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Local space-analyticity of solutions of certain singular parabolic equations. (English) Zbl 0784.35055
Summary: The equation \[ \begin{cases} u_ t-\Delta u^ m=0, & \text{ in } \Omega_ T, \\ u \in C \bigl( 0,T;L^ 2(\Omega) \bigr), & u^ m \in L^ 2 \bigl( 0,T;W_ 0^{1,2}(\Omega) \bigr), \\ u(\cdot,0)=u_ 0 \in L^{1+m}(\Omega), \end{cases} \tag{1} \] is singular at points where \(u=0\). We investigate the behaviour of the solution near these points of singularity, when (2) \((N-2)_ +/(N+2)<m<1\). It is shown that in spite of the singularity of the p.d.e., nonnegative solutions are analytic in the space variables and at least Lipschitz continuous in \(t\). We also establish sharp decay rates near the boundary of their domain of definition and near the extinction time. These results follow from accurate upper and lower bounds on the solutions that can be regarded as some sort of a global Harnack principle. The range in (2) is the best possible for such a Harnack principle to hold.

35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
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