Anisotropic Sobolev embeddings and the speed of propagation for parabolic equations

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.

v3 some typos corrected and theorem 2.6 reformulated. Comments welcome