Lévy Driven Stochastic Heat Equation with Logarithmic Nonlinearity: Well-Posedness and Large Deviation Principle

In this article, we study the well-posedness theory for solutions of the stochastic heat equations with logarithmic nonlinearity perturbed by multiplicative L & eacute;vy noise. By using Aldous tightness criteria and Jakubowski's version of the Skorokhod theorem on non-metric spaces along with the standard L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2$$\end{document}-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 term ulog(|u|)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u \log (|u|)$$\end{document} present in the underlying problem, the logarithmic Sobolev inequality and the nonlinear versions of Gronwall's inequalities play a crucial role.