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 $u\ge 0$, the boundedness condition for $|u|$ can be removed.