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

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.