Existence and stability of mild solutions to parabolic stochastic partial differential equations driven by Lévy space-time noise

Abstract

This paper is concerned with well-posedness and stability of parabolic stochastic partial differential equations. Firstly, we obtain some sufficient conditions ensuring the existence and uniqueness of mild solutions, and some $\mathcal{H}$-stability criteria for a class of parabolic stochastic partial differential equations driven by Lévy space-time noise under the local/non-Lipschitz condition. Secondly, we establish some existence-uniqueness theorems and present sufficient conditions ensuring the $\mathcal{H}'$-stability of mild solutions for a class of parabolic stochastic partial functional differential equations driven by Lévy space-time noise under the local/non-Lipschitz condition. These theoretical results generalize and improve some existing results. Finally, two examples are given to illustrate the effectiveness of our main results.

Keywords

• lévy space-time noise
• parabolic stochastic partial differential equation
• non-lipschitz
• well-posedness
• stability