# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Chiarenza F. - Serapioni R. , Degenerate Parabolic Equations and Harnack Inequality , to appear in Annali Mat. Pura e Appl. MR 772255 | Zbl 0573.35052 · Zbl 0573.35052  Chiarenza F. - Serapioni R. , Harnack Inequality for Degenerate Parabolic Equations , to appear in Comm. in P.D.E. Zbl 0546.35035 · Zbl 0546.35035  Chiarenza F. - Frasca M. , Boundedness for the Solutions of a Degenerate Parabolic Equation , to appear in Applicable Analysis . MR 757584 | Zbl 0522.35059 · Zbl 0522.35059  Coifman R. - Feferman C. , Weighted Norm Inequalities for Maximal Functions and Singular Integrals , Studia Math. , 51 ( 1974 ), pp. 241 - 250 . Article | MR 358205 | Zbl 0291.44007 · Zbl 0291.44007  Fabes E. - Kenig C. - Serapioni R. , The Local Regularity of Solutions of Degenerate Elliptic Equations , Comm. in P. D.E., 7 ( 1 ) ( 1982 ), pp. 77 - 116 . MR 643158 | Zbl 0498.35042 · Zbl 0498.35042  Ford W. - Waid M. , On Maximum Principle for Some Degenerate Parabolic Operators , J. Math. Anal. Appl. , 40 ( 1972 ), pp. 271 - 277 . MR 320530 | Zbl 0261.35046 · Zbl 0261.35046  Lady O. - Solonnikov V. - Ural’ceva N. , Linear and Quasilinear Equations of Parabolic Type , A.M.S. , Providence ( 1968 ). MR 241822  Moser J. , A Harnack Inequality for Parabolic Differential Equations , Comm. Pure Appl. Math. , 17 ( 1964 ), pp. 101 - 134 . MR 159139 | Zbl 0149.06902 · Zbl 0149.06902  Moser J. , On a Pointwise Estimate for Parabolic Differential Equations , Comm. Pure Appl. Math. , 24 ( 1971 ), pp. 727 - 740 . MR 288405 | Zbl 0227.35016 · Zbl 0227.35016  Waid M. , Second Order Time Degenerate Parabolic Equations , Trans. A.M.S. , 170 ( 1972 ), pp. 31 - 55 . MR 304860 | Zbl 0248.35064 · Zbl 0248.35064  Waid M. , Strong Maximum Principle for Time Degenerate Parabolic Operators , SIAM J. Appl. Math. , 26 ( 1974 ), pp. 196 - 202 . MR 336077 | Zbl 0291.47025 · Zbl 0291.47025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.