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On a nonlocal problem for partial differential equations. (Russian) Zbl 0621.35051

The author considers the following problem: \[ (P)\quad u_ y=u_{xx}\quad in\quad D=\{(x,y),\quad 0<x<a,\quad 0<y<b\},\quad u\in C(\bar D),\quad u(x,0)=\tau (x),\quad 0\leq x\leq a, \]
\[ (\partial /\partial y)\int^{\alpha}_{a}u(x,y)dx=\phi (y),\quad 0\leq y\leq b,\quad \psi (y)=(\partial /\partial y)\int^{a}_{\beta}u(x,y)dx,\quad 0\leq y\leq b, \] where \(\tau\), \(\phi\), \(\Psi\) are given continuous functions and \(\alpha\) and \(\beta\) are given numbers such that \(0\leq \alpha \leq a\), \(0\leq \beta <a\), \((\alpha -a)^ 2+\beta^ 2\neq 0.\)
The main results are contained in the theorems: Theorem 1. If u is a solution of the problem (P) for \(\tau (x)=0\), \(\phi (y)=0\), \(\psi (y)=0\) such that \(u,u_ x\in C(\bar D)\) and \(u_{xy}\in C(D_ 1)\), \(D_ 1=\{(x,y)\), \(0<x<a,0<y\leq b\}\), then \(u\equiv 0.\)
Theorem 2. If \(\tau \in C^ 1[0,a]\), \(\phi\) and \(\psi\in C[a,b]\), then the problem (P) admits solutions.
Reviewer: N.Luca

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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