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

This study focuses on the Rosenau-Burgers equation u(t) + u(xxxxt) - alpha u(xx) + 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 alpha > 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 alpha = 0, the unique solution oscillates around the initial average all the time. Finally, the numerical simulations are reported to confirm the theoretical results. (c) 2006 Elsevier Ltd. All rights reserved.