LONG TIME BEHAVIOUR OF THE SOLUTIONS OF NONLINEAR WAVE EQUATION

Jianjun Liu; Duohui Xiang

doi:10.11948/20230401

2024 Volume 14 Issue 5

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Jianjun Liu, Duohui Xiang. LONG TIME BEHAVIOUR OF THE SOLUTIONS OF NONLINEAR WAVE EQUATION[J]. Journal of Applied Analysis & Computation, 2024, 14(5): 2777-2787. doi: 10.11948/20230401

Citation:

Jianjun Liu, Duohui Xiang. LONG TIME BEHAVIOUR OF THE SOLUTIONS OF NONLINEAR WAVE EQUATION[J]. Journal of Applied Analysis & Computation, 2024, 14(5): 2777-2787. doi: 10.11948/20230401

LONG TIME BEHAVIOUR OF THE SOLUTIONS OF NONLINEAR WAVE EQUATION

Jianjun Liu^1,,
Duohui Xiang^1, ,

1.
College of Mathematics, Sichuan University, Chengdu 610065, Sichuan, China

Author Bio: Email: jianjun.liu@scu.edu.cn(J. Liu)
Corresponding author: Email: duohui.xiang@outlook.com(D. Xiang)
Fund Project: The author was supported by National Natural Science Foundation of China (Grant Nos. 11971299, 12090010, 12090013)

Abstract

Abstract

In this paper, we consider the nonlinear wave equation
$ u_{tt}-\Delta u+mu+f(x,u)=0,\ x\in \mathbb{T}^{d}:=( \mathbb{R}/2\pi \mathbb{Z})^{d}, $
where $ m>0 $ and $ f $ is an analytic function of order at least two in $ u $. The long time behaviour of its solutions is proved by Birkhoff normal form.
- Long time behaviour /
- nonlinear wave equation /
- Birkhoff normal form
MSC: 37K45, 37K55

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References

[1]	D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys., 2003, 234(2), 253–285. doi: 10.1007/s00220-002-0774-4 CrossRef Google Scholar
[2]	D. Bambusi, A Birkhoff normal form theorem for some semilinear PDEs, in Hamiltonian dynamical systems and applications, NATO Sci. Peace Secur. Ser. B Phys. Biophys., Springer, Dordrecht, 2008, 213–247. Google Scholar
[3]	D. Bambusi, J. M. Delort, B. Grébert and J. Szeftel, Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Comm. Pure Appl. Math., 2007, 60(11), 1665–1690. doi: 10.1002/cpa.20181 CrossRef Google Scholar
[4]	D. Bambusi and B. Grébert, Birkhoff normal form for partial differential equations with tame modulus, Duke Math. J., 2006, 135(3), 507–567. Google Scholar
[5]	J. Bernier, E. Faou and B. Grébert, Long time behavior of the solutions of NLW on the $d$-dimensional torus, Forum Math. Sigma, 2020, 8, 12–26. doi: 10.1017/fms.2020.8 CrossRef $d$-dimensional torus" target="_blank">Google Scholar
[6]	J. Bernier, E. Faou and B. Grébert, Rational normal forms and stability of small solutions to nonlinear Schrödinger equations, Ann. PDE, 2020, 6(2), 14. doi: 10.1007/s40818-020-00089-5 CrossRef Google Scholar
[7]	J. Bernier and B. Grébert, Long time dynamics for generalized Korteweg-de Vries and Benjamin-Ono equations, Arch. Ration. Mech. Anal., 2021, 241(3), 1139–1241. doi: 10.1007/s00205-021-01666-z CrossRef Google Scholar
[8]	M. Berti and J. M. Delort, Almost global solutions of capillary-gravity water waves equations on the circle, volume 24 of Lecture Notes of the Unione Matematica Italiana, Springer, Cham, Unione Matematica Italiana, [Bologna], 2018. Google Scholar
[9]	L. Biasco, J. E. Massetti and M. Procesi, An abstract Birkhoff normal form theorem and exponential type stability of the 1d NLS, Comm. Math. Phys., 2020, 375(3), 2089–2153. doi: 10.1007/s00220-019-03618-x CrossRef Google Scholar
[10]	J. Bourgain, Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Geom. Funct. Anal., 1996, 6(2), 201–230. doi: 10.1007/BF02247885 CrossRef Google Scholar
[11]	J. Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDE, J. Anal. Math., 2000, 80, 1–35. doi: 10.1007/BF02791532 CrossRef Google Scholar
[12]	Q. Chen, H. Cong, L. Meng and X. Wu, Long time stability result for 1-dimensional nonlinear Schrödinger equation, J. Differential Equations, 2022, 315, 90–121. doi: 10.1016/j.jde.2022.01.032 CrossRef Google Scholar
[13]	H. Cong, C. Liu and P. Wang, A Nekhoroshev type theorem for the nonlinear wave equation, J. Differential Equations, 2020, 269(4), 3853–3889. doi: 10.1016/j.jde.2020.03.015 CrossRef Google Scholar
[14]	H. Cong, J. Liu and X. Yuan, Stability of KAM tori for nonlinear Schrödinger equation, Mem. Amer. Math. Soc., 2016, 239(1134), vii+85. Google Scholar
[15]	H. Cong, L. Mi and P. Wang, A Nekhoroshev type theorem for the derivative nonlinear Schrödinger equation, J. Differential Equations, 2020, 268(9), 5207–5256. doi: 10.1016/j.jde.2019.11.005 CrossRef Google Scholar
[16]	H. Cong, L. Mi, X. Wu and Q. Zhang, Exponential stability estimate for the derivative nonlinear Schrödinger equation, Nonlinearity, 2022, 35(5), 2385–2423. doi: 10.1088/1361-6544/ac5c66 CrossRef Google Scholar
[17]	J. M. Delort, On long time existence for small solutions of semi-linear Klein-Gordon equations on the torus, J. Anal. Math., 2009, 107, 161–194. doi: 10.1007/s11854-009-0007-2 CrossRef Google Scholar
[18]	J. M. Delort, Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres, Mem. Amer. Math. Soc., 2015, 234(1103), vi+80. Google Scholar
[19]	E. Faou and B. Grébert, A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus, Anal. PDE, 2013, 6(6), 1243–1262. doi: 10.2140/apde.2013.6.1243 CrossRef Google Scholar
[20]	B. Grébert, R. Imekraz and É. Paturel, Normal forms for semilinear quantum harmonic oscillators, Comm. Math. Phys., 2009, 291(3), 763–798. doi: 10.1007/s00220-009-0800-x CrossRef Google Scholar
[21]	X. Yuan and J. Zhang, Long time stability of Hamiltonian partial differential equations, SIAM J. Math. Anal., 2014, 46(5), 3176–3222. doi: 10.1137/120900976 CrossRef Google Scholar
[22]	J. Zhang, Almost global solutions to Hamiltonian derivative nonlinear Schrödinger equations on the circle, J. Dynam. Differential Equations, 2020, 32(3), 1401–1455. doi: 10.1007/s10884-019-09773-y CrossRef Google Scholar
[23]	Q. Zhang, Long-time existence for semi-linear Klein-Gordon equations with quadratic potential, Comm. Partial Differential Equations, 2010, 35(4), 630–668. doi: 10.1080/03605300903509112 CrossRef Google Scholar