Asymptotic behavior of solutions to the Rosenau-Burgers equation with a periodic initial boundary

This study focuses on the Rosenau-Burgers equation ut + ux x x x t - α ux x + f (u)x = 0 with a periodic initial boundary condition. It is proved that with smooth initial value the global solution uniquely exists. Furthermore, for α > 0, the global solution converges time asymptotically to the average of the initial value in an exponential form, and the convergence rate is optimal; while for α = 0, the unique solution oscillates around the initial average all the time. Finally, the numerical simulations are reported to confirm the theoretical results. © 2006 Elsevier Ltd. All rights reserved.