A remark for characterizing blowup introduced by Giga and Kohn

Abstract

Giga and Kohn studied the blowup solutions for the equation $v_{t} - \Delta v - |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 $\Delta u - \frac{1}{2} x \cdot \nabla u + |u|^{p - 1} u - \beta u = 0$ where $\beta = \frac{1}{p - 1}$ 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.