Geodesic
Nonlocal Problem for a Parabolic-Hyperbolic Equation in a Rectangular Domain
Matematičeskie zametki, Tome 89 (2011) no. 4, pp. 596-602

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For an equation of mixed type, namely, $$ (1-\operatorname{sgn}t)u_{tt}+(1-\operatorname{sgn}t)u_{t}-2u_{xx}=0 $$ in the domain $\{(x,t)\mid0$, where $\alpha$, $\beta$ are given positive real numbers, we study the problem with boundary conditions $$ u(0,t)=u(1,t)=0,\quad -\alpha\le t\le\beta,\qquad u(x,-\alpha)-u(x,\beta)=\varphi(x),\quad 0\le x\le1. $$ We establish a uniqueness criterion for the solution constructed as the sum of Fourier series. We establish the stability of the solution with respect to its nonlocal condition $\varphi(x)$.
Keywords: parabolic-hyperbolic equation, Fourier series, initial boundary-value problem, differential equation, Weierstrass test.
Mots-clés : nonlocal condition 
@article{MZM_2011_89_4_a10,
     author = {K. B. Sabitov},
     title = {Nonlocal {Problem} for a {Parabolic-Hyperbolic} {Equation} in a {Rectangular} {Domain}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {596--602},
     publisher = {mathdoc},
     volume = {89},
     number = {4},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_89_4_a10/}
}
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K. B. Sabitov. Nonlocal Problem for a Parabolic-Hyperbolic Equation in a Rectangular Domain. Matematičeskie zametki, Tome 89 (2011) no. 4, pp. 596-602. http://geodesic.mathdoc.fr/item/MZM_2011_89_4_a10/