96-567 Jiahong Wu

Well-posedness of a Semilinear Heat Equation with Weak Initial Data (32K, Latex) Nov 13, 96

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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}).

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