Geodesic
Nonlocal initial-boundary-value problems for a~degenerate hyperbolic equation
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2009), pp. 49-58

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We consider the equation $y^mu_{xx}-u_{yy}-b^2y^mu=0$ in the rectangular area $\{(x,y)\mid0$, where $m>0$, $b\ge0$, $T>0$ are given real numbers. For this equation we study problems with initial conditions $u(x,0)=\tau(x)$, $u_y(x,0)=\nu(x)$, $0\le x\leq1$, and nonlocal boundary conditions $u(0,y)=u(1,y)$, $u_x(0,y)=0$ or $u_x(0,y)=u_x(1,y)$, $u(1,y)=0$ with $0\le y\le T$. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems.
Keywords: nonlocal problem, spectral method, completeness, sum of biorthogonal series. 
@article{IVM_2009_12_a5,
     author = {Yu. K. Sabitova},
     title = {Nonlocal initial-boundary-value problems for a~degenerate hyperbolic equation},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {49--58},
     publisher = {mathdoc},
     number = {12},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2009_12_a5/}
}
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Yu. K. Sabitova. Nonlocal initial-boundary-value problems for a~degenerate hyperbolic equation. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2009), pp. 49-58. http://geodesic.mathdoc.fr/item/IVM_2009_12_a5/