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

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