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

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. (C) 2022 Elsevier Inc. All rights reserved.

Automated summary

In this study, we investigate the large time behavior of solutions to the Cauchy problem for the BBM-Burgers equation. We show that the solution approaches the self-similar solution to the Burgers equation, known as the nonlinear diffusion wave. Additionally, we construct the appropriate second asymptotic profiles of the solutions based on the initial data. Through this discussion, we analyze the effect of the initial data on the long time behavior of the solution and derive the optimal asymptotic rate to the nonlinear diffusion wave. Moreover, we obtain the second asymptotic profiles of the solutions with slowly decaying data, which has not been studied previously.