On blow-up rate for the Hénon parabolic equation with Sobolev supercritical nonlinearity

TL;DR

This work analyzes the Hénon parabolic equation $u_t=\Delta u+|x|^{\sigma}u^p$ in a ball with Dirichlet boundary, focusing on Sobolev supercritical nonlinearity and the possibility of blow-up at the origin. The authors construct a radial threshold solution that blows up in finite time at the origin and develop a comprehensive blow-up-rate theory, proving that for $p_S(\sigma)<p<p_{JL}(\sigma)$ all blow-ups are Type I and obtaining matching upper and lower bounds, including for the weighted norm $|x|^{\sigma/(p-1)}u$. They also establish a threshold-solution framework for all $p>1+\sigma/N$ and classify possible asymptotic behaviors in subcritical and critical cases, showing that blow-up at the origin is inevitable in many regimes and that the rate is sharp. The results illuminate how the spatially varying weight $|x|^{\sigma}$ interacts with Sobolev-critical nonlinearities to shape blow-up dynamics, and they provide tools (singularity/decay estimates, weighted norms, zero-number methods) that may extend to other Hénon-type parabolic problems. The findings have implications for understanding singularity formation in weighted semilinear heat equations in bounded domains.

Abstract

We discuss the Hénon parabolic equation $\partial_t u = Δu + |x|^σu^p$ in a finite ball in $\mathbb{R}^N$ under the Dirichlet boundary condition, where $N\ge1$, $p>1$, and $σ>0$. We assume that the exponent $p$ is supercritical in the Sobolev sense. Since the spatial potential term $|x|^σ$ 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 $p>1+σ/N$.

Theorems & Definitions (23)

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• proof : Proof of Proposition \ref{['Proposition:SDE']}
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• proof : Proof of Theorem \ref{['Theorem:A']}
• proof : Proof of Theorem \ref{['Theorem:B']}
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