Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Blow up near higher modes of nonlinear wave equations


Author: Natalia Sternberg
Journal: Trans. Amer. Math. Soc. 296 (1986), 315-325
MSC: Primary 35L05; Secondary 35B35
DOI: https://doi.org/10.1090/S0002-9947-1986-0837814-7
MathSciNet review: 837814
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the instability properties of higher modes of the nonlinear wave equation $ {u_{tt}} - \Delta u - f(u) = 0$ defined on a smoothly bounded domain with Dirichlet boundary conditions. It is shown that they are unstable in the sense that in any neighborhood of a higher mode there exists a solution of the given equation which blows up in finite time.


References [Enhancements On Off] (What's this?)

  • [1] H. Amann, Nonlinear operators in ordered Banach spaces and some applications to nonlinear boundary value problems, Nonlinear Operators and the Calculus of Variations (Bruxelles, 1975), Lecture Notes in Math., vol. 543, Springer, Berlin and New York, 1976. MR 0513051 (58:23793)
  • [2] A. Ambrosetti, On the existence of multiple solutions for a class of nonlinear boundary value problems, Rend. Sem. Univ. Padova 49 (1973), 195-204. MR 0336068 (49:844)
  • [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. MR 0370183 (51:6412)
  • [4] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, Arch. Rational Mech. Anal. 82 (1983), 313-375. MR 695535 (84h:35054a)
  • [5] F. E. Browder, Existence theorems for nonlinear partial differential equations, Proc. Sympos. Pure Math., vol. 16, Amer. Math. Soc., Providence, R. I., 1970, pp. 1-60. MR 0269962 (42:4855)
  • [6] D. C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J. 22 (1972), 65-74. MR 0296777 (45:5836)
  • [7] C. V. Coffman, A minimum-maximum principle for a class of nonlinear integral equations, J. Analyse Math. 22, (1969), 391-419. MR 0249983 (40:3224)
  • [8] J. A. Hempel, Multiple solutions for a class of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 20 (1971), 983-996. MR 0279423 (43:5145)
  • [9] L. A. Ljusternik and L. G. Schnirelman, Méthodes topologique dans les problèmes variationels, Actualités Sci. Indust., No. 188, Hermann, Paris, 1934.
  • [10] R. Nussbaum, Positive solutions of nonlinear elliptic boundary value problems, J. Math. Anal. Appl. (2) 51 (1975), 461-482. MR 0382850 (52:3732)
  • [11] R. S. Palais, Critical point theory and minimax principle, Proc. Sympos. Pure Math., vol. 15, Amer. Math. Soc., Providence, R. I., 1970, pp. 185-212. MR 0264712 (41:9303)
  • [12] -, Ljusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115-132. MR 0259955 (41:4584)
  • [13] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), 273-303. MR 0402291 (53:6112)
  • [14] P. H. Rabinowitz, Variational methods for nonlinear elliptic eigenvalue problems, Indiana Univ. Math. J. 23 (1973/74), 729-754. MR 0333442 (48:11767)
  • [15] J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys. 91 (1983), 313-327. MR 723756 (84m:35111)
  • [16] W. Strauss, Nonlinear invariant wave equations, Invariant Wave Equations (Erice, 1977), Lecture Notes in Physics, No. 73, Springer-Verlag, Berlin and New York, 1978, pp. 197-249. MR 498955 (80b:35090)
  • [17] H. Berestycki and P. L. Lions, Théorie des points critique et instabilité des ondes stationnaires pour des equations de Schrödinger non-linéaires, C. R. Acad. Sci. Paris, 1985. MR 786909 (86m:35014)
  • [18] H. Berestycki and T. Cazenave, Instabilité des états stationnaire dans les équations de Schrödinger et de Klein-Gordon non-linéaires, C. R. Acad. Sci., Paris, 1981, pp. 489-492. MR 646873 (84f:35120)
  • [19] D. H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Math., vol. 309, Springer, 1973. MR 0463624 (57:3569)
  • [20] J. Shatah, Unstable ground states of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc. 290 (1985). MR 792821 (86k:35088)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35L05, 35B35

Retrieve articles in all journals with MSC: 35L05, 35B35


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0837814-7
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society