Stability properties of solutions to convection-reaction equations with nonlinear diffusion

TL;DR

This paper analyzes the stability and long-term behavior of solutions to a nonlinear convection-reaction-diffusion equation, revealing conditions for stationary solutions, their stability, and the occurrence of metastable states through numerical simulations.

Contribution

It provides new insights into the existence, stability, and metastable behavior of solutions to nonlinear convection-reaction equations with diffusion, including numerical validation.

Findings

Existence of stationary solutions with at most one zero inside the interval.

Stability and instability conditions depending on the viscosity coefficient.

Numerical evidence of metastable behavior with exponentially long transition phases.

Abstract

In this paper we study a convection-reaction-diffusion equation of the form \begin{equation*} u_t=\varepsilon(h(u)u_x)_x-f(u)_x+f'(u), \quad t>0, \end{equation*} with a nonlinear diffusion in a bounded interval of the real line. In particular, we first focus our attention on the existence of stationary solutions with at most one zero inside the interval, studying their behavior with respect to the viscosity coefficient $\varepsilon>0$ and their stability/instability properties. Then, we investigate the large time behavior of the solutions for finite times and the asymptotic regime. We also show numerically that, for a particular class of initial data, the so-called metastable behavior occurs, meaning that the time-dependent solution persists for an exponentially long (with respect to $\varepsilon$) time in a transition non-stable phase, before converging to a stable configuration.