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


Submitted Paper

Inserted: 4 feb 2021
Last Updated: 4 feb 2021

Year: 2021


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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