Asymptotic behavior of solutions toward the rarefaction waves to the Cauchy problem for the generalized Benjamin-Bona-Mahony-Burgers equation with dissipative term

TL;DR

This paper analyzes the long-term behavior of solutions to a generalized Benjamin-Bona-Mahony-Burgers equation with dissipation, proving solutions tend to rarefaction waves over time and establishing their stability.

Contribution

It demonstrates the asymptotic stability of rarefaction waves for the generalized Benjamin-Bona-Mahony-Burgers equation with dissipative terms, extending results to related KdV-Burgers equations.

Findings

Solutions tend toward rarefaction waves as time approaches infinity.

Proves global asymptotic stability of rarefaction waves.

Extends stability results to generalized KdV-Burgers equations.

Abstract

In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem with the far field condisiton for the generalized Benjamin-Bona-Mahony-Burgers equation with a fourth-order dissipative term. When the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single rarefaction wave, it is proved that the solution of the Cauchy problem tends toward the rarefaction wave as time goes to infinity. We can further obtain the same global asymptotic stability of the rarefaction wave to the generalized Korteweg-de Vries-Benjamin-Bona-Mahony-Burgers equation with a fourth-order dissipative term as the former one.