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.
Highlights
IntroductionA vivid interest has been devoted to the challenging project of modeling tumor growth
In the last decades, a vivid interest has been devoted to the challenging project of modeling tumor growth
This paper provides a unified mathematical analysis of a family of non-local diffuse interface models for tumor growth describing evolutions driven by longrange interactions
Summary
A vivid interest has been devoted to the challenging project of modeling tumor growth. The two results deal with the asymptotic behavior as ε 0 and the respective error estimate: as a byproduct, these yield existence and uniqueness of solutions, as well as continuous dependence on the data, for the system (1.4)–(1.8) with ε = 0. For every ε ∈ (0, ε0), let the initial data (φ0,ετ , μ0,ετ , σ0,ετ ) satisfy assumptions (2.2) and (2.9), and denote by (φετ , μετ , σετ , ξετ ) the respective unique weak solution to the system (1.4)–(1.8) obtained from theorem 2.1. For every ε ∈ (0, ε0) and τ ∈ (0, τ0), let the initial data (φ0,ετ , μ0,ετ , σ0,ετ ) satisfy (2.2) and (2.9), and denote by (φετ , μετ , σετ , ξετ ) the respective unique weak solution to the system (1.4)–(1.8) obtained from theorem 2.1. Let us recall that throughout this section ε, τ > 0 are fixed
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
