A remark for characterizing blowup introduced by Giga and Kohn

TL;DR

The paper sharpens the Liouville-type result underlying the blowup analysis for the semilinear heat equation by showing that nonnegative solutions of $\Delta u - \tfrac{1}{2} x\cdot\nabla u + |u|^{p-1}u - \beta u = 0$ with $\beta = \tfrac{1}{p-1}$ must satisfy $\nabla u \equiv 0$ for $1<p<p_*$, removing the previous $|u|$-boundedness assumption. It introduces a global estimate via a differential identity and integration by parts, and employs Gaussian-weighted energy methods together with Pohozaev-type identities to control the solution without global bounds. The approach is motivated by Gidas–Spruck and Quittner-type Liouville arguments and relies on a careful analysis of a weighted energy and a positivity condition arising from a verified discriminant. The result strengthens the link between nonnegativity and rigidity in the stationary equation, with direct implications for characterizing blowup profiles in the original parabolic problem.

Abstract

Giga and Kohn studied the blowup solutions for the equation $v_{t} - Δv - |v|^{p - 1} v = 0 $ and characterized the asymptotic behavior of $v$ near a singularity. In the proof, they reduced the problem to a Liouville theorem for the equation $Δu - \frac{1}{2} x \cdot \nabla u + |u|^{p - 1} u - βu = 0$ where $β= \frac{1}{p - 1}$ and $|u|$ is bounded. This article is a remark for their work and we will show when $u \geq 0$, the boundedness condition for $|u|$ can be removed.

Theorems & Definitions (13)

• Theorem 1.1: Proposition 2 in MR784476
• Theorem 1.2
• Lemma 2.1
• proof : Proof of Lemma \ref{['Thm1']}
• claim 1
• proof : Proof of Claim \ref{['Delta']}
• Lemma 2.2
• proof : Proof of Lemma \ref{['0927lemma1']}
• Lemma 2.3
• proof : Proof of Lemma \ref{['0927lemma2']}