Global asymptotic stability of the rarefaction waves to the Cauchy problem for the generalized Rosenau–Korteweg–de Vries–Burgers equation

In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem with the far field condition for the generalized Rosenau–Korteweg–de Vries–Burgers equation. 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 Rosenau–Benjamin–Bona–Mahony–Burgers equation with a third-order dispersive term as the former one.

Keywords

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

2010 Mathematics Subject Classification

Primary 35Q35. Secondary 35B40, 35G20, 35G25, 35L65, 35Q53.

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This work is supported in part by Grant-in-Aid for Scientific Research (C) 22K03371, Japan.

Received 27 May 2022

Accepted 13 February 2023

Published 21 July 2023