Anisotropic 𝑝-Laplacian Evolution of Fast Diffusion Type

Abstract

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 L1L^{1}-L∞L^{\infty} 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.

Keywords

• nonlinear parabolic equations
• anisotropic equation
• symmetrization
• fundamental solutions
• asymptotic behaviour
• 35k55
• 35k65
• 35a08
• 35b40