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.

我们考虑粘性Fornberg-Whitham方程的初值问题，该方程是非线性和非局部色散-耗散方程之一。在本文中，我们建立了解的整体存在性，并研究了其渐近行为。我们表明，由于粘度项的耗散效应，该问题的解收敛到了称为非线性扩散波的Burgers方程的自相似解。此外，我们通过构造高阶渐近曲线来分析非线性扩散波的最佳渐近率和解的详细结构。此外，我们研究了非局部色散项如何影响解的渐近行为，并将结果与​​KdV–Burgers方程的结果进行比较。