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Zbl 0588.35013
Chiarenza, Filippo; Serapioni, Raul
A remark on a Harnack inequality for dengerate parabolic equations. (English)
[J] Rend. Sem. Mat. Univ. Padova 73, 179-190 (1985). ISSN 0041-8994

This paper uses a carefully constructed sequence of technical lemmas to prove an a priori estimate (a Harnack-type inequality) for positive solutions of the degenerate parabolic equation $$ \sum\sp{n}\sb{j=1}(\partial /\partial x\sb j)(\sum\sp{n}\sb{i=1}a\sb{ij}(x)(\partial u/\partial x\sb i)=(\partial /\partial t)(w(x)u),\quad x\in \Omega \subseteq {\bbfR}\sp m,\quad t>0. $$ Here $\Omega$ is open and bounded and $m\ge 3.$ \par The main result shows that the solution u(x,t) in $$ D\sb{x\sb 0,t\sb 0}(\rho)=\{(x,t):\quad x\in \Omega,\quad 0<t<T,\quad \vert t-t\sb 0\vert <\rho\sp 2,\quad \vert x-x\sb 0\vert <2\rho \} $$ satisfies, for some $\gamma$ $$ \sup\sb{D\sp-\sb{x\sb 0,t\sb 0}(\rho)} u(x,t)\le \gamma \inf\sb{D\sp+\sb{x\sb 0,t\sb 0}(\rho)} u(x,t), $$ for all $\rho$ such that $D\sb{x\sb 0,t\sb 0}(\rho)\subseteq Q=\{(x,t):$ $x\in \Omega,0<t<T\}$. Here $$ D\sp-\sb{x\sb 0,t\sb 0}(\rho)=\{(x,t)\in Q:\quad t\sb 0-3\rho\sp 2/4<t<t\sb 0+\rho\sp 2/4,\quad \vert x-x\sb 0\vert <\rho /2\}, $$ $$ D\sp+\sb{x\sb 0,t\sb 0}(\rho)=\{(x,t)\in Q:\quad t\sb 0+3\rho\sp 2/4<t<t\sb 0+\rho\sp 2,\quad \vert x-x\sb 0\vert <\rho /2\}.$$
[G.C.Wake]

MSC 2000:: *35B45 A priori estimates
35K05 Heat equation
35K65 Parabolic equations of degenerate type

Keywords: heat equation; a priori estimate; positive solutions; degenerate parabolic equation

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