Strong well-posedness and inverse identification problem of a non-local phase field tumor model with degenerate mobilities

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.