A wave equation with a possibly jumping nonlinearity

Author:
J. R. Ward

Journal:
Proc. Amer. Math. Soc. **92** (1984), 209-214

MSC:
Primary 35B10; Secondary 35L70

DOI:
https://doi.org/10.1090/S0002-9939-1984-0754705-4

MathSciNet review:
754705

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Abstract | References | Similar Articles | Additional Information

Abstract: Existence of a doubly periodic solution to a forced semilinear wave equation is established. The nonlinearity may "jump" across any finite number of eigenvalues of finite multiplicity.

**[C-H]**S. N. Chow and J. K. Hale,*Methods of bifurcation theory*, Springer, Berlin and New York, 1982. MR**660633 (84e:58019)****[L]**H. Lovicarova,*Periodic solutions of a weakly nonlinear wave equation in one dimension*, Czechoslovak. Math. J.**19**(1969), 324-342. MR**0247249 (40:518)****[M.l]**J. Mawhin,*Periodic solutions of nonlinear dispersive wave equations*, Constructive Methods for Nonlinear Boundary Value Problems and Nonlinear Oscillations (Albrecht, Collatz and Kirchgässner, eds.), Birkhäuser, Basel, 1979, pp. 102-109. MR**565644 (81b:35003)****[M.2]**-,*Compacité, monotonie et convexité dans l'etude de problèmes aux limites semi-linéaires*, Séminaire d'Analyse Moderne, No. 19, Univ. de Sherbrooke, 1981.**[M-W.l]**J. Mawhin and J. Ward,*Asymptotic nonuniform non-resonance conditions in the periodic Dirichlet problem for semi-linear wave equations*, J. Math. Pures Appl. (to appear).**[M-W.2]**-,*Nonuniform non-resonance conditions in the periodic-Dirichlet problem for semi-linear wave equations with jumping nonlinearities*(to appear).**[Wi]**M. Willem,*Periodic solutions of wave equations with jumping nonlinearities*, J. Differential Equations**36**(1980), 20-27. MR**571124 (82f:35013)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1984-0754705-4

Keywords:
Nonlinear wave equation,
periodic solution

Article copyright:
© Copyright 1984
American Mathematical Society