On existence and some properties of Green function of adjoining problem for mixed equations (in Ukrainian)

НLet $\Omega$ be a domain in $\mathbb{R}^n$, which is bounded $n-1$-dimensional closed hypesurface $\partial\Omega$ from the class $C^\infty$, $\Omega^\circ$ is closed infinte differentiable hyperspace within $\Omega$, $\Omega = \Omega_1 \cup \Omega_2$, $\partial \Omega_1 = \Omega^\circ \cup \partial \Omega$, $\Omega_2 = \Omega \setminus \overline{\Omega_1}$, $Q = \Omega \times (0;T)$, $Q_{\sigma T} = \Omega \times (0;T)$, $Q_{1T} = \partial \Omega \times (0;T)$, $Q_{2T} = \Omega^\circ \times (0;T)$, $r=1,2$. In $Q$ there is considered a problem $$L^{(r)}u = \frac{\partial}{\partial t} u^{(r)} - k^{(r)}(x,t)u^{(r)}_{tt} + a^{(r)}(x,t)u^{(r)} + A^{(r)}u^{(r)} = F^{(r)} \quad (x,t) \in Q_{\sigma T},$$ $$B_1 u = \lambda^{(1)}(x,t)u^{(1)} - \lambda^{(2)}(x,t)u^{(2)} = F^1 \quad (x,t) \in Q_{2T},$$ $$B_2 u = \frac{\partial u^{(1)}}{\partial N^{(1)}} - \frac{\partial u^{(2)}}{\partial N^{(2)}} = F^2 \quad (x,t) \in Q_{2T}, (1)$$ $$u^{(2)}(x,t) = F \quad (x,t) \in Q_{2T},$$ $$u^{(r)}|_{t=0} = F^{(r)}_0, \quad u^{(r)}|_{\partial Q_r} = F^{(r)}_1, \quad r=1,2,$$ where $$A^{(r)}u = \sum_{i,j=1}^n a_{ij}^{(r)}(x,t) u_{x_i x_j} + \sum_{i=1}^n a_i^{(r)}(x,t)u_{x_i} + a^{(r)}(x,t)u,$$ are uniformly elliptic operators, $\frac{\partial}{\partial N^{(r)}} = \sum_{i=1}^n a_{ij}^{(r)}(x,t) \nu_j \frac{\partial}{\partial x_i}$, $\vec{\nu}$ is a normal to $Q_{1T}$ ($Q_{2T}$), which is external to $Q_{0T}$, ($Q$), all coefficients of the operators are infinite differentiable in the domains, $k^{(r)}(x,t)$ arbitrarily change the sign within $Q$, $k^{(r)}(x,0) \le 0$, $k^{(r)}(x,T) \le 0$, $P_\sigma = \{(x,0): x \in \Omega_r$, $k^{(r)}(x,0)>0\}$, by analogy we define $P_{\sigma T}^-$, $r=1,2$, $u(x_0, t) = u^N(x,t)$, $(x,t) \in \overline{Q_{\sigma r}}$.