On the effect of randomization on supercritical heat equations

TL;DR

This paper explores how different types of randomization, including forcing noise and initial condition randomization, influence the well-posedness and uniqueness of solutions to supercritical heat equations.

Contribution

It demonstrates that additive noise does not improve solution uniqueness, while initial condition randomization can affect well-posedness in supercritical heat equations.

Findings

Additive white-in-time, colored-in-space forcing does not ensure uniqueness.

Non-uniqueness persists with additive noise in local solutions.

Randomizing initial conditions impacts the well-posedness of the equation.

Abstract

Recently, in \cite{glogic2025non}, it has been shown that the focusing power nonlinearity heat equation \begin{equation}\label{Eq:Heat_abstract}\tag{NLH} \partial_t u -\Delta u = |u|^{p-1}u, \quad p>1, \end{equation} in dimensions $d \geq 3$ has non-unique local solutions in $L^q(\mathbb{R}^d)$ for $q < d(p-1)/2$ provided that $p < p_{JL}$, where $p_{JL}$ denotes the Joseph-Lundgren exponent. In this paper we investigate the effect of different randomizations on the well-posedness of the equation. First we show that adding a forcing term white in time and colored in space in \eqref{Eq:Heat_abstract} is not sufficient to improve the solution theory: namely, we prove non-uniqueness for local-in-time mild solutions of \eqref{Eq:Heat_abstract} with additive noise. Second, we discuss how randomizing the initial conditions of \eqref{Eq:Heat_abstract} affects its well-posedness.