Asymptotic attraction with algebraic rates toward fronts of dispersive-diffusive Burgers equations
Milena Stanislavova, Atanas G. Stefanov
TL;DR
This work addresses the long-time behavior of fronts in dispersive-diffusive Burgers-type equations, focusing on heteroclinic fronts and their attraction under large perturbations. Building on BBHY, the authors bootstrap a spectral-stability framework with new energy estimates, including weighted $L^2$-energies and diffusive semigroup techniques, to obtain explicit algebraic decay rates toward fronts for both the KdV-Burgers and fractional Burgers models. The main contributions are quantitative decay results: for the KdV-Burgers model, $\|v(t)\|_{L^2} \lesssim t^{-1/2}$ and $\|v(t)\|_{L^p}$ decays with precise exponents (including $p>2$ and $1<p<2$ with small $\delta$), while for the fractional Burgers equation, odd fronts satisfy decay like $\|v(t)\|_{L^p} \lesssim (\ln t / t)^{(1-1/p)/2}$ for $1<p<2$ and $\|v(t)\|_{L^q} \lesssim (\ln t / t)^{7/24-1/(12q)}$ for $2<q\le \infty$. These results are derived under explicit spectral conditions and show that convergence to fronts occurs with algebraic (and logarithmic) rates, providing a rigorous quantification of asymptotic attraction in these dispersive-diffusive systems.
Abstract
Burgers equation is a classic model, which arises in numerous applications. At its very core it is a simple conservation law, which serves as a toy model for various dynamics phenomena. In particular, it supports explicit heteroclinic solutions, both fronts and backs. Their stability has been studied in details. There has been substantial interest in considering dispersive and/or diffusive modifications, which present novel dynamical paradigms in such simple setting. More specificaly, the KdV-Burgers model has been showed to support unique fronts (not all of them monotone!) with fixed values at $\pm \infty$. Many articles, among which \cite{Pego}, \cite{NS1}, \cite{NS2}, have studied the question of stability of monotone (or close to monotone) fronts. In a breakthrough paper, \cite{BBHY}, the authors have extended these results in several different directions. They have considered a wider range of models. The fronts do not need to be monotone, but are subject of a spectral condition instead. Most importantly the method allows for large perturbations, as long as the heteroclinic conditions at $\pm \infty$ are met. That is, there is asymptotic attraction to the said fronts or equivalently the limit set consist of one point. The purpose of this paper is to extend the results of \cite{BBHY} by providing explicit algebraic rates of convergence as $t\to \infty$. We bootstrap these results from the results in \cite{BBHY} using additional energy estimates for two important examples namely KdV-Burgers and the fractional Burgers problem. These rates are likely not optimal, but we conjecture that they are algebraic nonetheless.