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

Article type: Research Article

Authors: Yoshida, Natsumi^{; *}

Affiliations: Graduate Faculty of Interdisciplinary Research Faculty of Education, University of Yamanashi, Kofu, Yamanashi 400-8510, Japan

Correspondence: [*] Corresponding author. E-mail: 14v00067@gmail.com.

Abstract: In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem with the far field condititon 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.

Keywords: Benjamin–Bona–Mahony–Burgers equation, Korteweg–de Vries–Benjamin–Bona–Mahony–Burgers equation, convex flux, asymptotic behavior, rarefaction wave

DOI: 10.3233/ASY-211747

Journal: Asymptotic Analysis, vol. 130, no. 1-2, pp. 167-185, 2022