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 L1-L∞ 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.
Anisotropic p-Laplacian Evolution of Fast Diffusion Type
Feo, Filomena;Volzone, Bruno
2021-01-01
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 L1-L∞ 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
