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

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

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.**A. Bahri and H. Brézis,*Periodic solution of a nonlinear wave equation*, Proc. Roy. Soc. Edinburgh Sect. A 1-D**85**(1980), 313-320. MR**82f:35011****2.**A. Bamberger, G. Chavent and P. Lailly,*About the stability of the inverse problem in wave equations; applications to the interpretation of seismic profiles*, Appl. Math. Optimiz.**5**(1979), 1-47. MR**80b:86002****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.**H. Brézis,*Periodic solutions of nonlinear vibrating strings and duality principles*, Bull. AMS**8**(1983), 409-426. MR**84e:35010****5.**H. Brézis and L. Nirenberg,*Forced vibrations for a nonlinear wave equation*, Comm. Pure Appl. Math.**31**(1978), 1-30. MR**81i:35112****6.**R. C. Brown, D. B. Hinton and S. Schwabik,*Applications of a one-dimensional Sobolev inequality to eigenvalue problems*, Differential Integral Equations**9**(1996), 481-498. MR**96k:34180****7.**M. Fe\v{c}kan,*Periodic solutions of certain abstract wave equations*, Proc. AMS**123**(1995), 456-471. MR**95c:35030****8.**P. J. McKenna,*On solutions of a nonlinear wave equation when the ratio of the period to the length of the intervals is irrational*, Proc. AMS**93**(1985), 59-64. MR**86f:35017****9.**P. Rabinowitz,*Free vibrations for a semilinear wave equation*, Comm. Pure Appl. Math.**31**(1978), 31-68. MR**84i:35109****10.**K. Yosida,*Functional analysis*, 6th ed., Springer-Verlag, Berlin, 1980. MR**82i:46002**

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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