On time periodic solutions to the generalized BBM-burgers equation with time-dependent periodic external force
Abstract
In this paper, we consider the generalized BBM-Burgers equation with periodic external force in Rn. Existence and uniqueness of time periodic solutions that have the same period as the external force are established in some suitable function space for the space dimension n≥ 3. Moreover, we also discuss the time asymptotic stability of the time periodic solution. The proof is mainly based on the contraction mapping theorem and continuous argument.
Keyword : generalized BBM-Burgers equation, existence of time periodic solutions, asymptotic stability
How to Cite
Wang, Y. (2020). On time periodic solutions to the generalized BBM-burgers equation with time-dependent periodic external force. Mathematical Modelling and Analysis, 25(2), 184-197. https://doi.org/10.3846/mma.2020.10319
This work is licensed under a Creative Commons Attribution 4.0 International License.
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S.M. Zheng. Nonlinear Evolution Equations. Monographs and Surveys in Pure and Applied Mathematics, 133, Chapan Hall/CRC, 2004.
G. Chen and H. Xue. Global existence of solution of Cauchy problem for nonlinear pseudo-parabolic equation. J. Differ. Equ., 245(10):2705–2722, 2008. https://doi.org/10.1016/j.jde.2008.06.040
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S. Kinami, M. Mei and S. Omata. Convergence to diffusion waves of the solutions for Benjamin-Bona-Mahony-Burgers equations. Appl. Anal., 75(3–4):317–340, 2000. https://doi.org/10.1080/00036810008840852
T.T. Li and Y.M. Chen. Global Classical Solutions for Nonlinear Evolution Equations. Pitman Monographs and Surveys in Pure and Applied Mathematics, 45, New York: Longman Scientific Technical, 1992.
L.A. Medeiros and G. Perla Menzala. Existence and uniqueness for periodic solutions of the Benjamin-Bona-Mahony equations. SIAM J. Math. Anal., 8(5):792– 799, 1977. https://doi.org/10.1137/0508062
M. Mei. Large time behavior of solution for generalized BenjaminBona-Mahony equations. Nonlinear Anal., 33(7):699–714, 1998. https://doi.org/10.1016/S0362-546X(97)00674-3
M. Mei. Lq decay rates of solutions for Benjamin-Bona-Mahony equations. J. Differ. Equ., 158(2):314–340, 1999. https://doi.org/10.1006/jdeq.1999.3638
S. Wang and H. Xu. On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term. J. Differ. Equ., 252(7):4243– 4258, 2012. https://doi.org/10.1016/j.jde.2011.12.016
Y.X. Wang. On the Cauchy problem for one dimension generalized Boussinesq equation. Int. J. Math., 26(03), 2015. https://doi.org/10.1142/S0129167X15500238
Y.X. Wang and G.H. Feng. Large-time behavior of solutions to the Rosenau equation with damped term. Math. Meth. Appl. Sci., 40(6):1986–2004, 2017. https://doi.org/10.1002/mma.4114
Y.Z. Wang and S. Chen. Asymptotic profile of solutions to the double dispersion equation. Nonlinear Anal., 134(March 2016):236–254, 2016. https://doi.org/10.1016/j.na.2016.01.009
Y.Z. Wang and K.Y. Wang. Large time behavior of solutions to the nonlinear pseudo-parabolic equation. J. Math. Anal. Appl., 417(1):272–292, 2014. https://doi.org/10.1016/j.jmaa.2014.03.030
Y.Z. Wang and H.J. Zhao. Pointwise estimates of global small solutions to the generalized double dispersion equation. J. Math. Anal. Appl., 448(1):672–690, 2017. https://doi.org/10.1016/j.jmaa.2016.11.030
L. Zhang. Decay of solutions of generalized Benjamin-Bona-Mahony-Burger equations in n-space dimensions. Nonlinear Anal., 25(12):1343–1369, 1995. https://doi.org/10.1016/0362-546X(94)00252-D
H.J. Zhao. Optimal temporal decay estimates for the solution to the multidimensional generalized BBM-Burgers equations with dissipative term. Appl. Anal., 75(1–2):85–105, 2000. https://doi.org/10.1080/00036810008840837
H.J. Zhao and R.A. Adams. Existence and convergence of solutions for the generalized BBM-Burgers equations with dissipative term ii: the multidimensional case. Appl. Anal., 75(1–2):107–135, 2010. https://doi.org/10.1080/00036810008840838
H.J. Zhao and B.J. Xuan. Existence and convergence of solutions for the generalized BBM-Burgers equations with dissipative term. Nonlinear Anal.: TMA, 28(11):1835–1849, 1997. https://doi.org/10.1016/S0362-546X(95)00237-P
S.M. Zheng. Nonlinear Evolution Equations. Monographs and Surveys in Pure and Applied Mathematics, 133, Chapan Hall/CRC, 2004.