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Zbl 0573.35053
Struwe, Michael
A counterexample in regularity theory for parabolic systems. (English)
[J] Czech. Math. J. 34(109), 183-188 (1984). ISSN 0011-4642; ISSN 1572-9141/e

The authors study the degenerate parabolic partial differential equation $$-(\partial /\partial x\sb i)(a\sb{ij}(x,t)(\partial u/\partial x\sb j))+(\partial u/\partial t)=f $$ under suitable assumptions on the coefficients, deriving global properties of the solutions, including the $L\sp 2$ continuity of solutions to the Cauchy-Dirichlet problem. They construct examples to show that, when $f=0$, the equation fails to have local $L\sp{\infty}$ estimates unlike the elliptic case. Other examples are provided to establish that Harnack's inequality cannot hold.
[O.T.Haimo]

MSC 2000:: *35K65 Parabolic equations of degenerate type
35B65 Smoothness of solutions of PDE
35D10 Regularity of generalized solutions of PDE
35K20 Second order parabolic equations, boundary value problems

Keywords: continuity; Cauchy-Dirichlet problem; local $L\sp{\infty }$ estimates; Harnack's inequality

Cited in: Zbl 0625.35047

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