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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