Correct solvability of parabolic conjugation problems in the space of growing functions (Q1107029)

Denote \(Q_0\subset R^{n+1}\), \(Q_0=Q^1_0 \cup Q^2_0\), \(Q^1_0\cap Q^2_0=\emptyset\). The following conjugation problem is considered: \[ \begin{multlined} \sum^{N^{\nu}}_{j=1} (A^\nu_{ij} u_ j^\nu)(t,x) \equiv D^{n_ i^{\nu_ t}} u_ i^\nu(t,x) - \sum^{N^\nu}_{j=1} \sum_{|\bar k|\leq2bn_ j^\nu} a_ k^{\nu ij} (t,x) D^{\bar k}_{t,x} u_ j^\nu (t,x) = f^{\nu}_{0i}(t,x); \\ (t,x)\in Q_0^\nu,\quad i=1,\ldots,N^\nu,\quad \nu =1,2 \end{multlined}\tag{1} \] \[ \sum^2_{\nu =1} \sum^{N^\nu}_{j=1} S^\nu_{ij} u_ j^\nu |_{Q^1_1} \equiv \sum^2_{\nu =1} \sum^{N^\nu}_{j=1} \sum_{|\bar k|\leq2bn_ j^\nu+\sigma_ i^1} s_{\bar k}^{\nu ij} (t,x) D^{\bar k}_{t,x} u_ j^{\nu}|_{Q^1_1} = f^1_{1i}; \quad i=1,\ldots,m^1+m^2 \tag{2} \] \[ \sum^{N^2}_{j=1} B_{ij} u^2_ j |_{Q^2_1} = \sum^{N^2}_{j=1} \sum_{|\bar k|\leq2bn^2_ j+\sigma^2_ i} b_{\bar k}^{ij} (t,x) D^{\bar k}_{t,x} u^2_ j |_{Q^2_1} = f^2_{1i}, \qquad i = 1,\ldots,m^2 \tag{3} \] \[ D^{\varrho^\nu_ t} u^\nu_ i (t,x) |_{t=0} = \varphi^{\nu\varrho^\nu} (x), \quad x\in\Omega^0_{0\nu}, \quad \varrho^\nu = 0,\ldots,n^\nu_ i-1, \quad i=1,\ldots,N^\nu, \quad \nu=1,2 \tag{4} \] where \(\bar k=(k_0,\ldots,k_ n)\), \(|\bar k|=2bk_0+ | k|\), \(D^{\bar k}_{t,x}=D^{k_0}_ t D^ k_ x\) and \(n^\nu_ j\), \(\sigma^\nu_ i\) are integers. Under certain circumstances the system (1)--(4) admits a unique solution which is written in an integro-differential representation. One indicates sufficient conditions of correct solvability for the above problem.