Non-uniqueness of mild solutions to supercritical heat equations
Irfan Glogić, Martina Hofmanová, Theresa Lange, Eliseo Luongo
TL;DR
This work demonstrates non-uniqueness of mild $L^q$-solutions for the focusing nonlinear heat equation in the supercritical range $1+\frac{2}{d}<p<p_{JL}$ and for all $1\le q<q_c$, where $q_c=\frac{d(p-1)}{2}$. The authors adapt the Jia–Šverák framework by constructing radial forward self-similar expanders with unstable similarity modes, proving the existence of unstable eigenvalues for $p<p_{JL}$, and using unstable manifolds to produce ancient solutions. They then implement a localization and truncation procedure to convert the singular self-similar data into two distinct mild $L^q$-solutions arising from the same initial datum, thereby establishing non-uniqueness in the unforced, radial setting. The results identify $q_c$ as the sharp threshold for unconditional uniqueness in this regime and illustrate how unstable self-similar structures govern solution multiplicity in nonlinear parabolic equations.
Abstract
We consider 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 $d \geq 3$. 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-Šverák 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-Šverák method to a nonlinear parabolic equation without forcing.