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Scattering theory in the energy space for a class of non-linear wave equations

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Abstract

We study the asymptotic behaviour in time of the solutions and the theory of scattering in the energy space for the non-linear wave equation

$$\square \varphi + f(\varphi ) = 0$$

in ℝn,n≧3. We prove the existence of the wave operators, asymptotic completeness for small initial data and, forn≧4, asymptotic completeness for arbitrarily large data. The assumptions onf cover the case wheref behaves slightly better than a single powerp=1+4/(n−2), both near zero and at infinity (see (1.5), (1.6) and (1.8)).

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Authors and Affiliations

  1. Laboratoire de Physique Théorique et Hautes Energies, Université de Paris XI, Bâtiment 211, F-91405, Orsay, France

    J. Ginibre

  2. Dipartimento di Fisica, Università di Bologna and INFN, Sezione di Bologna, Italy

    G. Velo

Authors
  1. J. Ginibre
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  2. G. Velo
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Communicated by A. Jaffe

Laboratoire associé au Centre National de la Recherche Scientifique

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Ginibre, J., Velo, G. Scattering theory in the energy space for a class of non-linear wave equations. Commun.Math. Phys. 123, 535–573 (1989). https://doi.org/10.1007/BF01218585

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  • DOI: https://doi.org/10.1007/BF01218585

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