Non-uniqueness of mild solutions to supercritical heat equations

TL;DR

This paper demonstrates that for supercritical heat equations, uniqueness of solutions fails below a certain integrability threshold, extending the Jia-vere1k method to nonlinear parabolic equations.

Contribution

It proves non-uniqueness of mild solutions for supercritical heat equations below the critical integrability exponent, using a novel adaptation of the Jia-vere1k spectral method.

Findings

Non-uniqueness of solutions when q < d(p-1)/2 for p < p_{JL}

Verification of spectral assumptions for the heat equation case

Extension of Jia-vere1k method to nonlinear parabolic equations

Abstract

We consider 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 $d \geq 3$. It is well-known that if $p$ is large enough then \eqref{Eq:Heat_abstract} is unconditionally locally well-posed in $L^q(\mathbb{R}^d)$ for $q \geq d(p-1)/2$. We prove that this result is optimal in the sense that uniqueness of local solutions fails when $q < d(p-1)/2$ as long as $p < p_{JL}$, where $p_{JL}$ stands for the Joseph-Lundgren exponent. Our proof is based on the method that Jia-\v{S}ver\'ak proposed in \cite{JiaSve15} to show non-uniqueness of Leray solutions to incompressible 3d Navier-Stokes equations. In particular, we rigorously verify for \eqref{Eq:Heat_abstract} the (analogue of the) spectral assumption made in \cite{JiaSve15}. To our knowledge, this is the first rigorous implementation of the Jia-\v{S}ver\'ak method to a nonlinear parabolic equation without forcing.