Anisotropic Sobolev embeddings and the speed of propagation for parabolic equations

TL;DR

This paper investigates how solutions to anisotropic quasilinear parabolic equations expand over time, establishing optimal bounds on the speed of support propagation in different spatial directions based on initial data localization.

Contribution

It introduces bounds on the directional support expansion for anisotropic parabolic equations, highlighting optimality for large times.

Findings

Directional velocity bounds depend on initial data localization.

Expansion rate is proven to be optimal asymptotically.

Results apply to orthotropic anisotropic diffusion scenarios.

Abstract

We consider a quasilinear parabolic Cauchy problem with spatial anisotropy of orthotropic type and study the spatial localization of solutions. Assuming the initial datum is localized with respect to a coordinate having slow diffusion rate, we bound the corresponding directional velocity of the support along the flow. The expansion rate is shown to be optimal for large times.