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On nonlinear problems of parabolic type with implicit constitutive equations involving flux
Type of publication: Article
Citation:
Publication status: Published
Journal: Math. Models Methods Appl. Sci.
Volume: 31
Number: 10
Year: 2021
Pages: 2039--2090
DOI: 10.1142/S0218202521500457
Abstract: We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first order divergence operator acting on a flux function, which is related to the spatial gradient of the unknown through an additional implicit equation. This setting, broad enough in terms of applications, significantly expands the paradigm of nonlinear parabolic problems. Formulating four conditions concerning the form of the implicit equation, we first show that these conditions describe a maximal monotone $p$-coercive graph. We then establish the global-in-time and large-data existence of (weak) solution and its uniqueness. Towards this goal, we adopt and significantly generalize the Minty method of monotone mappings. A unified theory, containing several novel tools, is developed in a way to be tractable numerically.
Preprint project: NCMM
Preprint year: 2020
Preprint number: 07
Preprint ID: NCMM/2020/07
Keywords: existence, Implicit constitutive theory, nonlinear parabolic systems, uniqueness, weak solutions
Authors Bulíček, Miroslav
Málek, Josef
Maringová, Erika
Added by: [MB]
Total mark: 0
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