  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.
