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

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