Asymptotic behavior of solutions to the generalized KdV-Burgers equation with slowly decaying data

TL;DR

This paper investigates the long-term behavior of solutions to the generalized KdV-Burgers equation with slowly decaying initial data, establishing the optimal rate at which solutions approach a nonlinear diffusion wave.

Contribution

It derives the optimal asymptotic decay rate of solutions to the generalized KdV-Burgers equation with slowly decaying initial data, highlighting the impact of initial decay rate changes.

Findings

Established the optimal asymptotic rate to the nonlinear diffusion wave.

Analyzed how the decay rate of initial data influences the asymptotic behavior.

Demonstrated the relationship between initial decay and long-term solution convergence.

Abstract

We consider the asymptotic behavior of the global solutions to the initial value problem for the generalized KdV-Burgers equation. It is known that the solution to this problem converges to a self-similar solution to the Burgers equation called a nonlinear diffusion wave. In this paper, we derive the optimal asymptotic rate to the nonlinear diffusion wave when the initial data decays slowly at spatial infinity. In particular, we investigate that how the change of the decay rate of the initial value affects the asymptotic rate to the nonlinear diffusion wave.