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Well-posedness results for triply nonlinear degenerate parabolic equations.

hal.structure.identifierLaboratoire de Mathématiques de Besançon (UMR 6623) [LMB]
dc.contributor.authorANDREIANOV, Boris
hal.structure.identifierCentro de Investigación en Ingeniería Matemática [Concepción] [CI²MA]
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorBENDAHMANE, Mostafa
hal.structure.identifierCenter of Mathematics for Applications [Oslo] [CMA]
dc.contributor.authorKARLSEN, Kenneth Hvistendahl
hal.structure.identifierLAME Ouagadougou, Burkina-Faso [LAME]
dc.contributor.authorOUARO, Stanislas
dc.date.accessioned2024-04-04T02:29:07Z
dc.date.available2024-04-04T02:29:07Z
dc.date.issued2009
dc.identifier.issn0022-0396
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/190149
dc.description.abstractEnWe study well-posedness of triply nonlinear degenerate elliptic–parabolic–hyperbolic problems of the kind b(u)t −div a(u,∇φ(u))+ ψ(u) = f , u|t=0 = u0 in a bounded domain with homogeneous Dirichlet boundary conditions. The nonlinearities b,φ and ψ are supposed to be continuous non-decreasing, and the nonlinearity ˜a falls within the Leray–Lions framework. Some restrictions are imposed on the dependence of a(u,∇φ(u)) on u and also on the set where φ degenerates. A model case is a(u,∇φ(u)) = f(b(u),ψ(u),φ(u)) + k(u)a0(∇φ(u)), with a nonlinearity φ which is strictly increasing except on a locally finite number of segments, and the nonlinearity a0 which is of the Leray–Lions kind. We are interested in existence, uniqueness and stability of L∞ entropy solutions. For the parabolic–hyperbolic equation (b = Id), we obtain a general continuous dependence result on data u0, f and nonlinearities b,ψ,φ, a. Similar result is shown for the degenerate elliptic problem, which corresponds to the case of b ≡ 0 and general non-decreasing surjective ψ. Existence, uniqueness and continuous dependence on data u0, f are shown in more generality. For instance, the assumptions [b + ψ](R) = R and the continuity of φ ◦([b + ψ]^{−1}) permit to achieve the well-posedness result for bounded entropy solutions of this triply nonlinear evolution problem.
dc.language.isoen
dc.publisherElsevier
dc.subjectDegenerate hyperbolic–parabolic equation
dc.subjectConservation law
dc.subjectLeray–Lions type operator
dc.subjectNon-Lipschitz flux
dc.subjectEntropy solution
dc.subjectExistence
dc.subjectUniqueness
dc.subjectStability
dc.title.enWell-posedness results for triply nonlinear degenerate parabolic equations.
dc.typeArticle de revue
dc.identifier.doi10.1016/j.jde.2009.03.001
dc.subject.halMathématiques [math]/Equations aux dérivées partielles [math.AP]
dc.identifier.arxiv0810.2326
bordeaux.journalJournal of Differential Equations
bordeaux.page277–302
bordeaux.volume247
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue1
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-00475758
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-00475758v1
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