A problem with nonlocal integral condition of the second kind for one-dimensional hyperbolic equation.

*(Russian. English summary)*Zbl 1413.35007Summary: In this paper, we consider a problem for a one-dimensional hyperbolic equation with nonlocal integral condition of the second kind. Uniqueness and existence of a generalized solution are proved. In order to prove this statement we suggest a new approach. The main idea of it is that given nonlocal integral condition is equivalent with a different condition, nonlocal as well but this new condition enables us to derive a priori estimates of a required solution in Sobolev space. By means of derived estimates we show that a sequence of approximate solutions constructed by Galerkin procedure is bounded in Sobolev space. This fact implies the existence of weakly convergent subsequence. Finally, we show that the limit of extracted subsequence is the required solution to the problem.

##### MSC:

35A01 | Existence problems for PDEs: global existence, local existence, non-existence |

35L10 | Second-order hyperbolic equations |

35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |

##### Keywords:

hyperbolic equation; nonlocal integral conditions; generalized solution; Sobolev space; Galerkin procedure##### References:

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