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Time decay for nonlinear wave equations in two space dimensions

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Abstract

The Cauchy Problem for the equation utt−Δu+|u|p−1u=0 (x∈ℝ2, t>0, ρ>1) is studied. Smooth Cauchy data is prescribed, and no smallness condition is imposed. For ρ>5, it is shown that the maximum amplitude of such a wave decays at the expected rate t−1/2 as t→∞. For 1+√8<ρ≦5, the maximum amplitude still decays, but at a slower rate. These results are then used to demonstrate the existence of the scattering operator when ρ>ρo, where ρo is the root of the cubic equation ρ3-2ρ2-7ρ-8=0; thus ρo≅4.15.

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Authors and Affiliations

  1. Department of Mathematics, Indiana University, 47405, Bloomington, IN., USA

    Robert Glassey

  2. Fachbereich Mathematik, Universität Wuppertal, D-5600, Wuppertal 1, Fed. Rep. of Germany

    Hartmut Pecher

Authors
  1. Robert Glassey
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  2. Hartmut Pecher
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Alfred P. Sloan Research Fellow

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Glassey, R., Pecher, H. Time decay for nonlinear wave equations in two space dimensions. Manuscripta Math 38, 387–400 (1982). https://doi.org/10.1007/BF01170934

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  • DOI: https://doi.org/10.1007/BF01170934

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