On a nonlocal boundary value problem for a parabolic-hyperbolic equation with two perpendicular lines of type change

The paper addresses a boundary value problem with a shift for a parabolic-hyperbolic equation involving a spectral parameter and characterized by the presence of two mutually perpendicular lines of type change. The domain under consideration is composed of subdomains in which the equation changes type—being parabolic in some regions and hyperbolic in others—making the problem particularly interesting and analytically challenging. Additionally, the equation includes discontinuous coefficients, which necessitates a special formulation of the transmission (gluing) conditions at the interfaces between regions of different types. The study aims to formulate and justify the wellposedness (existence and uniqueness of the solution) of the boundary value problem with a shift in a complex geometrical domain for an arbitrary real value of the spectral parameter $\lambda$. The authors employ the method of integral transforms and operator theory, including integral and integro-differential operators of the form $A^{n,\lambda}_{mx}$, $B^{n,\lambda}_{mx}$, and $C^{n,\lambda}_{mx}$, whose properties play a key role in the analysis. It is shown that the original problem can be reduced to an equivalent system of Fredholm integral equations of the second kind with continuous kernels. The solvability of this system is established using the Fredholm alternative theorem. Furthermore, it is demonstrated that the solution can be represented explicitly in terms of integral formulas involving Bessel functions and special kernels that capture the behavior of the solution in various parts of the domain. Matching conditions play a crucial role in ensuring the correct joining of solutions across regions where the type of the equation changes and along the line of coefficient discontinuity.