On a class of non-local phase-field models for tumor growth with possibly singular potentials, chemotaxis, and active transport

Abstract

Abstract This paper provides a unified mathematical analysis of a family of non-local diffuse interface models for tumor growth describing evolutions driven by long-range interactions. These integro-partial differential equations model cell-to-cell adhesion by a non-local term and may be seen as non-local variants of the corresponding local model proposed by Garcke et al (2016). The model in consideration couples a non-local Cahn–Hilliard equation for the tumor phase variable with a reaction–diffusion equation for the nutrient concentration, and takes into account also significant mechanisms such as chemotaxis and active transport. The system depends on two relaxation parameters: a viscosity coefficient and parabolic-regularization coefficient on the chemical potential. The first part of the paper is devoted to the analysis of the system with both regularizations. Here, a rich spectrum of results is presented. Weak well-posedness is first addressed, also including singular potentials. Then, under suitable conditions, existence of strong solutions enjoying the separation property is proved. This allows also to obtain a refined stability estimate with respect to the data, including both chemotaxis and active transport. The second part of the paper is devoted to the study of the asymptotic behavior of the system as the relaxation parameters vanish. The asymptotics are analyzed when the parameters approach zero both separately and jointly, and exact error estimates are obtained. As a by-product, well-posedness of the corresponding limit systems is established.