The nonlinear heat equation involving highly singular initial values and new blowup and life span results

In this paper we prove local existence of solutions to the nonlinear heat equation $u_t = \Delta u +a |u|^\alpha u, ; t\in(0,T),; x=(x_1,,\cdots,, x_N)\in {\mathbb R}^N,; a = \pm 1,; \alpha>0;$withinitialvalue$u(0)\in L^1_{\rm{loc}}\left({\mathbb R}^N\setminus{0}\right)$, anti-symmetric with respect to $x_1,x_2,\cdots ,x_m$ and $|u(0)|\leq C(-1)^m\partial_{1}\partial_{2}\cdot \cdot \cdot \partial_{m}(|x|^{-\gamma})$for$x_1>0,; \cdots,; x_m>0,$where$C>0$isaconstant,$m\in {1,; 2,; \cdots,; N},$ $0<\gamma <N$ and $0<\alpha <2\mathrm{/}(\gamma +m)\mathrm{.}$ This gives a local existence result with highly singular initial values. As an application, for $a=1,$ we establish new blowup criteria for $0<\alpha \le 2\mathrm{/}(\gamma +m)$, including the case $m=0.$ Moreover, if $(N-4)\alpha <2,$ we prove the existence of initial values $u_0=\lambda f,$ for which the resulting solution blows up in finite time $T_{\max }(\lambda f),$ if $\lambda >0$ is sufficiently small. We also construct blowing up solutions with initial data ${\lambda }_{n}f$ such that $\lambda_n^{[({1\over \alpha}-{\gamma+m\over 2})^{-1}]}T_{\max}(\lambda_n f)$ has different finite limits along different sequences ${\lambda }_n\to 0$. Our result extends the known “small lambda” blow up results for new values of $\alpha$ and a new class of initial data.