Explosive solutions of parabolic stochastic partial differential equations with L$\acute{e}$vy noise

TL;DR

This paper investigates conditions under which solutions to certain parabolic stochastic PDEs with Lévy noise blow up in finite time or exist globally, providing theoretical results and illustrative examples.

Contribution

It introduces new criteria for explosion and global existence of solutions to stochastic PDEs driven by Lévy noise, including a Lyapunov functional approach.

Findings

Positive solutions blow up in finite time under certain conditions.

Global solutions exist for specific stochastic Allen-Cahn equations.

Theoretical criteria are supported by multiple examples.

Abstract

In this paper, we study the explosive solutions to a class of parbolic stochastic semilinear differential equations driven by a L$\acute{\mbox{e}}$vy type noise. The sufficient conditions are presented to guarantee the existence of a unique positive solution of the stochastic partial differential equation under investigation. Moreover, we show that the positive solutions will blow up in finite time in mean $L^{p}$-norm sense, provided that the initial data, the nonlinear term and the multiplicative noise satisfies some conditions. Several examples are presented to illustrated the theory. Finally, we establish a global existence theorem based on a Lyapunov functional and prove that a stochastic Allen-Cahn equation driven by L$\acute{\mbox{e}}$vy noise has a global solution.