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We consider a diffuse interface model for tumour growth consisting of aCahn--Hilliard equation with source terms coupled to a reaction-diffusionequation. The coupled system of partial differential equations models a tumourgrowing in the presence of a nutrient species and surrounded by healthy tissue.The model also takes into account transport mechanisms such as chemotaxis andactive transport. We establish well-posedness results for the tumour model anda variant with a quasi-static nutrient. It will turn out that the presence ofthe source terms in the Cahn--Hilliard equation leads to new difficulties whenone aims to derive a priori estimates. However, we are able to prove continuousdependence on initial and boundary data for the chemical potential and for theorder parameter in strong norms.

Well-posedness of a Cahn--Hilliard system modelling tumour growth with chemotaxis and active transport