Space-time-dependent source identification problem for a subdiffusion equation

In this paper, we investigate the inverse problem of determining the right-hand side of a subdiffusion equation with a Caputo time derivative, where the right-hand side depends on both time and certain spatial variables. Similar inverse problems have been previously explored for hyperbolic and parabolic equations, with some studies establishing the existence and uniqueness of generalized solutions, while others proved the uniqueness of classical solutions. However, such inverse problems for fractional-order equations have not been addressed prior to this work. Here, we establish the existence and uniqueness of the weak solution to the considered inverse problem. To solve it, we employ the Fourier method with respect to the variable independent of the unknown right-hand side, followed by the method of successive approximations to compute the Fourier coefficients of the solution. Notably, the results obtained are also novel for parabolic equations.