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Harnack estimates and extinction profile for weak solutions of certain singular parabolic equations. (English) Zbl 0772.35006
Summary: We establish an intrisic Harnack estimate for nonnegative weak solutions of the singular equation $u_ t-\Delta u^ m=0,\qquad 0<m<1,\qquad \text{ in } D'(\Omega_ T),$ $u\in C(0,T;L^ 2_{loc}(\Omega)),\qquad u^ m\in L^ 2(0,T;W^{1,2}_{loc}(\Omega)),$ for $$m$$ in the optimal range $$((N- 2)_ +/N,1)$$. Intrinsic means that, due to the singularity, the space- time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself. Consequences are, sharp supestimates on the solutions and decay rates as $$t$$ approaches the extinction time. Analogous results are shown for $$p$$-Laplacian type equations.

##### MSC:
 35B45 A priori estimates in context of PDEs 35K65 Degenerate parabolic equations 35K55 Nonlinear parabolic equations 35K99 Parabolic equations and parabolic systems
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