Inverse problem for a multi-term time-fractional diffusion equation with the Caputo derivatives

Abstract

This paper investigates an inverse source problem for a multi-term time-fractional diffusion equation with Caputo derivatives. The source term is separable as \(f(x)g(t)\), with the unknown spatial component \(f(x)\) reconstructed from an overdetermination condition at interior time \(t_0 \in (0, T]\). The elliptic part is governed by a self-adjoint positive differential operator \(A(x, D)\) of order \(m \ge 2\). The solution features a spectral representation using the multinomial Mittag-Leffler function, for which we derive novel precise asymptotic expansions. These asymptotics provide a uniform lower bound for the solution's characteristic denominator, enabling sufficient conditions for the existence of a classical solution. Uniqueness of the reconstructed source holds under natural assumptions on the data and \(g(t)\). Despite the problem's ill-posedness, high-regularity classical solutions are achievable under suitable structural conditions.