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Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients

Kwon, Dohyun and Mészáros, Alpár Richárd (2021) 'Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients.', Journal of the London Mathematical Society., 104 (2). pp. 688-746.


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.

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Publisher statement:© 2021 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.
Date accepted:06 February 2021
Date deposited:07 October 2021
Date of first online publication:22 February 2021
Date first made open access:07 October 2021

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