This paper is concerned about the existence of periodic solutions of the viscous Burgers' equation when a forced oscillation is prescribed. We establish the existence theory by contraction mapping in $ H^s[0,1] $ with $ s\ge 0 $. Asymptotical periodicity is obtained as well, and the periodic solution is achieved by selecting a suitable function as initial data to generate a solution and passing time limit to infinity. Moreover, uniqueness and global stability is achieved for this periodic solution.
Citation: |
[1] |
J. Bona, S. Sun and B. Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane, Trans. Amer. Math. Soc., 354 (2002), 427-490.
doi: 10.1090/S0002-9947-01-02885-9.![]() ![]() ![]() |
[2] |
J. Bona, S. Sun and B. Zhang, Forced oscillations of a damped Korteweg-de Vries equation in a quarter plane, Comm. Cont. Math., 5 (2003), 369-400.
doi: 10.1142/S021919970300104X.![]() ![]() ![]() |
[3] |
J. Bona, S. Sun and B. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. PDE, 28 (2003), 1391-1436.
doi: 10.1081/PDE-120024373.![]() ![]() ![]() |
[4] |
H. Brézis, Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc., 8 (1993), 409-426.
doi: 10.1090/S0273-0979-1983-15105-4.![]() ![]() ![]() |
[5] |
H. Brézis and L. Nirenberg, Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math., 31 (1978), 1-30.
doi: 10.1002/cpa.3160310102.![]() ![]() ![]() |
[6] |
S. Chen, C. Hsia, C. Jung and B. Kwon, Asymptotic stability and bifurcation of time-periodic solutions for the viscous Burgers' equation, J. Math. Anal. Appl., 445 (2017), 655-676.
doi: 10.1016/j.jmaa.2016.08.018.![]() ![]() ![]() |
[7] |
W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498.
doi: 10.1002/cpa.3160461102.![]() ![]() ![]() |
[8] |
J. M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite-dimensional dynamical system in the long time, J. Diff. Equs., 74 (1988), 369-390.
doi: 10.1016/0022-0396(88)90010-1.![]() ![]() ![]() |
[9] |
R. Grimshaw, Nonlinear Ordinary Differential Equations, Applied Mathematics and Engineering Science Texts. CRC Press, Boca Raton, FL, 1993.
![]() ![]() |
[10] |
J. B. Keller and L. Ting, Periodic vibrations of systems governed by nonlinear partial differential equations, Comm. Pure Appl. Math., 19 (1966), 371-420.
doi: 10.1002/cpa.3160190404.![]() ![]() ![]() |
[11] |
G. Łukaszewicz, E. E. Ortega-Torres and M. A. Rojas-Medar, Strong periodic solutions for a class of abstract evolution equations, Nonlinear Anal., 54 (2003), 1045-1056.
doi: 10.1016/S0362-546X(03)00125-1.![]() ![]() ![]() |
[12] |
P. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math., 31 (1978), 31-68.
doi: 10.1002/cpa.3160310103.![]() ![]() ![]() |
[13] |
P. Rabinowitz, Periodic solutions of Hamiltonian systems: A survey, SIAM J. Math. Anal., 13 (1982), 343-352.
doi: 10.1137/0513027.![]() ![]() ![]() |
[14] |
G. R. Sell and Y. You, Inertial manifolds: The nonselfadjoint case, J. Diff. Eqns., 96 (1992), 203-255.
doi: 10.1016/0022-0396(92)90152-D.![]() ![]() ![]() |
[15] |
R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Reginal Confeences Series in Applied Math., 66, SIAM, Philadelphia, 1995.
doi: 10.1137/1.9781611970050.![]() ![]() ![]() |
[16] |
M. Usman and B. Zhang, Forced oscillations of a class of nonlinear dispersive wave equations and their stability, Jrl. Syst. Sci. & Comp., 20 (2007), 284-292.
doi: 10.1007/s11424-007-9025-2.![]() ![]() ![]() |
[17] |
M. Usman and B. Zhang, Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability, Disc. Cont. Dyn. Sys., 26 (2010), 1509-1523.
doi: 10.3934/dcds.2010.26.1509.![]() ![]() ![]() |
[18] |
O. Vejvoda, Partial Differential Equations: Time-Periodic Solutions, Martinus Nijhoff Publishers, 1982.
![]() |
[19] |
C. E. Wayne, Periodic solutions of nonlinear partial differential equations, Notices of Amer. Math. Soc., 44 (1997), 895-902.
![]() ![]() |
[20] |
B. Zhang, Forced oscillation of the Korteweg-de Vries-Burgers equation and its stability, Control of Nonlinear Distributed Parameter Systems (College Station, TX, 1999), 337–357, Lecture Notes in Pure and Appl. Math., 218, Dekker, New York, 2001.
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