Forced oscillation of viscous Burgers' equation with a time-periodic force

Taige Wang; Bing-Yu Zhang

doi:10.3934/dcdsb.2020160

Article Contents

2021, Volume 26, Issue 2: 1205-1221. Doi: 10.3934/dcdsb.2020160

This issue Previous Article On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model Next Article Collision-free flocking for a time-delay system

Forced oscillation of viscous Burgers' equation with a time-periodic force

Taige Wang^1, , and
Bing-Yu Zhang^1,

1.
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA

^* Corresponding author: Taige Wang
^* Corresponding author: Taige Wang

Received: November 30, 2019

Early access: May 2020

Published: February 2021

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35K55, 34K13.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[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.