Anisotropic p-Laplacian Evolution of Fast Diffusion Type

We study an anisotropic, possibly non-homogeneous version of the evolution p-Laplacian equation when fast diffusion holds in all directions. We develop the basic theory and prove symmetrization results from which we derive sharp estimates. We prove the existence of a self-similar fundamental solutio...

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Bibliographic Details
Published in:Advanced nonlinear studies 2021-08, Vol.21 (3), p.523-555
Main Authors: Feo, Filomena, Vázquez, Juan Luis, Volzone, Bruno
Format: Article
Language:English
Subjects:
35A08
35B40
35K55
35K65
Anisotropic Equation
Asymptotic Behaviour
Fundamental Solutions
Nonlinear Parabolic Equations
Symmetrization
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Summary:We study an anisotropic, possibly non-homogeneous version of the evolution p-Laplacian equation when fast diffusion holds in all directions. We develop the basic theory and prove symmetrization results from which we derive sharp estimates. We prove the existence of a self-similar fundamental solution of this equation in the appropriate exponent range, and uniqueness in a smaller range. We also obtain the asymptotic behaviour of finite mass solutions in terms of the self-similar solution. Positivity, decay rates as well as other properties of the solutions are derived. The combination of self-similarity and anisotropy is not common in the related literature. It is however essential in our analysis and creates mathematical difficulties that are solved for fast diffusions.
ISSN:1536-1365
2169-0375
DOI:10.1515/ans-2021-2136