Advertisement

Israel Journal of Mathematics

, Volume 22, Issue 3–4, pp 273–303 | Cite as

Saddle points and instability of nonlinear hyperbolic equations

  • L. E. Payne
  • D. H. Sattinger
Article

Abstract

A number of authors have investigated conditions under which weak solutions of the initial-boundary value problem for the nonlinear wave equation will blow up in a finite time. For certain classes of nonlinearities sharp results are derived in this paper. Extensions to parabolic and to abstract operator equations are also given.

Keywords

Weak Solution Saddle Point Euler Equation Finite Time Nonlinear Wave Equation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. T. Glassey,Blow-up theorems for nonlinear wave equations, Math. Z.132 (1973), 183–302.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    K. Jörgens,Nonlinear wave equations, Lecture Notes, University of Colorado, March, 1970.Google Scholar
  3. 3.
    J. B. Keller,On solutions of nonlinear wave equations, Comm. Pure Appl. Math.10 (1957), 523–530.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    R. J. Knops, H. A. Levine and L. E. Payne,Nonexistence, instability and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastrodynamics, Arch. Rational Mech. Anal.55 (1974), 52–72.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Puu=−Au+F(u), Trans. Amer. Soc.192 (1974), 1–21.zbMATHGoogle Scholar
  6. 6.
    H. A. Levine,Some additional remarks on nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal.5 (1974), 138–146.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    H. A. Levine,A note on a nonexistence theorem for some nonlinear wave equations, SIAM J. Math. Anal.5 (1974), 644–648.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put=−Au+F(u), Arch. Rational Mech. Anal.51 (1973), 371–386.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D. H. Sattinger,Stability of nonlinear hyperbolic equations, Arch. Rational Mech. Anal.28 (1968), 226–244.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D. H. Sattinger,On global solutions of nonlinear hyperbolic equations, Arch. Rational Mech. Anal.30 (1968), 148–172.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    D. H. Sattinger,Topics in Stability and Bifurcation Theory, Springer Lecture Notes in Mathematics, 309.Google Scholar
  12. 12.
    J. Serrin,Nonlinear Elliptic Equations of Second Order, AMS Symposium in Partial Differential Equations, Berkeley, Calif., August, 1971.Google Scholar
  13. 13.
    M. Tsutsumi,On solutions of semilinear differential equations in a Hilbert space, Math. Japon.17 (1972), 173–193.MathSciNetzbMATHGoogle Scholar
  14. 14.
    M. Tsutsumi,Existence and Nonexistence of Global Solutions for Nonlinear Parabolic Equations, Publications Research Institute for Mathematical Sciences, Kyoto University,8 (1972), 211–229.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    M. Tsutsumi,Some Nonlinear Evolution Equations of Second Order, Proc. Japan Acad.47 (1971), 450–955.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University 1976

Authors and Affiliations

  • L. E. Payne
    • 1
    • 2
  • D. H. Sattinger
    • 1
    • 2
  1. 1.Cornell UniversityUSA
  2. 2.University of MinnesotaUSA

Personalised recommendations