Problem with nonlocal boundary condition for a hyperbolic equation
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 3 (2017), pp. 26-33
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In this paper we consider an initial-boundary problem with nonlocal boundary condition for one-dimensional hyperbolic equation. Nonlocal condition is dynamic so as represents a relation between values of derivatives with respect of spacial variables of a required solution, first-order derivatives with respect to time variable and an integral of a required solution of spacial variable. We prove the existence and uniqueness of a generalized solution, which belongs to the Sobolev space. To prove uniquely solvability of the problem techniques developed specifically for research nonlocal problems are used. The application of these methods allowed us to obtain a priori estimates, through which the uniqueness of the solution is proved. The proof is based on the a priori estimates obtained in this paper and Galyorkin's procedure.
Keywords:
nonlocal boundary condition, hyperbolic equation, generalized solution, Sobolev space.
@article{VSGU_2017_3_a3,
author = {V. A. Kirichek},
title = {Problem with nonlocal boundary condition for a hyperbolic equation},
journal = {Vestnik Samarskogo universiteta. Estestvennonau\v{c}na\^a seri\^a},
pages = {26--33},
publisher = {mathdoc},
number = {3},
year = {2017},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/VSGU_2017_3_a3/}
}
V. A. Kirichek. Problem with nonlocal boundary condition for a hyperbolic equation. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 3 (2017), pp. 26-33. http://geodesic.mathdoc.fr/item/VSGU_2017_3_a3/