Optimal waiting time bounds for some flux-saturated diffusion equations

TL;DR

This paper establishes optimal conditions and bounds for the waiting time phenomenon in flux-saturated diffusion equations, advancing understanding of their initial data influence and temporal behavior.

Contribution

It provides the first optimal growth conditions and upper bounds for waiting times in flux-saturated diffusion equations using novel subsolutions and comparison principles.

Findings

Optimal growth condition for initial data discriminates waiting time occurrence.

Derived optimal upper bounds on waiting times.

Introduced new subsolutions and comparison results for flux-saturated equations.

Abstract

We consider the Cauchy problem for two prototypes of flux-saturated diffusion equations. In arbitrary space dimension, we give an optimal condition on the growth of the initial datum which discriminates between occurrence or nonoccurrence of a waiting time phenomenon. We also prove optimal upper bounds on the waiting time. Our argument is based on the introduction of suitable families of subsolutions and on a comparison result for a general class of flux-saturated diffusion equations.