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

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.