Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable

TL;DR

This paper establishes bounds on the modulus of continuity for solutions to quasilinear parabolic equations in one spatial dimension, linking solution regularity to initial conditions and elapsed time.

Contribution

It characterizes equations where Lipschitz constants of solutions can be controlled by initial oscillation and time, providing new bounds for solution regularity.

Findings

Bound the modulus of continuity in terms of initial data and time

Characterize equations with Lipschitz bounds based on initial oscillation

Provide explicit bounds for solution regularity over time

Abstract

We bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time. In particular we characterize those equations for which the Lipschitz constants of solutions can be bounded in terms of their initial oscillation and elapsed time.