Absolute continuity of the solution to stochastic generalized Burgers-Huxley equation

TL;DR

This paper investigates the existence, uniqueness, and absolute continuity of solutions to the stochastic generalized Burgers-Huxley equation driven by space-time white noise, using Malliavin calculus and comparison theorems.

Contribution

It establishes global solvability, a comparison theorem, and the absolute continuity of the solution's law, advancing understanding of stochastic nonlinear PDEs with multiplicative noise.

Findings

Proved existence of a unique local mild solution.

Established global solvability using bounds and stopping times.

Demonstrated absolute continuity of the solution's law.

Abstract

The present work deals with the global solvability as well as absolute continuity of the law of the solution to stochastic generalized Burgers-Huxley (SGBH) equation driven by multiplicative space-time white noise in a bounded interval of $\mathbb{R}$. We first prove the existence of a unique local mild solution to SGBH equation with the help of a truncation argument and contraction mapping principle. Then global solvability results are obtained by using uniform bounds of the local mild solution and stopping time arguments. Later, we establish a comparison theorem for the solution of SGBH equation having higher order nonlinearities and it plays a crucial role in this work. Then, we discuss the weak differentiability of the solution to SGBH equation in the Malliavin calculus sense. Finally, we obtain the absolute continuity of the law of the solution with respect to the Lebesgue measure on $\mathbb{R}$, and the existence of density with the aid of comparison theorem and weak differentiability of the solution.