Calculus of Variations and Geometric Measure Theory
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L. Giacomelli - S. Moll - F. Petitta

Nonlinear diffusion in transparent media

created by giacomelli on 04 Feb 2021
modified by petitta on 23 Sep 2021

[BibTeX]

Accepted Paper

Inserted: 4 feb 2021
Last Updated: 23 sep 2021

Journal: International Mathematics Research Notices IMRN
Year: 2021

Abstract:

We consider a prototypical nonlinear parabolic equation whose flux has three distinguished features: it is nonlinear with respect to both the unknown and its gradient, it is homogeneous, and it depends only on the direction of the gradient. For such equation, we obtain existence and uniqueness of entropy solutions to the Dirichlet problem, the homogeneous Neumann problem, and the Cauchy problem. Qualitative properties of solutions, such as finite speed of propagation and the occurrence of waiting-time phenomena, with sharp bounds, are shown. We also discuss the formation of jump discontinuities both at the boundary of the solutions' support and in the bulk.


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