Geodesic
On the Solvability of Certain Spatially Nonlocal Boundary-Value Problems for Linear Hyperbolic Equations of Second Order
Matematičeskie zametki, Tome 90 (2011) no. 2, pp. 254-268

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The present paper studies the solvability of spatially nonlocal boundary-value problems with Samarskii boundary condition with variable coefficients for linear hyperbolic equations of second order in the one-dimensional case. In the case of boundary conditions with constant coefficients for the equation $$ u_{tt}-u_{xx}+c(x)u=f(x,t), $$ similar problems were studied earlier by other authors; a significant aspect of their papers was the use of the Fourier method, which dictated a special form of the equation as well as the constancy of the coefficients of the boundary conditions. The method used here does not involve such constraints and allows us to study more general problems.
Keywords: linear hyperbolic equation of second order, spatially nonlocal boundary-value problem, Samarskii boundary condition, Young's inequality. 
@article{MZM_2011_90_2_a7,
     author = {A. I. Kozhanov},
     title = {On the {Solvability} of {Certain} {Spatially} {Nonlocal} {Boundary-Value} {Problems} for {Linear} {Hyperbolic} {Equations} of {Second} {Order}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {254--268},
     publisher = {mathdoc},
     volume = {90},
     number = {2},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_90_2_a7/}
}
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A. I. Kozhanov. On the Solvability of Certain Spatially Nonlocal Boundary-Value Problems for Linear Hyperbolic Equations of Second Order. Matematičeskie zametki, Tome 90 (2011) no. 2, pp. 254-268. http://geodesic.mathdoc.fr/item/MZM_2011_90_2_a7/