Existence and uniqueness of global solutions to the stochastic heat equation with super-linear drift on an unbounded spatial domain

TL;DR

This paper establishes the existence and uniqueness of global solutions for a stochastic heat equation with super-linear growth forcing on an unbounded domain, introducing a novel dynamic weighting method for control.

Contribution

It provides the first proof of global solutions under super-linear growth conditions satisfying an Osgood criterion on unbounded domains, with a new dynamic weighting technique.

Findings

Proved existence and uniqueness of solutions under super-linear growth conditions.

Developed a new dynamic weighting method for unbounded spatial solutions.

Extended analysis to equations with complex super-linear forcing terms.

Abstract

We prove the existence and uniqueness of global solutions to the semilinear stochastic heat equation on an unbounded spatial domain with forcing terms that grow superlinearly and satisfy an Osgood condition $\int 1/|f(u)|du = +\infty$ along with additional restrictions. For example, consider the forcing $f(u) = u \log(e^e + |u|)\log(\log(e^e+|u|))$. A new dynamic weighting procedure is introduced to control the solutions, which are unbounded in space.