Asymptotic stability of semigroups associated with linear weak dissipative systems with memory

Abstract

We study the asymptotic behavior of the solutions of a class of linear dissipative integral differential equations. We show in the abstract setting a necessary and sufficient condition to get an exponential decay of the solution. In the case of the lack of exponential decay, we find the polynomial rate of decay of the solution. Some examples are given.

General decay of solutions of a viscoelastic equation

“…This result is sharp (see [25,Theorem 12]). This result has been improved by Rivera et al [27], where the authors studied a more general abstract problem than (1.2) and established a necessary and sufficient condition to obtain an exponential decay. In the case of lack of exponential decay, a polynomial decay has been proved.…”

Asymptotic stability of abstract dissipative systems with infinite memory

“…It is well known, following a method devised in the pioneering paper [5] (see also [13,15,16]), that the system (1.1)-(1.2) can be formulated as the following abstract linear first-order system:…”

“…∀t, s ∈ R + , η 0 (s) = η 0 (s) = u 0 (0) − u 0 (s), ∀s ∈ R + (η t is the relative history of u, and it was introduced first in [ , (1.4) it is well known (see [13] for example) that H endowed with the inner product…”

“…In other words, for which class of kernels g we have (strong stability) lim t→∞ U (t) 2 H = 0, (1.5) and is it possible to get a decay estimation on U 2 This question was the subject of several works appeared in the last few years (see [3,[6][7][8][9]13,[15][16][17][18], and the references therein). To focus on our motivation, let us mention some known results in the literature related to the stabilization of abstract systems with past history (for further results of stabilization, we refer the reader to the list of references of this paper, which is not exhaustive, and the references therein).…”

General Decay of Solutions of a Weak Viscoelastic Equation

“…The main result, for the case 0 < α < 1, shows that the decay of solutions of (2) is polynomial even if the kernel g decays exponentially. A more general abstract problem was studied by Rivera et al [17] and a necessary and sufficient condition for the exponential decay was obtained.…”