Periodic solutions to nonlinear one dimensional

wave equation with -dependent coefficients

Authors:
V. Barbu and N. H. Pavel

Journal:
Trans. Amer. Math. Soc. **349** (1997), 2035-2048

MSC (1991):
Primary 35L70, 35B10, 35L05

MathSciNet review:
1373628

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with -periodicity and regularity of solutions to the one dimensional nonlinear wave equation with -dependent coefficients

**1.**Abbas Bahri and Haïm Brézis,*Periodic solution of a nonlinear wave equation*, Proc. Roy. Soc. Edinburgh Sect. A**85**(1980), no. 3-4, 313–320. MR**574025**, 10.1017/S0308210500011896**2.**A. Bamberger, G. Chavent, and P. Lailly,*About the stability of the inverse problem in 1-D wave equations—applications to the interpretation of seismic profiles*, Appl. Math. Optim.**5**(1979), no. 1, 1–47. MR**526426**, 10.1007/BF01442542**3.**V. Barbu and N. H. Pavel,*An inverse problem for the one dimensional wave equation*, SIAM J. Control and Optimiz.**35-5**(1997), to appear.**4.**Haïm Brézis,*Periodic solutions of nonlinear vibrating strings and duality principles*, Bull. Amer. Math. Soc. (N.S.)**8**(1983), no. 3, 409–426. MR**693957**, 10.1090/S0273-0979-1983-15105-4**5.**H. Brézis and L. Nirenberg,*Forced vibrations for a nonlinear wave equation*, Comm. Pure Appl. Math.**31**(1978), no. 1, 1–30. MR**470377**, 10.1002/cpa.3160310102**6.**R. C. Brown, D. B. Hinton, and Š. Schwabik,*Applications of a one-dimensional Sobolev inequality to eigenvalue problems*, Differential Integral Equations**9**(1996), no. 3, 481–498. MR**1371703****7.**Michal Fečkan,*Periodic solutions of certain abstract wave equations*, Proc. Amer. Math. Soc.**123**(1995), no. 2, 465–470. MR**1234625**, 10.1090/S0002-9939-1995-1234625-9**8.**P. J. McKenna,*On solutions of a nonlinear wave question when the ratio of the period to the length of the interval is irrational*, Proc. Amer. Math. Soc.**93**(1985), no. 1, 59–64. MR**766527**, 10.1090/S0002-9939-1985-0766527-X**9.**S. M. Khzardzhyan,*Closedness of eigenfunctions of nonselfadjoint homogeneous boundary value problems and its practical significance*, Vychisl. Prikl. Mat. (Kiev)**47**(1982), 50–55 (Russian). MR**700388****10.**Kôsaku Yosida,*Functional analysis*, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR**617913**

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Additional Information

**V. Barbu**

Affiliation:
Department of Mathematics, University of Iasi, Iasi, Romania

Email:
barbu@uaic.ro

**N. H. Pavel**

Affiliation:
Department of Mathematics, Ohio University, Athens, Ohio 45701

Email:
npavel@bing.math.ohiou.edu

DOI:
https://doi.org/10.1090/S0002-9947-97-01714-5

Keywords:
Forced vibrations of nonhomogeneous strings,
propagation of seismic waves,
eigenvalues and eigenfunctions,
Fourier series,
subdifferentials,
maximal monotone operators,
Sobolev spaces

Received by editor(s):
April 18, 1995

Received by editor(s) in revised form:
December 4, 1995

Additional Notes:
This research was carried out while the first author was visiting Ohio University

Article copyright:
© Copyright 1997
American Mathematical Society