Asymptotic behaviors for nonlinear dispersive equations with damping or dissipative terms
Séminaire Laurent Schwartz — EDP et applications (2017-2018), Talk no. 6, 11 p.

In this note, we will review our recent work on the asymptotic behaviors of nonlinear Klein-Gordon equation with damping terms and Landau-Lifschitz flows from Eucliedean spaces and hyperbolic spaces. By the method of concentration-compactness attractors, we prove that the global bounded solution will decouple into a finite number of equilibrium points with different shifts from the origin. For the Landau-Lifschitz flow from Euclidean spaces, we prove that the solution with energy below 4π will converge to some constant map in the energy space. While for the Landau-Lifschitz flow from two dimensional spaces, the solution will converge to some harmonic map.

Published online:
DOI: 10.5802/slsedp.120
Li, Ze 1; Zhao, Lifeng 2

1 Institute of Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 10019 China
2 Wu Wen-Tsun Key Laboratory of Mathematics Chinese Academy of Science and Department of Mathematics University of Science and Technology of China Hefei 230026 Anhui China
@article{SLSEDP_2017-2018____A6_0,
     author = {Li, Ze and Zhao, Lifeng},
     title = {Asymptotic behaviors for nonlinear dispersive equations with damping or dissipative terms},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:6},
     pages = {1--11},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2017-2018},
     doi = {10.5802/slsedp.120},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/slsedp.120/}
}
TY  - JOUR
AU  - Li, Ze
AU  - Zhao, Lifeng
TI  - Asymptotic behaviors for nonlinear dispersive equations with damping or dissipative terms
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:6
PY  - 2017-2018
SP  - 1
EP  - 11
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://archive.numdam.org/articles/10.5802/slsedp.120/
DO  - 10.5802/slsedp.120
LA  - en
ID  - SLSEDP_2017-2018____A6_0
ER  - 
%0 Journal Article
%A Li, Ze
%A Zhao, Lifeng
%T Asymptotic behaviors for nonlinear dispersive equations with damping or dissipative terms
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:6
%D 2017-2018
%P 1-11
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U http://archive.numdam.org/articles/10.5802/slsedp.120/
%R 10.5802/slsedp.120
%G en
%F SLSEDP_2017-2018____A6_0
Li, Ze; Zhao, Lifeng. Asymptotic behaviors for nonlinear dispersive equations with damping or dissipative terms. Séminaire Laurent Schwartz — EDP et applications (2017-2018), Talk no. 6, 11 p. doi : 10.5802/slsedp.120. http://archive.numdam.org/articles/10.5802/slsedp.120/

[1] N. Burq, G. Raugel, W. Schlag, Long time dynamics for damped Klein-Gordon equations, Ann. Sci. ENS 50 (2017), 1447-1498. | DOI | MR | Zbl

[2] T. Cazenave, Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Functional Analysis, 60 (1985), 36-55. | DOI | MR | Zbl

[3] R. Cote, On the soliton resolution for equivariant wave maps to the sphere, Comm. Pure. App. Math, 68 (2015), 1946-2004. | DOI | MR | Zbl

[4] R. Cote, C. Kenig, A. Lawrie, W. Schlag, Characterization of large energy solutions of the equivariant wave map problem: II , Amer. J. Math. 137 (2015), no. 1, 209-250. | DOI | MR | Zbl

[5] R. Cote, C. Kenig, A. Lawrie, W. Schlag, Characterization of large energy solutions of the equivariant wave map problem: I, Amer. J. Math. 137 (2015), no.1, 139-207. | DOI | MR | Zbl

[6] R. Cote, C. Kenig, A. Lawrie, W, Schlag, Profiles for the radial focusing 4d energy-critical wave equation, Math. Ann. 365 (2016), no. 1-2, 707-803. | DOI | MR | Zbl

[7] T. Duyckaerts, C. Kenig, F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation, Cambridge Journal of Mathematics 1 (2013), no. 1, 75-144. | DOI | MR | Zbl

