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

Equations of the form $u_t=\Delta (u^m)\pm u^p,\qquad x\in\mathbb{R}^N,\;t>0$ 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:

 35K65 Degenerate parabolic equations 35B60 Continuation and prolongation of solutions to PDEs 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations 35K15 Initial value problems for second-order parabolic equations

extinction
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