Global solutions to stochastic wave equations with superlinear coefficients

TL;DR

This paper establishes the existence and uniqueness of solutions to stochastic wave equations in low dimensions with superlinear coefficients involving logarithmic growth, expanding the class of solvable stochastic PDEs.

Contribution

It introduces novel methods for handling superlinear coefficients with logarithmic growth in stochastic wave equations, providing new existence and uniqueness results.

Findings

Proved existence and uniqueness for dimensions 1, 2, 3.

Developed sharp moment estimates for solutions.

Applied results to specific Gaussian noise models.

Abstract

We prove existence and uniqueness of a random field solution $(u(t,x); (t,x)\in [0,T]\times \mathbb{R}^d)$ to a stochastic wave equation in dimensions $d=1,2,3$ with diffusion and drift coefficients of the form $|z| \big( \ln_+(|z|) \big)^a$ for some $a>0$. The proof relies on a sharp analysis of moment estimates of time and space increments of the corresponding stochastic wave equation with globally Lipschitz coefficients. We give examples of spatially correlated Gaussian driving noises where the results apply.