A nonlinear heat equation with singular initial data
The authors take up the model problem of a semilinear heat equation with a powertype source on a bounded domain \(u_t-\Delta u=|u|^{p-1}u\) with Dirichlet conditions and an initial value \(u_0\in L_q\). They show that the number \(q^*= N(p-1)/2\) is critical in the sense that for \(q>q^*\) and \(q\geq 1\) (or \(q=q^*>1\)) a unique local classical solution exists, belonging to \(C([0,T],L_q)\); and this solution fulfills additionally \(\lim_{t\to0}t^{n/2q}|u(t)|_\infty=0\). For \(q>q^*\) and \(q\geq p\), the same applies to weak solutions (of the corresponding integral equation). In this context, remarkable results were proven by \textit{F. B. Weißler} [see, e.g., Indiana Univ. Math. J. 29, 79-102 (1980; Zbl 0443.35034) and Proc. Symp. Pure Math. 45/2, 545-551 (1986; Zbl 0631.35049)]. The authors present a lot of open and interesting problems, which are worth to study.
scientific article; zbMATH DE number 927203
scientific article