### 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).

Original language | English (US) |
---|---|

Pages (from-to) | 287-329 |

Number of pages | 43 |

Journal | Duke Mathematical Journal |

Volume | 150 |

Issue number | 2 |

DOIs | |

State | Published - Nov 2009 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Duke Mathematical Journal*,

*150*(2), 287-329. https://doi.org/10.1215/00127094-2009-053

**Scattering for the two-dimensional energy-critical wave equation.** / Ibrahim, Slim; Majdoub, Mohamed; Masmoudi, Nader; Nakanishi, Kenji.

Research output: Contribution to journal › Article

*Duke Mathematical Journal*, vol. 150, no. 2, pp. 287-329. https://doi.org/10.1215/00127094-2009-053

}

TY - JOUR

T1 - Scattering for the two-dimensional energy-critical wave equation

AU - Ibrahim, Slim

AU - Majdoub, Mohamed

AU - Masmoudi, Nader

AU - Nakanishi, Kenji

PY - 2009/11

Y1 - 2009/11

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=77957041378&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957041378&partnerID=8YFLogxK

U2 - 10.1215/00127094-2009-053

DO - 10.1215/00127094-2009-053

M3 - Article

VL - 150

SP - 287

EP - 329

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 2

ER -