  In this paper we prove local existence of solutions to the nonlinear heat
equation $u_t = \Delta u +a |u|^\alpha u, \; t\in(0,T),\; x=(x_1,\,\cdots,\,
x_N)\in {\mathbb R}^N,\; a = \pm 1,\; \alpha>0;$ with initial value $u(0)\in
L^1_{\rm{loc}}\left({\mathbb R}^N\setminus\{0\}\right)$, 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 $C>0$ is a constant, $m\in \{1,\; 2,\;
\cdots,\; N\},$ $0<\gamma<N$ and $0<\alpha<2/(\gamma+m).$ This gives a local
existence result with highly singular initial values.
  As an application, for $a=1,$ we establish new blowup criteria for
$0<\alpha\leq 2/(\gamma+m)$, 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.
