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

Abstract

In this article, we study the well-posedness theory for solutions of the stochastic heat equations with logarithmic nonlinearity perturbed by multiplicative Lévy noise. By using Aldous tightness criteria and Jakubowski’s version of the Skorokhod theorem on non-metric spaces along with the standard \(L^2\)-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 \( u \log (|u|)\) present in the underlying problem, the logarithmic Sobolev inequality and the nonlinear versions of Gronwall’s inequalities play a crucial role.