Continuation of blowup solutions of nonlinear heat equations in several space dimensions.

*(English)*Zbl 0874.35057Equations of the form

$${u}_{t}={\Delta}\left({u}^{m}\right)\pm {u}^{p},\phantom{\rule{2.em}{0ex}}x\in {\mathbb{R}}^{N},\phantom{\rule{4pt}{0ex}}t>0$$

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.

Reviewer: M.Fila (Bratislava)

##### MSC:

35K65 | Parabolic equations of degenerate type |

35B60 | Continuation of solutions of PDE |

35B40 | Asymptotic behavior of solutions of PDE |

35K55 | Nonlinear parabolic equations |

35K15 | Second order parabolic equations, initial value problems |