Forward and inverse problems for a mixed-type equation with the Caputo fractional derivative and Dezin-type non-local condition

This work is dedicated to the study of a mixed-type partial differential equation involving a Caputo fractional derivative in the time domain $t > 0$ and a classical parabolic equation in the domain $t < 0$, along with Dezin-type non-local boundary and gluing conditions. The forward and inverse problems are studied in detail. For the forward problem, the existence and uniqueness of solutions are established using the Fourier method, under appropriate assumptions on the initial data and the right-hand side. We also analyze the dependency of solvability on the parameter $\lambda$, from the Dezin-type condition. For the inverse problem, where the right-hand side is separable as $F(x,t) = f(x)g(t)$ (the unknown function is $f(x)$), the existence and uniqueness of a solution are proven under a certain condition on the function $g(t)$ (a constant sign is sufficient).