Levy driven stochastic heat equation with logarithmic nonlinearity: Well-posedness and Large deviation principle

TL;DR

This paper investigates the existence, uniqueness, and large deviation principles for solutions of stochastic heat equations with logarithmic nonlinearities driven by Levy noise, addressing challenges posed by non-Lipschitz nonlinearities.

Contribution

It establishes well-posedness and large deviation principles for a class of stochastic heat equations with logarithmic nonlinearities and Levy noise, using advanced probabilistic and analytical techniques.

Findings

Existence of pathwise unique strong solutions.

Large deviation principle for the solutions.

Handling of non-Lipschitz nonlinearities without linear growth.

Abstract

In this article, we study the well-posedness theory for solutions of the stochastic heat equations with logarithmic nonlinearity perturbed by multiplicative Levy noise. By using Aldous tightness criteria and Jakubowski version of the Skorokhod theorem on non-metric spaces along with the standard L2 method, we establish the existence of a path-wise unique strong solution. Moreover, by using a weak convergence method, we establish a large deviation principle for the strong solution of the underlying problem. Due to the lack of linear growth and locally Lipschitzness of the nonlinear term present in the underlying problem, the logarithmic Sobolev inequality and the nonlinear versions of Gronwall inequalities play a crucial role.