Stochastic heat equations with logarithmic nonlinearity

TL;DR

This paper proves the existence and uniqueness of solutions to stochastic heat equations with logarithmic nonlinearities on bounded domains, utilizing the logarithmic Sobolev inequality within an $L^2(D)$ framework.

Contribution

It establishes the well-posedness of stochastic heat equations with logarithmic nonlinearities for all initial data in $L^2(D)$, a novel extension in the field.

Findings

Existence and uniqueness of solutions are proven.

The results hold for all initial values in $L^2(D)$.

The logarithmic Sobolev inequality is key to the analysis.

Abstract

In this paper, we establish the existence and uniqueness of solutions to stochastic heat equations with logarithmic nonlinearity driven by Brownian motion on a bounded domain $D$ in the setting of $L^2(D)$ space. The result is valid for all initial values in $L^2(D)$. The logarithmic Sobolev inequality plays an important role.