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

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.

28 pages

References in corpus (4)

• $L^p$-$L^q$ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data
• The Cauchy problem for the nonlinear damped wave equation with slowly decaying data
• Asymptotic behavior of solutions to the generalized KdV-Burgers equation with slowly decaying data
• Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data