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Compactness Properties and General Stability Result for an Abstract Nonlinear Viscoelastic Equation

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Abstract

This paper develops a unified method to derive compactness properties and convergence to a steady state as well as decay estimates of global bounded solutions of the abstract semilinear second order integro-differential equation

$$\begin{aligned} u_{tt}(t)+ Au(t) -\int ^\tau _0 k(s) Au(t-s)ds+ f(u)=g(t), \ \tau \in \{t, \infty \}, \end{aligned}$$

with finite (\(\tau =t\)) or infinite (\(\tau =\infty \)) memory. Here, A is a continuous linear operator, k is a positive nonincreasing function, f is a nonlinear function such that \(A+f\) is the gradient of a potential satisfying the Łojasiewicz–Simon inequality, and the time-dependent right-hand side g decays to 0 at infinity. The general results are applied to particular semilinear evolution viscoelastic problems at the end of the paper.

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  1. Department of Mathematics, Faculty of Sciences, Lebanese University, Zahle, Lebanon

    Hassan Yassine

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Yassine, H. Compactness Properties and General Stability Result for an Abstract Nonlinear Viscoelastic Equation. J Dyn Diff Equat 34, 2229–2257 (2022). https://doi.org/10.1007/s10884-021-10014-4

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