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Excluding blowup at zero points of the potential by means of Liouville-type theorems

Jong-Shenq Guo, Philippe Souplet

TL;DR

This work analyzes blowup behavior for the semilinear heat equation with a spatially dependent nonlinearity $u_t=\Delta u+V(x)f(u)$, showing that blowup cannot occur at zero points of the potential $V$ for monotone-in-time solutions when $f(u)\sim u^p$ with subcritical $p$. It introduces a local version of Merle-Zaag's blowup-ODE behavior via Liouville-type arguments, yielding a local type I estimate and enabling blowup exclusion at zeros of $V$ under mild geometric conditions. The results extend to radial settings and include a detailed examination of weaker nonlinearities $f(u)=u[\log(1+u)]^a$, where blowup becomes global and type I across the domain, underscoring a qualitative difference from power nonlinearities. Overall, the paper combines Liouville-type rigidity, nondegeneracy, and careful rescaling to derive local blowup behavior and exclusion results for variable potentials.

Abstract

We prove a local version of a (global) result of Merle and Zaag about ODE behavior of solutions near blowup points for subcritical nonlinear heat equations. As an application, for the equation $u_t= Δu+V(x)f(u)$, we rule out the possibility of blowup at zero points of the potential $V$ for monotone in time solutions when $f(u)\sim u^p$ for large $u$, both in the Sobolev subcritical case and in the radial case. This solves a problem left open in previous work on the subject. Suitable Liouville-type theorems play a crucial role in the proofs.

Excluding blowup at zero points of the potential by means of Liouville-type theorems

TL;DR

This work analyzes blowup behavior for the semilinear heat equation with a spatially dependent nonlinearity , showing that blowup cannot occur at zero points of the potential for monotone-in-time solutions when with subcritical . It introduces a local version of Merle-Zaag's blowup-ODE behavior via Liouville-type arguments, yielding a local type I estimate and enabling blowup exclusion at zeros of under mild geometric conditions. The results extend to radial settings and include a detailed examination of weaker nonlinearities , where blowup becomes global and type I across the domain, underscoring a qualitative difference from power nonlinearities. Overall, the paper combines Liouville-type rigidity, nondegeneracy, and careful rescaling to derive local blowup behavior and exclusion results for variable potentials.

Abstract

We prove a local version of a (global) result of Merle and Zaag about ODE behavior of solutions near blowup points for subcritical nonlinear heat equations. As an application, for the equation , we rule out the possibility of blowup at zero points of the potential for monotone in time solutions when for large , both in the Sobolev subcritical case and in the radial case. This solves a problem left open in previous work on the subject. Suitable Liouville-type theorems play a crucial role in the proofs.

Paper Structure

This paper contains 5 sections, 11 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. A nondegeneracy result
  3. Proof of Theorem \ref{['th2']}
  4. Proof of Theorem \ref{['th1']}
  5. Blowup at zero points of the potential for weak nonlinearities

Key Result

Theorem 1.1

Assume hyp0-hyp3, with $\Omega$ bounded, $f$ of class $C^2$ and convex. Let $x_0\in\Omega$ be such that $V(x_0)=0$ and (i) Assume that $p<p_S$ and let $u$ be a nonnegative classical solution of e1 such that $u_t\ge 0$. Then $x_0$ is not a blowup point of $u$. (ii) Assertion (i) remains valid for any $p>1$, if we assume in addition that $u$ and $V$ are radially symmetric and $\Omega=B_R$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Proposition 5.1
  • Proposition 5.2
  • ...and 1 more