A Liouville theorem for supercritical Fujita equation and its applications

TL;DR

This paper establishes a Liouville theorem for ancient solutions of the supercritical Fujita equation, demonstrating that solutions close to a specific ODE solution are actually ODE solutions, and applies this to analyze singular sets of weak solutions.

Contribution

It proves a Liouville theorem for supercritical Fujita equation solutions and applies it to describe the structure of singular sets in weak solutions.

Findings

Solutions close to the ODE solution are actually ODE solutions.

The singular set at the end time decomposes into a rectifiable part and a smaller Hausdorff dimension part.

The rectifiable part is characterized by tangent functions being constant.

Abstract

We prove a Liouville theorem for ancient solutions to the supercritical Fujita equation \[\partial_tu-\Delta u=|u|^{p-1}u, \quad -\infty <t<0, \quad p>\frac{n+2}{n-2},\] which says if $u$ is close to the ODE solution $u_0(t):=(p-1)^{-\frac{1}{p-1}}(-t)^{-\frac{1}{p-1}}$ at large scales, then it is an ODE solution (i.e. it depends only on $t$). This implies a stability property for ODE blow ups in this problem. As an application of these results, we show that for a suitable weak solution, its singular set at the end time can be decomposed into two parts: one part is relatively open and $(n-1)$-rectifiable, and it is characterized by the property that tangent functions at these points are the two constants $\pm(p-1)^{-\frac{1}{p-1}}$; the other part is relatively closed and its Hausdorff dimension is not larger than $n-\left[2\frac{p+1}{p-1}\right]-1$.