Asymptotic behavior of solutions toward the constant state to the Cauchy problem for the non-viscous diffusive dispersive conservation law

TL;DR

This paper studies how solutions to a specific non-viscous diffusive dispersive conservation law behave over time, showing they tend toward a constant state as time approaches infinity.

Contribution

It establishes the asymptotic convergence of solutions to the constant state for the non-viscous diffusive dispersive conservation law.

Findings

Solutions tend to the constant state as time increases

Proved asymptotic behavior for the Cauchy problem

Analyzed the influence of initial conditions on long-term behavior

Abstract

In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem for the scalar non-viscous diffusive dispersive conservation laws where the far field states are prescribed. We proved that the solution of the Cauchy problem tends toward the constant state as time goes to infinity.