Classic and generalized solutions of the mixed problem for~wave~equation with a summable potential. Part~I.~Classic~solution of the mixed problem

V. P. Kurdyumov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 3, pp. 311-319

Résumé

The resolvent approach and the using of the idea of A. N. Krylov on the acceleration of convergence of Fourier series, the properties of a formal solution of a mixed problem for a homogeneous wave equation with a summable potential and a zero initial function are studied. This method makes it possible to obtain deep results on the convergence of a formal series with arbitrary boundary conditions and without overestimating the requirements for the smoothness of the initial data. The different-order boundary conditions considered in the article are such that the operator corresponding to the spectral problem may have an infinite set of multiple eigenvalues and their associated functions. A classical solution is obtained without overstating the requirements for the initial velocity $u'_t(x,0) = \psi(x)$. It is shown that for $\psi(x) \in L[0,1]$ the formal solution, being the uniform limit of the classical ones, is a generalized solution, and when $\psi(x) \in L_p[0,1], ~ 1 $, the formal solution has much smoother properties than the case $\psi(x) \in L[0,1]$.