Geodesic
About the classical solution of the mixed problem for the wave equation
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 1, pp. 56-66

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The classic solution of the mixed problem for a wave equation with a complex potential and minimal smoothness of initial data is established by the Fourier method. The resolvent approach consists of constructing formal solution with the help of the Cauchy–Poincaré method of integrating the resolvent of the corresponding spectral problem over spectral parameter. The method requires no information about eigen and associated functions and uses only the main part of eigenvalues asymptotics. Krylov's idea of accelerating the convergence of Fourier series is essentially employed. The boundary conditions of the mixed problem can produce multiple spectrum and infinite number of associated functions in the spectral problem, thus making more difficult the analysis of the formal solution. 
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A. P. Khromov. About the classical solution of the mixed problem for the wave equation. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 1, pp. 56-66. http://geodesic.mathdoc.fr/item/ISU_2015_15_1_a8/