C. Kenig et F. Merle ont montré que les solutions de l’équation des ondes focalisante quintique sur l’espace euclidien de dimension 3 ont un comportement linéaire en-dessous d’un certain seuil d’énergie. Ce comportement linéaire est caractérisé par la finitude de la norme dans les variables espace-temps. Dans cet exposé, je donnerai une estimation précise de cette norme globale pour les solutions dont l’énergie est proche de l’énergie seuil.
Duyckaerts, Thomas 1 ; Merle, Frank 1
@article{SEDP_2008-2009____A12_0,
author = {Duyckaerts, Thomas and Merle, Frank},
title = {D\'eg\'en\'erescence du comportement lin\'eaire pour l{\textquoteright}\'equation des ondes semi-lin\'eaire focalisante critique},
journal = {S\'eminaire Goulaouic-Schwartz},
note = {talk:12},
pages = {1--9},
year = {2008-2009},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
language = {fr},
url = {https://www.numdam.org/item/SEDP_2008-2009____A12_0/}
}
TY - JOUR AU - Duyckaerts, Thomas AU - Merle, Frank TI - Dégénérescence du comportement linéaire pour l’équation des ondes semi-linéaire focalisante critique JO - Séminaire Goulaouic-Schwartz N1 - talk:12 PY - 2008-2009 SP - 1 EP - 9 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://www.numdam.org/item/SEDP_2008-2009____A12_0/ LA - fr ID - SEDP_2008-2009____A12_0 ER -
%0 Journal Article %A Duyckaerts, Thomas %A Merle, Frank %T Dégénérescence du comportement linéaire pour l’équation des ondes semi-linéaire focalisante critique %J Séminaire Goulaouic-Schwartz %Z talk:12 %D 2008-2009 %P 1-9 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U https://www.numdam.org/item/SEDP_2008-2009____A12_0/ %G fr %F SEDP_2008-2009____A12_0
Duyckaerts, Thomas; Merle, Frank. Dégénérescence du comportement linéaire pour l’équation des ondes semi-linéaire focalisante critique. Séminaire Goulaouic-Schwartz (2008-2009), Exposé no. 12, 9 p.. https://www.numdam.org/item/SEDP_2008-2009____A12_0/
[BG99] Hajer Bahouri and Patrick Gérard. High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math., 121(1) :131–175, 1999. | Zbl | MR
[DM08] Thomas Duyckaerts and Frank Merle. Dynamics of threshold solutions for energy-critical wave equation. Int. Math. Res. Pap. IMRP, pages Art ID rpn002, 67, 2008. | Zbl | MR
[DM09] Thomas Duyckaerts and Frank Merle. Scattering norm estimate near the threshold for energy-critical focusing semilinear wave equation. Indiana Univ. Math. J., 58 :1971–2002, 2009.
[Gri90] Manoussos G. Grillakis. Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity. Ann. of Math. (2), 132(3) :485–509, 1990. | Zbl | MR
[GSV92] Jean Ginibre, Avy Soffer, and Giorgio Velo. The global Cauchy problem for the critical nonlinear wave equation. J. Funct. Anal., 110(1) :96–130, 1992. | Zbl | MR
[Kap94] Lev Kapitanski. Global and unique weak solutions of nonlinear wave equations. Math. Res. Lett., 1(2) :211–223, 1994. | Zbl | MR
[KM06] Carlos E. Kenig and Frank Merle. Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math., 166(3) :645–675, 2006. | Zbl | MR
[KST09] Joachim Krieger, Wilhelm Schlag, and Daniel Tataru. Slow blow-up solutions for the critical focusing semilinear wave equation. Duke Math. J., 147(1) :1–53, 2009. | MR
[Nak99] Kenji Nakanishi. Scattering theory for the nonlinear Klein-Gordon equation with Sobolev critical power. Internat. Math. Res. Notices, (1) :31–60, 1999. | Zbl | MR
[Pec84] Hartmut Pecher. Nonlinear small data scattering for the wave and Klein-Gordon equation. Math. Z., 185(2) :261–270, 1984. | Zbl | MR
[Rey90] Olivier Rey. The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent. J. Funct. Anal., 89(1) :1–52, 1990. | Zbl | MR
[SS93] Jalal Shatah and Michael Struwe. Regularity results for nonlinear wave equations. Ann. of Math. (2), 138(3) :503–518, 1993. | Zbl | MR
[SS94] Jalal Shatah and Michael Struwe. Well-posedness in the energy space for semilinear wave equations with critical growth. Internat. Math. Res. Notices, (7) :303ff., approx. 7 pp. (electronic), 1994. | Zbl | MR
[Str88] Michael Struwe. Globally regular solutions to the Klein-Gordon equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 15(3) :495–513 (1989), 1988. | Zbl | MR | Numdam
[Tao06] Terence Tao. Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions. Dyn. Partial Differ. Equ., 3(2) :93–110, 2006. | Zbl | MR
