Traveling phase interfaces in viscous forward-backward diffusion equations

TL;DR

This paper investigates traveling wave solutions in viscous regularizations of ill-posed bistable diffusion equations, revealing detailed phase boundary structures and analyzing regimes with different interface behaviors.

Contribution

It characterizes all monotone traveling wave solutions with a single interface in a specific viscous regularization model, advancing understanding of phase boundary dynamics.

Findings

Characterized all monotone traveling wave solutions connecting phases.

Analyzed sharp-interface regimes with vanishing viscosity and bilinear limits.

Provided insights into the multiscale evolution of phase boundaries.

Abstract

The viscous regularization of an ill-posed diffusion equation with bistable nonlinearity predicts a hysteretic behavior of dynamical phase transitions but a complete mathematical \mbox{understanding} of the intricate multiscale evolution is still missing. We shed light on the fine structure of \mbox{propagating} phase boundaries by carefully examining traveling wave solutions in a special case. Assuming a trilinear constitutive relation we characterize all waves that possess a monotone \mbox{profile} and connect the two phases by a single interface of positive width. We further study the two sharp-interface regimes related to either vanishing viscosity or the bilinear limit.