Scattering for the two-dimensional energy-critical wave equation

Scattering for the two-dimensional energy-critical wave equation

Slim Ibrahim, Mohamed Majdoub, Nader Masmoudi, and Kenji Nakanishi

Source: Duke Math. J. Volume 150, Number 2 (2009), 287-329.

Abstract

We investigate existence and asymptotic completeness of the wave operators for the nonlinear Klein-Gordon equation with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser-type inequality. We prove that if the energy is below or equal to the critical value, then the solution approaches a free Klein-Gordon solution at the time infinity. An interesting feature in the critical case is that the Strichartz estimate together with Sobolev-type inequalities cannot control the nonlinear term uniformly on each time interval: it crucially depends on how much the energy is concentrated. Thus we have to trace concentration of the energy along time, in order to set up favorable nonlinear estimates, and only after that we can apply Bourgain's induction argument (or any other similar one)

Primary Subjects: 35L70

Secondary Subjects: 35Q55, 35B40, 35B33, 37K05, 37L50

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1255699342
Digital Object Identifier: doi:10.1215/00127094-2009-053

back to Table of Contents

References

H. Bahouri and P. GéRard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), 131--175.

Mathematical Reviews (MathSciNet): MR1705001

Zentralblatt MATH: 0919.35089

Digital Object Identifier: doi:10.1353/ajm.1999.0001

A. A. Baraket, Local existence and estimations for a semilinear wave equation in two dimension space. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 7 (2004), 1--21.

Mathematical Reviews (MathSciNet): MR2044259

H. Bahouri and J. Shatah, Decay estimates for the critical semilinear wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 783--789.

Mathematical Reviews (MathSciNet): MR1650958

Zentralblatt MATH: 0924.35084

Digital Object Identifier: doi:10.1016/S0294-1449(99)80005-5

J. Bourgain, Scattering in the energy space and below for 3D NLS, J. Anal. Math. 75 (1998) 267--297.

Mathematical Reviews (MathSciNet): MR1655835

Zentralblatt MATH: 0972.35141

Digital Object Identifier: doi:10.1007/BF02788703

—, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), 145--171.

Mathematical Reviews (MathSciNet): MR1626257

Digital Object Identifier: doi:10.1090/S0894-0347-99-00283-0

JSTOR: links.jstor.org

P. Brenner, On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations, J. Differential Equations 56 (1985), 310--344.

Mathematical Reviews (MathSciNet): MR0780495

Zentralblatt MATH: 0513.35066

Digital Object Identifier: doi:10.1016/0022-0396(85)90083-X

H. BréZisandT. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal. 4 (1980), 677--681.

Mathematical Reviews (MathSciNet): MR0582536

Digital Object Identifier: doi:10.1016/0362-546X(80)90068-1

T. Cazenave, Equations de Schrödinger non linéaires en dimension deux, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 327--346.

Mathematical Reviews (MathSciNet): MR0559676

M. Christ, J. Colliander, and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, to appear in Ann. Henri Poincaré, preprint,\arxivmath/0311048v1[math.AP]

J. Colliander, M. Grillakis, and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, to appear in Comm. Pure Appl. Math., preprint,\arxiv0807.0871v2[math.AP]

Mathematical Reviews (MathSciNet): MR2527809

Digital Object Identifier: doi:10.1002/cpa.20278

J. Colliander, S. Ibrahim, M. Majdoub, and N. Masmoudi, Energy critical NLS in two space dimension, to appear in J. Hyperbolic Differ. Equ., preprint,\arxiv0806.2979v1[math.AP]

J. Ginibre, A. Soffer, and G. Velo, The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal. 110 (1992), 96--130.

Mathematical Reviews (MathSciNet): MR1190421

Zentralblatt MATH: 0813.35054

Digital Object Identifier: doi:10.1016/0022-1236(92)90044-J

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z. 189 (1985), 487--505.

Mathematical Reviews (MathSciNet): MR0786279

Zentralblatt MATH: 0549.35108

Digital Object Identifier: doi:10.1007/BF01168155

—, Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 43 (1985), 399--442.

Mathematical Reviews (MathSciNet): MR0824083

—, Scattering theory in the energy space for a class of nonlinear wave equations, Comm. Math. Phys. 123 (1989), 535--573.

Mathematical Reviews (MathSciNet): MR1006294

Zentralblatt MATH: 0698.35112

Digital Object Identifier: doi:10.1007/BF01218585

Project Euclid: euclid.cmp/1104178983

M. G. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. (2) 132 (1990), 485--509.

Mathematical Reviews (MathSciNet): MR1078267

Digital Object Identifier: doi:10.2307/1971427

JSTOR: links.jstor.org

—, Regularity for the wave equation with a critical nonlinearity, Comm. Pure Appl. Math. 45 (1992), 749--774.

Mathematical Reviews (MathSciNet): MR1162370

Zentralblatt MATH: 0785.35065

Digital Object Identifier: doi:10.1002/cpa.3160450604

S. Ibrahim and M. Majdoub, Solutions globales de l'équation des ondes semi-linéaire critique à coefficients variables, Bull. Soc. Math. France 131 (2003), 1--22.

Mathematical Reviews (MathSciNet): MR1975803

S. Ibrahim, M. Majdoub, and N. Masmoudi, Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity, Comm. Pure Appl. Math. 59 (2006), 1639--1658.

Mathematical Reviews (MathSciNet): MR2254447

Zentralblatt MATH: 1117.35049

Digital Object Identifier: doi:10.1002/cpa.20127

—, Double logarithmic inequality with a sharp constant, Proc. Amer. Math. Soc. 135 (2007), 87--97.

