Conjugation boundary value problem on the plane. II (Q2761537)

Let \(\mathbb{R}^2=\overline\Omega_1\cup\overline\Omega_2\), \(\partial\overline\Omega_1\cap\partial\overline\Omega_2=l\), where \(\Omega_1\) is unbounded and doubly-connected; \(\Omega_2\) is a simple one-connected compact with the boundary \(\partial\Omega=l\), \(l\) is a smooth closed contour. The authors study the conjugation problem for semilinear parabolic type partial differential equation \(u_{t}=L^{(i)}u+F_{i}(u,u_{x})\), \(i=1,2\) with boundary conditions \(u|_{t=t_0}=\phi_{i}(x), x\in\Omega_{i}, i=1,2\); continuity conditions on \(l\subset \mathbb{R}^{2}:\) NEWLINE\[NEWLINE\lim_{x\to x_0\in l}u(t;x)|_{x\in\Omega_1}=\lim_{x\to x_0\in l}u(t;x)|_{x\in\Omega_2},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\lim_{x\to x_0\in l}\langle A^{(1)}\vec n_{x},{\text{grad}} u \rangle|_{x\in\Omega_1}=\lim_{x\to x_0\in l}\langle A^{(2)}\vec n_{x},{\text{grad}} u \rangle|_{x\in\Omega_2}NEWLINE\]NEWLINE for any \(x_0\in l\), where \(\vec n_{x}\) is a unit vector of the normal to \(l\) at the point \(x\in l\). Here \(L^{(i)}=\langle\vec\partial, A^{(i)}\vec\partial\rangle\), \(i=1,2\) are elliptic differential operators with constant coefficients; \(F_{i}\in C^{(1)}, i=1,2\) are some Frechet differentiable nonlinear mappings. The existence and uniqueness of solution of the considered problem is proved. The integral representation of solutions is obtained. The generalization of this result for \(n\)-connected domain in \(\mathbb{R}^2\) is presented. NEWLINENEWLINENEWLINEFor part I see ibid. 5, 166-171 (2000; Zbl 0979.35079).