Higher-order asymptotic profiles of the solutions to the viscous Fornberg-Whitham equation

TL;DR

This paper investigates the long-term behavior of solutions to the viscous Fornberg-Whitham equation, establishing their convergence to a nonlinear diffusion wave and analyzing the influence of nonlocal dispersion on this asymptotic behavior.

Contribution

The paper introduces higher-order asymptotic profiles for solutions, revealing detailed structure and the impact of nonlocal dispersion, extending understanding beyond classical models.

Findings

Solutions converge to nonlinear diffusion wave due to viscosity.

Higher-order asymptotic profiles describe detailed solution structure.

Nonlocal dispersion affects the asymptotic rate and structure.

Abstract

We consider the initial value problem for the viscous Fornberg-Whitham equation which is one of the nonlinear and nonlocal dispersive-dissipative equations. In this paper, we establish the global existence of the solutions and study its asymptotic behavior. We show that the solution to this problem converges to the self-similar solution to the Burgers equation called the nonlinear diffusion wave, due to the dissipation effect by the viscosity term. Moreover, we analyze the optimal asymptotic rate to the nonlinear diffusion wave and the detailed structure of the solution by constructing higher-order asymptotic profiles. Also, we investigate how the nonlocal dispersion term affects the asymptotic behavior of the solutions and compare the results with the ones of the KdV-Burgers equation.