Existence of Lipschitz continuous solutions to the Cauchy–Dirichlet problem for anisotropic parabolic equations

The Cauchy–Dirichlet and the Cauchy problem for the degenerate and singular quasilinear anisotropic parabolic equations are considered. We show that the time derivative ut of a solution u belongs to L∞ under a suitable assumption on the smoothness of the initial data. Moreover, if the domain satisfi...

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Bibliographic Details
Published inJournal of functional analysis Vol. 272; no. 10; pp. 3965 - 3986
Main Authors Tersenov, Alkis S., Tersenov, Aris S.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.05.2017
Subjects
Degenerate parabolic equations
Singular parabolic equations
35K20
Singular parabolic equations
35B45
Degenerate parabolic equations
35K65
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Summary:The Cauchy–Dirichlet and the Cauchy problem for the degenerate and singular quasilinear anisotropic parabolic equations are considered. We show that the time derivative ut of a solution u belongs to L∞ under a suitable assumption on the smoothness of the initial data. Moreover, if the domain satisfies some additional geometric restrictions, then the spatial derivatives uxi belong to L∞ as well. In the singular case we show that the second derivatives uxixj of a solution of the Cauchy problem belong to L2.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2017.02.014