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

Abstract This study focuses on the Rosenau–Burgers equation u t + u x x x x t − α u x 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.