On solvability of a conjugation problem with second-order derivative with respect to the spatial variable in the conjugation condition for second-order parabolic equation (Q2735901)

Let \(\Omega_s\) be two bounded domains in \(\mathbb{R}^2\) and its boundaries consist from two curves \(\Gamma_s\), \(\Gamma_3\), which do not have common points and self-intersections \((s=1,2)\). The problem under consideration is to find the triple of functions \(u_1\), \(u_2\), \(u\), that are defined on \(\Omega_1\), \(\Omega_2\), \(\Gamma_3\) accordingly such that NEWLINE\[NEWLINE\frac {\partial u_s} {\partial t}- L_s u_s= f_{os}, \quad (x,t)\in \Omega_{sT},NEWLINE\]NEWLINE NEWLINE\[NEWLINEq(\omega,t) \frac{\partial u_1} {\partial n} -q(\omega,t) \frac{\partial u_2} {\partial n}+ \beta(\omega,t) \frac{\partial^2 u} {\partial \omega^2}= f_1, \quad (x,t)\in \Gamma_{3T},NEWLINE\]NEWLINE NEWLINE\[NEWLINEu_1= u_2=u, \qquad (x,t)\in \Gamma_{3T},NEWLINE\]NEWLINE NEWLINE\[NEWLINEu_s(x,0)= \psi_s(x,t), \quad (x,t) \in \Gamma_{sT}, NEWLINE\]NEWLINE here \(\frac {\partial} {\partial t}-L_s\) is the linear uniformly parabolic operator, \(\omega\) is some coordinate on \(\Gamma_3\). NEWLINENEWLINENEWLINEThe main result is a theorem of existence and uniqueness of smooth solutions in appropriate Hölder classes.