New general decay rates of solutions for an abstract semilinear stochastic evolution equation with an infinite memory

Semilinear stochastic evolution equation with an infinite memory

Abstract

Our interest in this research work is to analyse the asymptotic stability of the second-order stochastic evolution equation:\begin{equation*}u_{tt}(t)+Au(t) - \displaystyle{\int_{0}^{+\infty}}h(s)A^{\alpha}u(t-s) ds +f(u(t))= \sigma (t)W_t(t) , \mskip14mu \forall t \geq 0.\end{equation*}We first show that the system is well-posed by using the semi-group theory. Secondly, by assuming the general condition: $$h'(t) \leq - \xi(t) h(t) , \mskip14mu \forall t \geq 0 ,$$ where $\xi$ is a positive function which is not necessarily monotone, we establish two stability results with decay rates depending on $\alpha$ and on the regularity of the initial data. Finally, we give some applications in order to illustrate our abstract results. This study improves and generalizes many previous ones in the literature.