Problem With Shift for a "Pointwise" Loaded Hyperbolic-Parabolic Equation
The paper considers a loaded equation model of mixed hyperbolic-parabolic type. The load is a linear combination of the desired solution at fixed points in its domain. A distinguishing feature of the considered equation is that the loaded terms can occur on the domain boundary and the line of changing type, as well as on the interior points in the domain. When describing physical and biological processes, "pointwise" loaded terms determine an effect of a discrete character at the points of "loading". For simplification, the coefficients of the loaded terms are constants, nevertheless variable coefficients do not affect the results. For the considered equation, a nonlocal boundary value problem with a displacement condition in the hyperbolic part of the domain and a problem with displacement derivatives are investigated. Conditions for the existence and uniqueness of a regular solution to the problem are found. Special cases of the equation are considered with the solution to the problem written out explicitly.