Non existence of global solutions of parabolic equation in conical domains
Toshihiko Hamada · Tsukuba Journal of Mathematics · 1995
D\times(0, T)$ , (P)where $\sigma\geqq 0,$ $p>1,$ $u_{0}\geqq 0,$ $\langle x\rangle^{\sigma/(p\leftrightarrow 1)}u_{0}(\langle x\rangle=\sqrt{1+|x|^{2})}$ is continuous and bounded in $\overline{D}$ and $u_{0}=0$ on $\partial D$ .When $D=R^{N}$ and $\sigma=0$ , Fujita [1] and Weissler [2] proved that if $ 1 0$ and $p=1+(2+\sigma)/(N+\gamma_{+})$ are valid.Moreover we can prove that when $D=R^{N}$ there is no nontrivial global solution if $1+\sigma/(N-2)\leqq p\leqq 1+$ $(2+\sigma)/N$ and $\sigma>0$ .DEFINITION 1.1.For $T>0,$ $u=u(x, t)$ is called a solution of (P) in $(0, T)$ , if (A) $u$ is continuous in $\overline{D}\times[0, T$ ), (B) $u_{l},$ $u_{x_{i}}$ and $u_{x_{i^{x}j}}(i, j=1, \cdots, N)$ are continuous in $D\times(O, T)$ , (C) $\Vert u(t)\Vert_{\sigma/(p-1)}$ is finite for each $t\in[0, T$ ), (D) $u$ satisfies (P), where $\Vert u(t)\Vert_{\sigma/(p-1)}$ $:=\sup_{D}\langle x\rangle^{\sigma/(p-1)}|u(u, t)|$ .