Local space-analyticity of solutions of certain singular parabolic equations. (English) Zbl 0784.35055

DiBenedetto, Emmanuele; Kwong, Y.; Vespri, V.

Indiana Univ. Math. J. 40, No. 2, 741-765 (1991).

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.

MSC:

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

Keywords:

weak solutions; boundary value problem; locally analytic solution; sharp decay rates near the boundary; extinction time; upper and lower bounds; global Harnack principle