Large deviations of stochastic heat equations with logarithmic nonlinearity

TL;DR

This paper proves a large deviation principle for solutions to stochastic heat equations with logarithmic nonlinearity driven by Brownian motion, addressing challenges due to non-Lipschitz and non-monotone nonlinearities.

Contribution

It introduces new techniques using nonlinear Gronwall and Log-Sobolev inequalities to handle the non-Lipschitz, non-monotone logarithmic nonlinearities in stochastic heat equations.

Findings

Established a large deviation principle for the equations.

Developed methods to handle non-Lipschitz nonlinearities.

Extended large deviation theory to new classes of stochastic PDEs.

Abstract

In this paper, we establish a large deviation principle for the solutions to the stochastic heat equations with logarithmic nonlinearity driven by Brownian motion, which is neither locally Lipschitz nor locally monotone. Nonlinear versions of Gronwall's inequalities and Log-Sobolev inequalities play an important role.