Jiahong Wu
Well-posedness of a Semilinear Heat Equation
with Weak Initial Data
(32K, Latex)

ABSTRACT.  This paper mainly consists of two parts. In the first part
the initial value problem (IVP)of the semilinear heat equation
$$
\partial_t u- \Delta u =|u|^{k-1}u,
\quad \mbox{on \quad ${\Bbb R}^n\times (0,\infty)$},\quad k\ge 2
$$
$$
u(x,0)=u_0(x),\qquad \qquad x\in{\Bbb R}^n\qquad\qquad\quad
$$
with initial data in $\dot{L}_{r,p}$ are studied. We prove the
well-posedness when
$$
1<p<\infty, \quad
\frac{2}{k(k-1)} <  \frac{n}{p}\le \frac{2}{k-1},\quad\mbox{and}\quad
r=\frac{n}{p}-\frac{2}{k-1}
\quad (\le 0)
$$
and construct non-unique solutions for
$$
1< p< \frac{n(k-1)}{2} <k+1,
\quad\mbox{and}\quad
r< \frac{n}{p} -\frac{2}{k-1}.
$$
In the second part the well-posedness of the above IVP
for $k=2$ with $u_0\in H^s({\Bbb R}^n)$ is proved if
$$
-1< s,\quad \mbox{for $n=1$},\qquad \frac{n}{2}-2<s,\quad
 \mbox{for
$n\ge 2$}.
$$
The well-posedness result in the second part is not covered in the
first part. By taking special values of $r,p,s$ and $u_0$, 
these well-posedness results
reduce to some of those previously obtained by other authors
(\cite{bf},\cite{w1}).