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

Pengyu Chen

doi:10.3934/dcds.2020383

Article Contents

2021, Volume 41, Issue 6: 2725-3737. Doi: 10.3934/dcds.2020383

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

Pengyu Chen

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received: March 31, 2020

Revised: June 30, 2020

Early access: November 2020

Published: June 2021

Research supported by the National Natural Science Foundations of China (No. 12061063, No. 11661071), Project of NWNU-LKQN2019-3 and China Scholarship Council (No. 201908625016)

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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.

Keywords:

Non-autonomous stochastic evolution equations,
measure of noncompactness,
nonlocal initial conditions,
Wiener process,
noncompact evolution family.

Mathematics Subject Classification: Primary: 34F05; Secondary: 60H15.

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