly weakened by Mustafa and Messaoudi [38] and an explicit and general decay rate formula was obtained.…”