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


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.


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    BERGH, J. and LÖFSTRÖM, J.: Interpolation Spaces. Berlin, Heidelberg, New York: Springer 1976Google Scholar
  2. [2]
    BRENNER, P. and von WAHL, W.: Global Classical Solutions of Non-linear Wave Equations. Math.Z. 176 (1981), 87–121Google Scholar
  3. [3]
    GLASSEY, R. and STRAUSS, W.: Decay of a Yang-Mills Field Coupled to a Scalar Field. Comm. Math. Phys. 67 (1979), 51–67Google Scholar
  4. [4]
    MORAUETZ, C.: Appendix 3 in Lax, P. and Phillips, R.: Scattering Theory. New York, London: Academic Press, 1967Google Scholar
  5. [5]
    PECHER, H.: Lp-Abschätzungen und klassische Lösungen für nicht-lineare Wellengleichungen, I. Math. Z. 150 (1976), 159–183Google Scholar
  6. [6]
    PECHER, H.: Decay of Solutions of Nonlinear Wave Equations in Three Space Dimensions. To appear in Journal of Functional AnalysisGoogle Scholar
  7. [7]
    PECHER, H.: Decay and Asymptotics for Higher Dimensional Non-linear Wave Equations. To appear in Journal of Differential EquationsGoogle Scholar
  8. [8]
    STRAUSS, W.: Decay and Asymptotics for Ou=F(u). Journal of Functional Analysis 2 (1968), 409–457Google Scholar
  9. [9]
    STRAUSS, W.: Nonlinear Invariant Wave Equations. In: Invariant Wave Equations, Lecture Notes in Physics, No. 73, 1978Google Scholar

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

Personalised recommendations