Large time behavior of solutions to the Cauchy problem for the BBM-Burgers equation

TL;DR

This paper analyzes the long-term behavior of solutions to the BBM-Burgers equation, showing convergence to a nonlinear diffusion wave and constructing second asymptotic profiles based on initial data, including slowly decaying cases.

Contribution

It introduces the first analysis of second asymptotic profiles for solutions with slowly decaying initial data in the BBM-Burgers equation.

Findings

Solutions converge to the nonlinear diffusion wave over time

Constructed second asymptotic profiles depending on initial data

Derived optimal asymptotic rates for convergence

Abstract

We consider the large time behavior of the solutions to the Cauchy problem for the BBM-Burgers equation. We prove that the solution to this problem goes to the self-similar solution to the Burgers equation called the nonlinear diffusion wave. Moreover, we construct the appropriate second asymptotic profiles of the solutions depending on the initial data. Based on that discussion, we investigate the effect of the initial data on the large time behavior of the solution, and derive the optimal asymptotic rate to the nonlinear diffusion wave. Especially, the important point of this study is that the second asymptotic profiles of the solutions with slowly decaying data, whose case has not been studied, are obtained.