On blow-up rate for the H\'{e}non parabolic equation with Sobolev supercritical nonlinearity

TL;DR

This paper investigates the blow-up behavior of solutions to a supercritical Hénon parabolic equation, establishing conditions for Type I blow-up and analyzing the blow-up rate and classification of solutions.

Contribution

It constructs blow-up solutions at the origin for the supercritical Hénon equation and analyzes their blow-up rates, including classification results for threshold solutions.

Findings

Solutions can blow up at the origin despite the potential vanishing there.

All blow-ups are of Type I when p is below the Joseph–Lundgren exponent.

A lower bound for the blow-up rate matching Type I is established for certain solutions.

Abstract

We discuss the H\'{e}non parabolic equation $\partial_t u = \Delta u + |x|^\sigma u^p$ in a finite ball in $\mathbb{R}^N$ under the Dirichlet boundary condition, where $N\ge1$, $p>1$, and $\sigma>0$. We assume that the exponent $p$ is supercritical in the Sobolev sense. Since the spatial potential term $|x|^\sigma$ vanishes at the origin, solutions seem less likely to blow up at the origin. We construct a solution that blows up at the origin and also carry out an analysis of blow-up rate of solutions. In particular, if $p$ is less than the Joseph--Lundgren exponent, all blow-ups are shown to be of Type I. The lower bound corresponding to Type I rate is also shown for some particular blow-up solutions. As by products, we present a basic result on classification to threshold solutions for every $p>1+\sigma/N$.