Long time dynamics for damped Klein-Gordon equations

Burq, Nicolas; Raugel, Geneviève; Schlag, Wilhelm

doi:10.24033/asens.2349

Long time dynamics for damped Klein-Gordon equations
[Dynamique en temps grand des solutions de l'équation de Klein-Gordon amortie]

Burq, Nicolas ; Raugel, Geneviève ; Schlag, Wilhelm

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 6, pp. 1447-1498.

Résumé
Abstract

Nous démontrons que toute solution radiale d'énergie finie d'une classe générale d'équations de Klein-Gordon amorties ou bien explose en temps positif fini ou bien converge en temps positif vers une solution stationnaire dans $H^{1} \times L^{2}$ . En particulier, toute solution globale en temps positif est bornée en temps positif. Ce résultat s'applique aux non-linéarités focalisantes, sous-critiques pour l'énergie, ${| u |}^{p - 1} u$ , $1 < p < (d + 2) / (d - 2)$ , comme à toute non-linéarité, sous-critique pour l'énergie, remplissant une condition de signe de type Ambrosetti-Rabinowitz. La preuve fait appel, à la fois, à des techniques propres aux équations non linéaires dispersives et à des arguments de systèmes dynamiques (variétés invariantes dans des espaces de Banach et théorèmes de convergence).

For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^{1} \times L^{2}$ . In particular, any global in positive times solution is bounded in positive times. The result applies to standard energy subcritical focusing nonlinearities ${| u |}^{p - 1} u$ , $1 < p < (d + 2) / (d - 2)$ as well as to any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEs and dynamical systems (invariant manifold theory in Banach spaces and convergence theorems).

DOI : 10.24033/asens.2349

Classification : 35B.., 35B40, 35L05, 35L71, 37L10, 37L50, 37L45.
Keywords: Klein-Gordon equation with dissipation, subcritical focusing nonlinearity, radial solutions, convergence, invariant manifolds, center manifolds, Ambrosetti-Rabinowitz condition, Strichartz estimates.
Mot clés : Équation de Klein-Gordon amortie, non-linéarité sous-critique focalisante, solutions radiales, convergence, variétés invariantes, variétés centrales, condition d'Ambrosetti-Rabinowitz, estimations de Strichartz.

@article{ASENS_2017__50_6_1447_0,
     author = {Burq, Nicolas and Raugel, Genevi\`eve and Schlag, Wilhelm},
     title = {Long time dynamics for damped {Klein-Gordon} equations},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1447--1498},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 50},
     number = {6},
     year = {2017},
     doi = {10.24033/asens.2349},
     mrnumber = {3742197},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2349/}
}

TY  - JOUR
AU  - Burq, Nicolas
AU  - Raugel, Geneviève
AU  - Schlag, Wilhelm
TI  - Long time dynamics for damped Klein-Gordon equations
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2017
SP  - 1447
EP  - 1498
VL  - 50
IS  - 6
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2349/
DO  - 10.24033/asens.2349
LA  - en
ID  - ASENS_2017__50_6_1447_0
ER  -

%0 Journal Article
%A Burq, Nicolas
%A Raugel, Geneviève
%A Schlag, Wilhelm
%T Long time dynamics for damped Klein-Gordon equations
%J Annales scientifiques de l'École Normale Supérieure
%D 2017
%P 1447-1498
%V 50
%N 6
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2349/
%R 10.24033/asens.2349
%G en
%F ASENS_2017__50_6_1447_0

Burq, Nicolas; Raugel, Geneviève; Schlag, Wilhelm. Long time dynamics for damped Klein-Gordon equations. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 6, pp. 1447-1498. doi : 10.24033/asens.2349. http://www.numdam.org/articles/10.24033/asens.2349/

Bibliographie
Cité par

Aulbach, B., Lecture Notes in Math., 1058, Springer, 1984, 142 pages (ISBN: 3-540-13329-1) | DOI | MR | Zbl

Burchard, A.; Deng, B.; Lu, K. Smooth conjugacy of centre manifolds, Proc. Roy. Soc. Edinburgh Sect. A, Volume 120 (1992), pp. 61-77 (ISSN: 0308-2105) | DOI | MR | Zbl

Bates, P. W.; Jones, C. K. R. T., Dynamics reported, Vol. 2 (Dynam. Report. Ser. Dynam. Systems Appl.), Volume 2, Wiley, 1989, pp. 1-38 | DOI | MR | Zbl

Berestycki, H.; Lions, P.-L. Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., Volume 82 (1983), pp. 313-345 (ISSN: 0003-9527) | DOI | MR | Zbl

Berestycki, H.; Lions, P.-L. Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., Volume 82 (1983), pp. 347-375 (ISSN: 0003-9527) | DOI | MR | Zbl

Brunovský, P.; Poláčik, P. On the local structure of $ω$ -limit sets of maps, Z. Angew. Math. Phys., Volume 48 (1997), pp. 976-986 (ISSN: 0044-2275) | DOI | MR | Zbl

Burq, N.; Raugel, G.; Schlag, W. Long time dynamics for damped Klein-Gordon equations II (preprint) | MR

Carr, J., Applied Mathematical Sciences, 35, Springer, 1981, 142 pages (ISBN: 0-387-90577-4) | MR | Zbl

Cazenave, T. Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., Volume 60 (1985), pp. 36-55 (ISSN: 0022-1236) | DOI | MR | Zbl

Cortázar, C.; García-Huidobro, M.; Yarur, C. S. On the uniqueness of sign changing bound state solutions of a semilinear equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 28 (2011), pp. 599-621 (ISSN: 0294-1449) | DOI | Numdam | MR | Zbl

