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Asymptotic behavior of solutions to the Rosenau-Burgers equation with a periodic initial boundary. (English) Zbl 1123.35057
Summary: 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, numerical simulations are reported to confirm the theoretical results.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35B40 Asymptotic behavior of solutions to PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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