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manuscripta mathematica

, Volume 38, Issue 3, pp 387–400 | Cite as

Time decay for nonlinear wave equations in two space dimensions

  • Robert Glassey
  • Hartmut Pecher
Article

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.

Keywords

Wave Equation Cauchy Problem Slow Rate Number Theory Time Decay 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Robert Glassey
    • 1
  • Hartmut Pecher
    • 2
  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA
  2. 2.Fachbereich MathematikUniversität WuppertalWuppertal 1Fed. Rep. of Germany

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