Chow, S. N.; Hale, J. K., Grundl. math. Wiss., 251, Springer, 1982, 515 pages (ISBN: 0-387-90664-9) | MR | Zbl

Chen, X.-Y.; Hale, J. K.; Tan, B. Invariant foliations for $C^{1}$ semigroups in Banach spaces, J. Differential Equations, Volume 139 (1997), pp. 283-318 (ISSN: 0022-0396) | DOI | MR | Zbl

Chen, C. C.; Lin, C. S. Uniqueness of the ground state solutions of $Δ u + f (u) = 0$ in $𝐑^{n}, n \geq 3$ , Comm. Partial Differential Equations, Volume 16 (1991), pp. 1549-1572 (ISSN: 0360-5302) | DOI | MR | Zbl

Chow, S.-N.; Lin, X.-B.; Lu, K. Smooth invariant foliations in infinite-dimensional spaces, J. Differential Equations, Volume 94 (1991), pp. 266-291 (ISSN: 0022-0396) | DOI | MR | Zbl

Coffman, C. V. Uniqueness of the ground state solution for $Δ u - u + u^{3} = 0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., Volume 46 (1972), pp. 81-95 (ISSN: 0003-9527) | DOI | MR | Zbl

Duyckaerts, T.; Kenig, C.; Merle, F. Classification of radial solutions of the focusing, energy-critical wave equation, Camb. J. Math., Volume 1 (2013), pp. 75-144 (ISSN: 2168-0930) | DOI | MR | Zbl

Feireisl, E. Convergence to an equilibrium for semilinear wave equations on unbounded intervals, Dynam. Systems Appl., Volume 3 (1994), pp. 423-434 (ISSN: 1056-2176) | MR | Zbl

Feireisl, E. Finite energy travelling waves for nonlinear damped wave equations, Quart. Appl. Math., Volume 56 (1998), pp. 55-70 (ISSN: 0033-569X) | DOI | MR | Zbl

Henry, D., Lecture Notes in Math., 840, Springer, 1981, 348 pages (ISBN: 3-540-10557-3) | MR | Zbl

Haraux, A.; Jendoubi, M. A. On the convergence of global and bounded solutions of some evolution equations, J. Evol. Equ., Volume 7 (2007), pp. 449-470 (ISSN: 1424-3199) | DOI | MR | Zbl

Haraux, A.; Jendoubi, M. A. Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations, Volume 9 (1999), pp. 95-124 (ISSN: 0944-2669) | DOI | MR | Zbl

Hale, J. K.; Massatt, P., Dynamical systems, II (Gainesville, Fla., 1981), Academic Press, New York, 1982, pp. 85-101 | MR | Zbl

Hirsch, M. W.; Pugh, C. C.; Shub, M., Lecture Notes in Math., 583, Springer, 1977, 149 pages | MR | Zbl

Hale, J. K.; Raugel, G. Convergence in gradient-like systems with applications to PDE, Z. Angew. Math. Phys., Volume 43 (1992), pp. 63-124 (ISSN: 0044-2275) | DOI | MR | Zbl

Ionescu, A. D.; Schlag, W. Agmon-Kato-Kuroda theorems for a large class of perturbations, Duke Math. J., Volume 131 (2006), pp. 397-440 (ISSN: 0012-7094) | DOI | MR | Zbl

Keller, C. Large-time asymptotic behavior of solutions of nonlinear wave equations perturbed from a stationary ground state, Comm. Partial Differential Equations, Volume 8 (1983), pp. 1073-1099 (ISSN: 0360-5302) | DOI | MR | Zbl

Keel, M.; Tao, T. Endpoint Strichartz estimates, Amer. J. Math., Volume 120 (1998), pp. 955-980 http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.5keel.pdf (ISSN: 0002-9327) | DOI | MR | Zbl

Matano, H. Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ., Volume 18 (1978), pp. 221-227 (ISSN: 0023-608X) | DOI | MR | Zbl

McLeod, K. Uniqueness of positive radial solutions of $Δ u + f (u) = 0$ in $𝐑^{n}$ . II, Trans. Amer. Math. Soc., Volume 339 (1993), pp. 495-505 (ISSN: 0002-9947) | DOI | MR | Zbl

Nakanishi, K.; Schlag, W., Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2011, 253 pages (ISBN: 978-3-03719-095-1) | DOI | MR | Zbl

Palis, J. J.; de Melo, W., Springer, 1982, 198 pages (ISBN: 0-387-90668-1) | MR | Zbl

Payne, L. E.; Sattinger, D. H. Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., Volume 22 (1975), pp. 273-303 (ISSN: 0021-2172) | DOI | MR | Zbl

Raugel, G., Dynamical systems (Montecatini Terme, 1994) (Lecture Notes in Math.), Volume 1609, Springer, 1995, pp. 208-315 | DOI | MR | Zbl

Simon, L., Lectures in Mathematics ETH Zürich, Birkhäuser, 1996, 152 pages (ISBN: 3-7643-5397-X) | DOI | MR | Zbl

Strauss, W. A. Existence of solitary waves in higher dimensions, Comm. Math. Phys., Volume 55 (1977), pp. 149-162 http://projecteuclid.org/euclid.cmp/1103900983 (ISSN: 0010-3616) | DOI | MR | Zbl

Zelenjak, T. I. Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differencial $^{'}$ nye Uravnenija, Volume 4 (1968), pp. 34-45 ; English transl. Diff. Equations 4 (1968), 17–22 (ISSN: 0374-0641) | MR

Cité par Sources :