Blow up near higher modes of nonlinear wave equations

Sternberg, Natalia

doi:10.1090/S0002-9947-1986-0837814-7

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
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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.

[1] Herbert Amann, Nonlinear operators in ordered Banach spaces and some applications to nonlinear boundary value problems, Nonlinear operators and the calculus of variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975) Springer, Berlin, 1976, pp. 1–55. Lecture Notes in Math., Vol. 543. MR 0513051
[2] Antonio Ambrosetti, On the existence of multiple solutions for a class of nonlinear boundary value problems, Rend. Sem. Mat. Univ. Padova 49 (1973), 195–204. MR 336068
[3] Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, https://doi.org/10.1016/0022-1236(73)90051-7
[4] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345. MR 695535, https://doi.org/10.1007/BF00250555
[5] Felix E. Browder, Existence theorems for nonlinear partial differential equations, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 1–60. MR 0269962
[6] David C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J. 22 (1972/73), 65–74. MR 296777, https://doi.org/10.1512/iumj.1972.22.22008
[7] Charles V. Coffman, A minimum-maximum principle for a class of non-linear integral equations, J. Analyse Math. 22 (1969), 391–419. MR 249983, https://doi.org/10.1007/BF02786802
[8] Joachim A. Hempel, Multiple solutions for a class of nonlinear boundary value problems, Indiana Univ. Math. J. 20 (1970/71), 983–996. MR 279423, https://doi.org/10.1512/iumj.1971.20.20094
[9] L. A. Ljusternik and L. G. Schnirelman, Méthodes topologique dans les problèmes variationels, Actualités Sci. Indust., No. 188, Hermann, Paris, 1934.
[10] Roger Nussbaum, Positive solutions of nonlinear elliptic boundary value problems, J. Math. Anal. Appl. 51 (1975), no. 2, 461–482. MR 382850, https://doi.org/10.1016/0022-247X(75)90133-X
[11] Richard S. Palais, Critical point theory and the minimax principle, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif, 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 185–212. MR 0264712
[12] Richard S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115–132. MR 259955, https://doi.org/10.1016/0040-9383(66)90013-9
[13] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), no. 3-4, 273–303. MR 402291, https://doi.org/10.1007/BF02761595
[14] Paul H. Rabinowitz, Variational methods for nonlinear elliptic eigenvalue problems, Indiana Univ. Math. J. 23 (1973/74), 729–754. MR 333442, https://doi.org/10.1512/iumj.1974.23.23061
[15] Jalal Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys. 91 (1983), no. 3, 313–327. MR 723756
[16] Walter A. Strauss, Nonlinear invariant wave equations, Invariant wave equations (Proc. “Ettore Majorana” Internat. School of Math. Phys., Erice, 1977) Lecture Notes in Phys., vol. 73, Springer, Berlin-New York, 1978, pp. 197–249. MR 498955
[17] Henri Berestycki and Pierre-Louis Lions, Théorie des points critiques et instabilité des ondes stationnaires pour des équations de Schrödinger non linéaires, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 10, 319–322 (French, with English summary). MR 786909
[18] Henri Berestycki and Thierry Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 9, 489–492 (French, with English summary). MR 646873
[19] David H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Mathematics, Vol. 309, Springer-Verlag, Berlin-New York, 1973. MR 0463624
[20] Jalal Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc. 290 (1985), no. 2, 701–710. MR 792821, https://doi.org/10.1090/S0002-9947-1985-0792821-7