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A Class of Impulsive Stochastic Parabolic Functional Differential Equations and Their Asymptotics

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Abstract

This paper is devoted to the study of the initial value problem for a class of semilinear impulsive stochastic parabolic functional differential equations. Incorporating certain positive operators, these equations have as archetype impulsive stochastic functional heat differential equations supplemented by homogeneous Dirichlet boundary conditions. By the classical Galerkin’s method, we prove a well-posedness result for the initial value problem for a class of linear random evolution equation which is necessary in developing our main theory. Using this well-posedness result and the classical contraction mapping argument, we prove that the initial value problem under consideration is globally well-posed. Employing the technique of Razumikhin, we prove under some additional assumptions that the trivial solution of the equation in question is mean-square exponentially stable.

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Notes

  1. Consult [3, pp. 293–308] for frequently-used results related to the complexification of real Hilbert spaces.

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Acknowledgements

We are grateful to the anonymous reviewers for their valuable suggestions which led an interesting improvement of the manuscript. The work is supported by NSFC (#11471231, #11571244 and #11401404), such support is deeply appreciated.

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  1. School of Mathematics, Sichuan University, Chengdu, 610064, P.R. China

    Chengqiang Wang

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Wang, C. A Class of Impulsive Stochastic Parabolic Functional Differential Equations and Their Asymptotics. Acta Appl Math 146, 163–186 (2016). https://doi.org/10.1007/s10440-016-0063-4

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