2011
DOI: 10.1016/j.jmaa.2011.04.079
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Asymptotic stability of abstract dissipative systems with infinite memory

Abstract: We consider in this paper the problem of asymptotic behavior of solutions to an abstract linear dissipative integrodifferential equation with infinite memory (past history) modeling linear viscoelasticity. We show that the stability of the system holds for a much larger class of the convolution kernels than the one considered in the literature, and we provide a relation between the decay rate of the solutions and the growth of the kernel at infinity. Some applications are also given.

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Cited by 107 publications
(125 citation statements)
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“…We also point out that kernels satisfying (5.38) has been considered by Messaoudi and Said-Houari [19] for Timoshenko systems. Recently, the assumption (5.38) has been further weakened by Guesmia [12], where he studied an abstract hyperbolic system with past history.…”
mentioning
confidence: 99%
“…We also point out that kernels satisfying (5.38) has been considered by Messaoudi and Said-Houari [19] for Timoshenko systems. Recently, the assumption (5.38) has been further weakened by Guesmia [12], where he studied an abstract hyperbolic system with past history.…”
mentioning
confidence: 99%
“…They provided the well‐posedness and exponential stability, while memory kernel g ( t ) is exponentially goes to zero. Guesmia considered the following system: {arrayutt+Au(t)0+g(s)Bu(ts)ds=0,t>0,arrayu(t)=u0(t),t0. Here, A and B are two self‐adjoint positive definite operators with D(A)D(B); g ( s ) satisfies: g:R+R+ is a differentiable nonincreasing function satisfying 0<0+g(s)ds<1a0, there exists a positive, increasing, convex function G:R+R+ satisfying G(0)=G(0)=0,limtG(t)=+, …”
Section: Introductionmentioning
confidence: 99%
“…On the one hand, to create the negative counterparts of the terms in the energy, we combine the fireworks of [4], [14] and [19] with necessary modifications. On the other hand, to estimate the infinite integral terms in (4.20) below, we use the approach which was first proved by Guesmia [10] and used by many researchers (see [11,14]). This paper is organized as follows.…”
Section: Introductionmentioning
confidence: 99%