[8] T. Duyckaerts, C. Kenig, F. Merle, Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 3, 533-599. | DOI | Zbl

[9] T. Duyckaerts, C. Kenig, F. Merle, Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation: the nonradial case, J. Eur. Math. Soc. (JEMS) 14 (2011), 1389-1454. | Zbl

[10] T., Duyckaerts, C., Kenig, F., Merle, Scattering for radial, bounded solutions of focusing supercritical wave equations, Int. Math. Res. Not. IMRN 2014, no. 1, 224-258. | DOI | MR | Zbl

[11] E. Feireisl, Convergence to an equilibrium for semilinear wave equations on unbounded intervals, Dynam. Syst. Appl. 3 (1994), 423-434. | Zbl

[12] E. Feireisl, Finite energy travelling waves for nonlinear damped wave equations, Quarterly Journal of Applied mathematics LVI(1998), 55-70. | DOI | MR | Zbl

[13] H., Jia, B. Liu, G. Xu, Long time dynamics of defocusing energy critical 3 + 1 dimensional wave equation with potential in the radial case, Comm. Math. Phy., 339(20-15), 353-384. | DOI | MR | Zbl

[14] H. Jia, C. Kenig, Asymptotic decomposition for semilinear wave and equivariant wave map equations, Amer. J. Math. 139 (2017), 1521-1603. | DOI | MR | Zbl

[15] C., Kenig, A., Lawrie, W., Schlag, Relaxation of wave maps exterior to a ball to harmonic maps for all data, Geom. Funct. Anal. (GAFA). 24 (2014), no. 2, 610-647. | DOI | MR | Zbl

[16] C., Kenig, A., Lawrie, B.P., Liu, W., Schlag, Stable soliton resolution for exterior wave maps in all equivariance classes, Advances in Math. 285 (2015), 235-300. | DOI | MR | Zbl

[17] C., Kenig, A., Lawrie, B.P., Liu, W., Schlag, Channels of energy for the linear radial wave equation, Advances in Math. 285 (2015), 877-936. | DOI | MR | Zbl

[18] A. Lawrie, S.J. Oh, S. Shahshahani. Profile decompositions for wave equations on hyperbolic space with applications, Math. Ann., 365(1-2), 707-803, 2016. | DOI | MR | Zbl

[19] A. Lawrie, S.J. Oh, S. Shahshahani. Gap eigenvalues and asymptotic dynamics of geometric wave equations on hyperbolic space, 271(11), 3111-3161, 2016. | DOI | MR | Zbl

[20] A. Lawrie, S.J. Oh, S. Shahshahani. Stability of stationary equivariant wave maps from the hyperbolic plane, , 2014. | arXiv | DOI | MR | Zbl

[21] A. Lawrie, S.J. Oh, S. Shahshahani. The Cauchy problem for wave maps on hyperbolic space in dimensions d4, , 2015. | arXiv | DOI | Zbl

[22] M. Lemm, V. Markovic. Heat flows on hyperbolic spaces, , 2015. | arXiv | DOI | MR | Zbl

[23] Z. Li, L. Zhao, Asymptotic decomposition for nonlinear damped Klein-Gordon equations, , 2015. | arXiv

[24] Z. Li, L. Zhao, Asymptotic behaviors of Landau-Lifshitz flows from 2 to Kähler manifolds, Cal.Var.PDE (2017), 56-96. | Zbl

[25] Z. Li, L. Zhao, Convergence to harmonic maps for the Landau-Lifshitz flows on two dimensional hyperbolic spaces, . | arXiv | Zbl

[26] R. Schoen, S.T. Yau. Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature, Commentarii Mathematici Helvetici, 1976, 39: 333-341. | DOI | MR | Zbl

[27] A. Soffer, Soliton dynamics and scattering, ICM 2006 Vol3, 459-471. | DOI | Zbl

[28] T. Tao, A (concentration-) compact attractor for highdimensional nonlinear Schrödinger equations, Dynamics of Partial Differential Equations, 4 (2007) 1-53. | DOI | Zbl

Cited by Sources: