Our aim in this paper is to study a Cahn–Hilliard model with a symport term. This equation is proposed to model some energy mechanisms (e.g., lactate) in glial cells. The main difficulty is to prove the existence of a biologically relevant solution. This is achieved by considering a modified equation and taking a logarithmic nonlinear term. A second difficulty is to prove additional regularity on the solutions which is essential to prove a strict separation from the pure states 0 and 1 in one and two space dimensions. We also consider a second model, based on the Cahn–Hilliard–Oono equation.

中文翻译：

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胶质细胞的Cahn–Hilliard模型

我们本文的目的是研究带有同向项的Cahn-Hilliard模型。提出该方程以对神经胶质细胞中的某些能量机制（例如乳酸）建模。主要困难是证明存在生物学相关解决方案。这是通过考虑修改的方程式并采用对数非线性项来实现的。第二个困难是证明解决方案的附加规则性，这对于证明在一个和两个空间维度上与纯状态0和1严格分离至关重要。我们还考虑了基于Cahn–Hilliard–Oono方程的第二个模型。