In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn-Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3-24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn-Hilliard equation for the tumor cell fraction, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases.

中文翻译：

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带分数算子的肿瘤生长模型的渐近分析

在本文中，我们研究了一个由三个演化算子方程组成的系统，其中涉及具有紧解算的自伴随、单调、无界、线性算子的分数幂。该系统构成了最初由 Hawkins-Daarud 等人提出的 Cahn-Hilliard 型模拟肿瘤生长的相场系统的广义和松弛版本。(Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3-24)。最近由本作者和 E. Rocca 合着的论文研究了原始相场系统及其某些宽松版本。该模型由用于肿瘤细胞分数的 Cahn-Hilliard 方程组成，并与代表富含营养的细胞外水体积分数的函数 S 的反应扩散方程相结合。由于流体运动的影响被忽略。受控制模型不同成分演化的扩散机制可能具有不同（例如，分数）类型的可能性的推动，本作者在最近的一篇笔记中研究了上述作品中研究的系统的概括。在相当一般的假设下，已显示适定性和规律性结果。特别是，通过以一般变分不等式的形式编写控制化学势演化的方程，还可以承认对数或双障碍类型对能量密度的奇异或非光滑贡献。在本说明中，我们对控制系统进行渐近分析，因为两个（小）松弛参数分别并同时趋近于零。对各个情况建立相应的适定性和正则性结果；特别是，我们详细讨论了在每个发生的情况下必须假设哪些关于可容许非线性的假设。