Mathematical Reviews (MathSciNet): MR2280178

Zentralblatt MATH: 1130.46018

Digital Object Identifier: doi:10.1090/S0002-9939-06-08240-2

—, Ill-posedness of $H\sp 1$-supercritical waves, C. R. Math. Acad. Sci. Paris 345 (2007), 133--138.

Mathematical Reviews (MathSciNet): MR2344811

Zentralblatt MATH: 1127.35073

Digital Object Identifier: doi:10.1016/j.crma.2007.06.008

—, Well and ill-posedness issues for energy super critical waves, preprint,\arxiv0906.3092v1[math.AP]

L. Kapitanski, Global and unique weak solutions of nonlinear wave equations, Math. Res. Lett. 1 (1994), 211--223.

Mathematical Reviews (MathSciNet): MR1266760

Zentralblatt MATH: 0841.35067

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math. 201 (2008), 147--212.

Mathematical Reviews (MathSciNet): MR2461508

Zentralblatt MATH: 05612700

Digital Object Identifier: doi:10.1007/s11511-008-0031-6

—, Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications, to appear in Amer. J. Math., preprint,\arxiv0810.4834v2[math.AP]

H. Kozono, T. Ogawa, and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z. 242 (2002), 251--278.

Mathematical Reviews (MathSciNet): MR1980623

Zentralblatt MATH: 1055.35087

Digital Object Identifier: doi:10.1007/s002090100332

J. F. Lam, B. Lippmann, and F. Tappert, Self-trapped laser beams in plasma, Phys. Fluids 20 (1977), 1176--1179.

G. Lebeau, Non linear optic and supercritical wave equation, Bull. Soc. Roy. Sci. Liège 70 (2001), 267--306.

Mathematical Reviews (MathSciNet): MR1904059

—, Perte de régularité pour les équations d'ondes sur critiques, Bull. Soc. Math. France 133 (2005), 145--157.

Mathematical Reviews (MathSciNet): MR2145023

H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations. J. Funct. Anal. 130 (1995), 357--426.

Mathematical Reviews (MathSciNet): MR1335386

Zentralblatt MATH: 0846.35085

Digital Object Identifier: doi:10.1006/jfan.1995.1075

N. Masmoudi and F. Planchon, On uniqueness for the critical wave equation Comm. Partial Differential Equations 31 (2006), 1099--1107.

Mathematical Reviews (MathSciNet): MR2254606

Zentralblatt MATH: 1106.35035

Digital Object Identifier: doi:10.1080/03605300500358012

M. Nakamura and T. Ozawa, The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order, Discrete Contin. Dynam. Systems 5 (1999), 215--231.

Mathematical Reviews (MathSciNet): MR1664497

Zentralblatt MATH: 0958.35011

—, Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth, Math. Z. 231 (1999), 479--487.

Mathematical Reviews (MathSciNet): MR1704989

Zentralblatt MATH: 0931.35107

Digital Object Identifier: doi:10.1007/PL00004737

K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions $1$ and $2$, J. Funct. Anal. 169 (1999), 201--225.

Mathematical Reviews (MathSciNet): MR1726753

Digital Object Identifier: doi:10.1006/jfan.1999.3503

—, Scattering theory for the nonlinear Klein-Gordon equation with Sobolev critical power, Internat. Math. Res. Notices 1999, no. 1, 31--60.

Mathematical Reviews (MathSciNet): MR1666973

Zentralblatt MATH: 0933.35166

Digital Object Identifier: doi:10.1155/S1073792899000021

F. Planchon, On uniqueness for semilinear wave equations, Math. Z. 244 (2003), 587--599.

Mathematical Reviews (MathSciNet): MR1992026

Zentralblatt MATH: 1023.35079

F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 261--290.

Mathematical Reviews (MathSciNet): MR2518079

Zentralblatt MATH: 05564563

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb R\sp 2$. J. Funct. Anal. 219 (2005), 340--367.

Mathematical Reviews (MathSciNet): MR2109256

Zentralblatt MATH: 1119.46033

Digital Object Identifier: doi:10.1016/j.jfa.2004.06.013

J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices 1994, no. 7, 303--309.

Mathematical Reviews (MathSciNet): MR1283026

Zentralblatt MATH: 0830.35086

Digital Object Identifier: doi:10.1155/S1073792894000346

W. A. Strauss, On weak solutions of semi-linear hyperbolic equations, An. Acad. Brasil. Ci. 42 (1970), 645--651.

Mathematical Reviews (MathSciNet): MR0306715

—, Nonlinear Wave Equations, CBMS Regional Conf. Ser. in Math. 73, Amer. Math. Soc., Providence, 1989.

Mathematical Reviews (MathSciNet): MR1032250

M. Struwe, Globally regular solutions to the $u^5$ Klein-Gordon equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), 495--513.

Mathematical Reviews (MathSciNet): MR1015805

Zentralblatt MATH: 0728.35072

—, Uniqueness for critical nonlinear wave equations and wave maps via the energy inequality, Comm. Pure Appl. Math. 52 (1999), 1179--1188.

Mathematical Reviews (MathSciNet): MR1692140

Zentralblatt MATH: 0933.35141

Digital Object Identifier: doi:10.1002/(SICI)1097-0312(199909)52:9<1179::AID-CPA6>3.0.CO;2-E

B. Wang, On existence and scattering for critical and subcritical nonlinear Klein-Gordon equations in $H\sp s$, Nonlinear Anal. 31 (1998), 573--587.

Mathematical Reviews (MathSciNet): MR1487847

previous :: next