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
HTML articles powered by AMS MathViewer

by Natalia Sternberg PDF

Trans. Amer. Math. Soc. 296 (1986), 315-325 Request permission

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

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) Lecture Notes in Math., Vol. 543, Springer, Berlin, 1976, pp. 1–55. MR 0513051
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
Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI 10.1016/0022-1236(73)90051-7
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, DOI 10.1007/BF00250555
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
David C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J. 22 (1972/73), 65–74. MR 296777, DOI 10.1512/iumj.1972.22.22008
Charles V. Coffman, A minimum-maximum principle for a class of non-linear integral equations, J. Analyse Math. 22 (1969), 391–419. MR 249983, DOI 10.1007/BF02786802
Joachim A. Hempel, Multiple solutions for a class of nonlinear boundary value problems, Indiana Univ. Math. J. 20 (1970/71), 983–996. MR 279423, DOI 10.1512/iumj.1971.20.20094

Méthodes topologique dans les problèmes variationels

Roger Nussbaum, Positive solutions of nonlinear elliptic boundary value problems, J. Math. Anal. Appl. 51 (1975), no. 2, 461–482. MR 382850, DOI 10.1016/0022-247X(75)90133-X
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
Richard S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115–132. MR 259955, DOI 10.1016/0040-9383(66)90013-9
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, DOI 10.1007/BF02761595
Paul H. Rabinowitz, Variational methods for nonlinear elliptic eigenvalue problems, Indiana Univ. Math. J. 23 (1973/74), 729–754. MR 333442, DOI 10.1512/iumj.1974.23.23061
Jalal Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys. 91 (1983), no. 3, 313–327. MR 723756
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
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
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
David H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Mathematics, Vol. 309, Springer-Verlag, Berlin-New York, 1973. MR 0463624
Jalal Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc. 290 (1985), no. 2, 701–710. MR 792821, DOI 10.1090/S0002-9947-1985-0792821-7