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A remark on a Harnack inequality for dengerate parabolic equations. (English) Zbl 0588.35013
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 \[ \sum^{n}_{j=1}(\partial /\partial x_ j)(\sum^{n}_{i=1}a_{ij}(x)(\partial u/\partial x_ i)=(\partial /\partial t)(w(x)u),\quad x\in \Omega \subseteq {\mathbb{R}}^ m,\quad t>0. \] Here \(\Omega\) is open and bounded and \(m\geq 3.\)
The main result shows that the solution u(x,t) in \[ D_{x_ 0,t_ 0}(\rho)=\{(x,t):\quad x\in \Omega,\quad 0<t<T,\quad | t-t_ 0| <\rho^ 2,\quad | x-x_ 0| <2\rho \} \] satisfies, for some \(\gamma\) \[ \sup_{D^-_{x_ 0,t_ 0}(\rho)} u(x,t)\leq \gamma \inf_{D^+_{x_ 0,t_ 0}(\rho)} u(x,t), \] for all \(\rho\) such that \(D_{x_ 0,t_ 0}(\rho)\subseteq Q=\{(x,t):\) \(x\in \Omega,0<t<T\}\). Here \[ D^-_{x_ 0,t_ 0}(\rho)=\{(x,t)\in Q:\quad t_ 0-3\rho^ 2/4<t<t_ 0+\rho^ 2/4,\quad | x-x_ 0| <\rho /2\}, \]
\[ D^+_{x_ 0,t_ 0}(\rho)=\{(x,t)\in Q:\quad t_ 0+3\rho^ 2/4<t<t_ 0+\rho^ 2,\quad | x-x_ 0| <\rho /2\}. \]
Reviewer: G.C.Wake

MSC:
35B45 A priori estimates in context of PDEs
35K05 Heat equation
35K65 Degenerate parabolic equations
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