A remark for characterizing blowup introduced by Giga and Kohn

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