Asymptotic behavior of interface solutions to quasilinear parabolic equations with nonlinear forcing terms

TL;DR

This paper studies the long-term behavior of interface solutions in quasilinear parabolic equations, revealing their slow convergence to equilibrium through linearization and reduced dynamics.

Contribution

It introduces a novel analysis method for the asymptotic behavior of interface solutions, focusing on metastability and slow convergence in nonlinear parabolic equations.

Findings

Interface solutions exhibit exponential slow convergence to equilibrium.

Linearization around approximate steady states simplifies the dynamics.

Reduced one-dimensional motion describes interface convergence.

Abstract

We investigate the asymptotic behavior of solutions for quasilinear parabolic equations in bounded intervals. In particular, we are concerned with a special class of solutions, called interface solutions, which exhibit e metastable behavior, meaning that their convergence towards the asymptotic configuration of the system is exponentially slow. The key of our analysis is a linearization around an approximation of the steady state of the problem, and the reduction of the dynamics to a one-dimensional motion, describing the slow convergence of the interfaces towards the equilibrium.