Global well-posedness for the nonlinear generalized parabolic Anderson model equation

TL;DR

This paper establishes the global existence and uniqueness of solutions for a singular nonlinear parabolic Anderson model on two-dimensional tori using paracontrolled distributions and renormalization techniques.

Contribution

It introduces a novel approach combining paracontrolled distributions and renormalization to prove global well-posedness for the nonlinear parabolic Anderson model.

Findings

Proved global existence of solutions.

Established uniqueness via energy estimates.

Applied paracontrolled distribution method.

Abstract

We study the global existence of the singular nonlinear parabolic Anderson model equation on $2$-dimensional tours $\mathbb{T}^2$. The method is based on paracontrolled distribution and renormalization. After split the original nonlinear parabolic Anderson model equation into two simple equations, we prove the global well-posedness by some a priori estimates and smooth approximations. Furthermore, we prove the uniqueness of the solution by using classical energy estimates.