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A nonlinear heat equation with singular initial data. (English) Zbl 0868.35058

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 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.
Reviewer: M.Wiegner (Aachen)

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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