Nonlinear diffusion in transparent media

TL;DR

This paper studies a complex nonlinear parabolic equation modeling diffusion in transparent media, establishing existence, uniqueness, and qualitative behaviors like finite speed propagation and discontinuities.

Contribution

It introduces new existence and uniqueness results for entropy solutions of a nonlinear, gradient-dependent diffusion equation with detailed qualitative analysis.

Findings

Proved existence and uniqueness of entropy solutions.

Demonstrated finite speed of propagation and waiting-time phenomena.

Analyzed formation of jump discontinuities in solutions.

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.