On the Inverse Problem for a Degenerate Parabolic Equation with the Hadamard–Caputo Derivative Under Samarskii–Ionkin Conditions

Abstract

This paper investigates an inverse problem of recovering the spatial component of a source term for a degenerate ultraslow diffusion equation with a Hadamard–Caputo fractional derivative. Nonlocal boundary conditions of the Samarskii–Ionkin type are imposed. A distinctive feature of the problem is that the system of eigenfunctions of the associated spectral problem, although complete and minimal, does not form an unconditional basis in L2(0,1). Hence the standard eigenfunction Fourier expansion cannot be applied directly. To overcome this difficulty, we construct a special auxiliary system from normalized linear combinations of asymptotically close eigenfunctions and prove that it forms a Riesz basis. Expansion with respect to this basis reduces the inverse problem to a cascade system of fractional differential equations for the modal coefficients. Using spectral asymptotics, estimates for the Le Roy function, and two-sided Riesz basis inequalities, we prove the existence, uniqueness, and conditional Lipschitz stability of a strong generalized solution and rigorously justify the convergence of the resulting series. We also quantify the high-frequency instability of the inverse mapping in the weaker L2 data topology. An analytical two-mode illustration shows explicitly how neglecting the coupling between asymptotically close spectral modes changes the reconstructed source coefficient.