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Single point blow-up for a general semilinear heat equation. (English) Zbl 0597.35057

This paper is concerned with the behaviour of solutions to the semilinear heat equation which is formally equivalent to the integral equation \[ u(t)=e^{tQ}f+\int^{t}_{0}e^{(t-s)Q} F(u(s))ds, \] where \(Q=\Delta -\lambda I\), \(e^{tQ}=e^{-t\lambda} e^{t\Delta}\) and \(e^{t\Delta}\) is the heat semigroup with homogeneous Dirichlet boundary conditions on \(\Omega\). The main result of this paper is that, under certain conditions, solutions of this equation which blow up in finite time in fact blow up only at a single point.
Reviewer: V.Mustonen

MSC:

35K05 Heat equation
47D03 Groups and semigroups of linear operators
35B65 Smoothness and regularity of solutions to PDEs
45G10 Other nonlinear integral equations
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