  We extend previous weak well-posedness results obtained in Frigeri et al.
(2017) concerning a non-local variant of a diffuse interface tumor model
proposed by Hawkins-Daarud et al. (2012). The model consists of a non-local
Cahn--Hilliard equation with degenerate mobility and singular potential for the
phase field variable, coupled to a reaction-diffusion equation for the
concentration of a nutrient. We prove the existence of strong solutions to the
model and establish some high order continuous dependence estimates, even in
the presence of concentration-dependent mobilities for the nutrient variable in
two spatial dimensions. Then, we apply the new regularity results to study an
inverse problem identifying the initial tumor distribution from measurements at
the terminal time. Formulating the Tikhonov regularised inverse problem as a
constrained minimisation problem, we establish the existence of minimisers and
derive first-order necessary optimality conditions.
