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Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients
Author(s) -
Kwon Dohyun,
Mészáros Alpár Richárd
Publication year - 2021
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12444
Subject(s) - uniqueness , sobolev space , mathematics , differentiable function , degenerate energy levels , nonlinear system , mathematical analysis , parabolic partial differential equation , diffusion , boundary (topology) , type (biology) , space (punctuation) , partial differential equation , physics , computer science , ecology , quantum mechanics , biology , thermodynamics , operating system
This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in particular on gradient flows in the space of probability measures equipped with the distance arising in the Monge–Kantorovich optimal transport problem. The associated internal energy functionals in general fail to be differentiable, therefore classical results do not apply directly in our setting. We study the combination of both linear and porous medium type diffusions and we show the existence and uniqueness of the solutions in the sense of distributions in suitable Sobolev spaces. Our notion of solution allows us to give a fine characterization of the emerging critical regions, observed previously in numerical experiments. A link to a three phase free boundary problem is also pointed out.