×

zbMATH — the first resource for mathematics

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
Keywords:
extinction
PDF BibTeX XML Cite
Full Text: DOI