  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.
