M. Bulicek, J. Malek, E. Maringova:

"On nonlinear problems of parabolic type with implicit constitutive equations involving flux";

in: "ASC Report 36/2020", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2020, ISBN: 978-3-902627-13-1, 1 - 38.

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.

nonlinear parabolic systems, implicit constitutive theory, weak solutions, existence, uniqueness.

http://www.asc.tuwien.ac.at/preprint/2020/asc36x2020.pdf

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