Asymptotic profiles of solutions for the generalized Fornberg-Whitham equation with dissipation

TL;DR

This paper analyzes the long-term behavior of solutions to a nonlinear, nonlocal dispersive-dissipative equation, showing convergence to a heat kernel and detailing how dispersion, dissipation, and nonlinearity influence asymptotics.

Contribution

It provides the first asymptotic profiles for solutions of the generalized Fornberg-Whitham equation with dissipation, including second-term asymptotics based on nonlinearity degree.

Findings

Solutions converge to a modified heat kernel over time.

Second asymptotic profiles depend on the nonlinearity degree.

Dispersion, dissipation, and nonlinearity significantly affect asymptotic behavior.

Abstract

We consider the Cauchy problem for the generalized Fornberg-Whitham equation with dissipation. This is one of the nonlinear, nonlocal and dispersive-dissipative equations. The main topic of this paper is an asymptotic analysis for the solutions to this problem. We prove that the solution to this problem converges to the modified heat kernel. Moreover, we construct the second term of asymptotics for the solutions depending on the degree of the nonlinearity. In view of those second asymptotic profiles, we investigate the effects of the dispersion, dissipation and nonlinear terms on the asymptotic behavior of the solutions.