Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families

Abstract

In this paper, we investigate the non-autonomous stochastic evolution equations of parabolic type with nonlinear noise and nonlocal initial conditions in Hilbert spaces, where the operators in linear part depend on time t and generate an noncompact evolution family. New existence result of mild solutions is established under some weaker growth and measure of noncompactness conditions on nonlinear functions and nonlocal term. The discussions are based on Sadovskii's fixed-point theorem as well as the theory of evolution family. At last, as a sample of application, the obtained abstract result is applied to a class of non-autonomous stochastic partial differential equations of parabolic type with nonlocal initial conditions. The result obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

Existence and asymptotic behavior of square-mean S-asymptotically periodic solutions of fractional stochastic evolution equations

“…In recent years, more and more mathematicians have paid attention to the qualitative study of solutions of semilinear stochastic evolution equations(see [3,16,17,36] and the references therein). Very recently, in [7,10,11], Chen and Zhang investigated the non-autonomous stochastic evolution equations with nonlinear noise and nonlocal initial conditions, and obtained the existence results of mild solutions under some weaker growth conditions on nonlinear functions. It is also worth noting that in [8], Chen et al established a sufficient condition to judge the relative compactness of abstract continuous function families on infinite intervals.…”