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.


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


Full Text: DOI