Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families

Abstract

This paper investigates the Cauchy problem to a class of stochastic non-autonomous evolution equations of parabolic type governed by noncompact evolution families in Hilbert spaces. Combining the theory of evolution families, the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness, we established some new existence results of mild solutions under the situation that the nonlinear function satisfy some appropriate local growth condition and a noncompactness measure condition. Our results generalize and improve some previous results on this topic, since the strong restriction on the constants in the condition of noncompactness measure is completely deleted, and also the condition of uniformly continuity of the nonlinearity is not required. At last, as samples of applications, we consider the Cauchy problem to a class of stochastic non-autonomous partial differential equation of parabolic type.