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

This paper investigates a mixed-type partial differential equation involving the Caputo fractional derivative of order ρ ∈ (0,1) for t > 0, and a classical parabolic equation for t < 0. The problem is studied in an arbitrary N-dimensional domain Ω with smooth boundary, subject to Dezin-type non-local boundary and gluing conditions. For the forward problem, existence and uniqueness of the classical solution are established under suitable assumptions on the data, employing the Fourier method. The influence of the parameter λ in the non-local boundary condition on solvability is analyzed. Additionally, an inverse problem is considered, where the source term is separable as F(x,t) = f(x)g(t), with known g(t) and unknown spatial function f(x). Under certain conditions on g(t), the uniqueness and existence of the solution are proven. This work extends previous results on mixed-type equations, highlighting the role of fractional derivatives and nonlocal conditions in both forward and inverse settings. The findings contribute to the theory of mixed-type and fractional differential equations, with potential applications in subdiffusion and related processes.