Explosive solutions of parabolic stochastic partial differential equations with lévy noise

Kexue Li; Jigen Peng; Junxiong Jia

doi:10.3934/dcds.2017221

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2017, Volume 37, Issue 10: 5105-5125. Doi: 10.3934/dcds.2017221

This issue Previous Article On parabolic external maps Next Article A global bifurcation theorem for a positone multiparameter problem and its application

Explosive solutions of parabolic stochastic partial differential equations with lévy noise

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China

* Corresponding author
* Corresponding author

Received: August 2016

Revised: May 2017

Published: October 2017

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Abstract

In this paper, we study the explosive solutions to a class of parbolic stochastic semilinear differential equations driven by a Lévy 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 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 illustrate the theory. Finally, we establish a global existence theorem based on a Lyapunov functional and prove that a stochastic Allen-Cahn equation driven by Lévy noise has a global solution.

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Mathematics Subject Classification: Primary: 60H15, 60J75.

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