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 etal (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.

本文对描述由长程相互作用驱动的进化的肿瘤生长的一系列非局部弥散界面模型进行了统一的数学分析。这些积分偏微分方程通过非局部项对细胞间粘附进行建模，并且可以看作是 Garcke等人提出的相应局部模型的非局部变体(2016)。所考虑的模型将肿瘤时相变量的非局部 Cahn-Hilliard 方程与营养浓度的反应扩散方程相结合，并且还考虑了诸如趋化性和主动转运等重要机制。该系统取决于两个松弛参数：化学势的粘度系数和抛物线正则化系数。论文的第一部分专门分析具有两种正则化的系统。在这里，展示了丰富的结果。首先解决弱适定性，也包括奇异势。然后，在合适的条件下，证明了具有分离性质的强解的存在。这也允许获得关于数据的精确稳定性估计，包括趋化性和主动转运。论文的第二部分致力于研究系统在松弛参数消失时的渐近行为。当参数分别和联合接近零时，分析渐近线，并获得精确的误差估计。作为副产品，建立了相应限制系统的适定性。