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

This study focuses on the Rosenau–Burgers equation ut+uxxxxt−αuxx+f(u)x=0ut+uxxxxt−αuxx+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>0α>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=0α=0, the unique solution oscillates around the initial average all the time. Finally, the numerical simulations are reported to confirm the theoretical results.