Abstract
In this paper, we prove the existence of time-periodic weak solutions for the wave equation with homogeneous boundary conditions. This paper deals with the cases where a nonlinear term has a superlinear and sublinear growth.
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References
A. Bahri and H. Brezis, “Periodic solution of a nonlinear wave equation,” Proc. Roy. Soc. Edinburgh. Sect. A, 85, 313–320 (1980).
H. Brezis and H. Nirenberg, “Forced vibrations for a nonlinear wave equations,” Comm. Pure Appl. Math., 31, No. 1, 1–30 (1978).
E. R. Fadell, S. Y. Husseini, and P. H. Rabinowitz, “Borsuk-Ulam theorems for arbitrary S 1 actions and applications,” Trans. Amer. Math. Soc., 274, No. 1, 345–360 (1982).
E. Feireisl, “On the existence of multiplicity periodic solutions of a semilinear wave equation with a superlinear forcing term,” Czechoslovak Math. J., 38, No. 1, 78–87 (1988).
E. Feireisl, “Time-periodic solutions to a semilinear beam equation,” Nonlinear Anal., 12, 279–290 (1988).
P. I. Plotnikov, “The existence of a countable set of periodic solutions of the problem of forced oscillations for a weakly nonlinear wave equation,” Mat. Sb., 136(178), No. 4 (8), 546–560 (1988).
I. A. Rudakov, “Nonlinear oscillations of a string,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 9–13 (1984).
I. A. Rudakov, “Nonlinear vibrations of a nonhomogeneous string,” Fundam. Prikl. Mat., 8, No. 3, 877–886 (2002).
I. A. Rudakov, “Time-periodic solutions of an equation of forced oscillations of a string with homogeneous boundary conditions,” Differ. Uravn., No. 11, 1550–1555 (2003).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 189–201, 2005.
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Rudakov, I.A. Periodic solutions of a quasilinear wave equation with homogeneous boundary conditions. J Math Sci 150, 2588–2597 (2008). https://doi.org/10.1007/s10958-008-0157-2
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DOI: https://doi.org/10.1007/s10958-008-0157-2