Global solution for superlinear stochastic heat equation on $\mathbb{R}^d$ under Osgood-type conditions

TL;DR

This paper establishes the existence of global solutions for a stochastic heat equation on ^d with Gaussian noise, under Osgood-type conditions, preventing finite-time blow-up and advancing understanding of SPDE behavior.

Contribution

It provides the first global existence results for SHE with Osgood-type drift conditions, extending previous blow-up analyses and handling more general growth conditions.

Findings

Existence of non-exploding solutions under Osgood conditions

Improved understanding of blow-up phenomena in SPDEs

Extension to equations with growth-related diffusion coefficients

Abstract

We study the \textit{stochastic heat equation} (SHE) on $\R^d$ subject to a centered Gaussian noise that is white in time and colored in space.The drift term is assumed to satisfy an Osgood-type condition and the diffusion coefficient may have certain related growth. We show that there exists random field solution which do not explode in finite time. This complements and improves upon recent results on blow-up of solutions to stochastic partial differential equations.