On the effect of randomization on supercritical heat equations
Eliseo Luongo
TL;DR
We study the focusing nonlinear heat equation $\partial_t u-\Delta u=|u|^{p-1}u$ in $d\ge 3$ with $p>1+\frac{2}{d}$, where the critical exponent is $q_c=\frac{d(p-1)}{2}$ and deterministic non-uniqueness occurs for $1\le q<q_c$ when $p<p_{JL}$. The authors investigate two randomization routes: additive noise and randomized initial data. They prove that additive noise with the considered regularity does not regularize the problem and preserves non-uniqueness, while randomizing the initial data yields almost surely local well-posedness for a broad class of initial data, via modulation-space framework and probabilistic estimates. These results clarify the limits of regularization by noise for supercritical parabolic PDEs and point toward potential selection principles among non-unique solutions.
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 -Δu = |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.