Geodesic
Mixed problem for simplest hyperbolic first order equations with involution
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 1, pp. 10-20.

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In this paper investigates the mixed problem for the first order differential equation with involution at the potential and with periodic boundary conditions. Using the received refined asymptotic formulas for eigenvalues and eigenfunctions of the corresponding spectral problem, the application of the Fourier method is substantiated. We used techniques, which allow to avoid investigation of the uniform convergence of the series, obtained by term by term differentiation of formal solution on method of Fourier. This allows to get a classical solution with minimal requirements on the initial data of the problem. 
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M. Sh. Burlutskaya; A. P. Khromov. Mixed problem for simplest hyperbolic first order equations with involution. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 1, pp. 10-20. http://geodesic.mathdoc.fr/item/ISU_2014_14_1_a1/

[1] Burlutskaya M. Sh., Khromov A. P., “Initial-boundary Value Problems for First Order Hyperbolic Equations with Involution”, Doklady Mathematics, 84:3 (2011), 783–786 | DOI | MR | Zbl

[2] Burlutskaya M. Sh., Khromov A. P., “Substantiation of Fourier Method in Mixed Problem with Involution”, Izv. Sarat. Univ. (N.S.), Ser. Math. Mech. Inform., 11:4 (2011), 3–12

[3] Krylov A. N., On Some Differential Equations of Mathematical Physics Having Application to Technical Problems, GITTL, Moscow–Leningrad, 1950, 368 pp.

[4] Chernyatin V. A., Justification of the Fourier method in the mixed boundary value problem for partial differential equations, Moscow Univ. Press, Moscow, 1991, 112 pp. | MR | Zbl