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Continuation of blowup solutions of nonlinear heat equations in several space dimensions. (English) Zbl 0874.35057

Equations of the form

u t =Δ(u m )±u p ,x N ,t>0

are studied. For the positive sign and p>1, the solutions may blow up in finite time. For the negative sign and p<1, extinction may occur in the sense that initially positive solutions vanish at some point in finite time. The possible continuation of solutions after the appearence of singularities is investigated. A classification is obtained in terms of the exponents m>0 and p. Some questions that had been open for a long time are answered here. It is obvious that the methods used in the paper have wider applicability.


MSC:
35K65Parabolic equations of degenerate type
35B60Continuation of solutions of PDE
35B40Asymptotic behavior of solutions of PDE
35K55Nonlinear parabolic equations
35K15Second order parabolic equations, initial value problems
Keywords:
extinction