Periodic solutions to nonlinear one dimensional wave equation with 𝑋-dependent coefficients

Barbu, V.; Pavel, N.

doi:10.1090/S0002-9947-97-01714-5

Periodic solutions to nonlinear one dimensional wave equation with $X$-dependent coefficients
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by V. Barbu and N. H. Pavel

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

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

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Abstract:

This paper deals with $t$-periodicity and regularity of solutions to the one dimensional nonlinear wave equation with $x$-dependent coefficients

References

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, DOI 10.1017/S0308210500011896
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, DOI 10.1007/BF01442542
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.
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, DOI 10.1090/S0273-0979-1983-15105-4
H. Brézis and L. Nirenberg, Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math. 31 (1978), no. 1, 1–30. MR 470377, DOI 10.1002/cpa.3160310102
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
Michal Fec̆kan, Periodic solutions of certain abstract wave equations, Proc. Amer. Math. Soc. 123 (1995), no. 2, 465–470. MR 1234625, DOI 10.1090/S0002-9939-1995-1234625-9
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, DOI 10.1090/S0002-9939-1985-0766527-X
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
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