细胞侵入细胞外基质的体积填充模型弱解的存在性
Existence of weak solutions for a volume-filling model of cell invasion into extracellular matrix
AI总结：
该研究针对细胞侵入细胞外基质的体积填充耦合模型，利用部分梯度流结构证明了弱解的存在性，并通过有限差分模拟验证行波解、分析降解率趋于无穷时的渐近行为。
摘要：
我们研究了细胞侵入细胞外基质（ECM）模型的弱解存在性，该模型由描述细胞密度的非线性偏微分方程，与描述ECM密度的常微分方程（ODE）耦合构成。模型包含跨物种密度依赖的扩散项和增殖项，这些项体现了ECM在细胞侵入过程中为细胞提供结构支撑，同时通过体积填充效应抑制细胞生长的作用。此外，模型还纳入了细胞对ECM的降解作用。我们通过巧妙利用该问题的部分梯度流结构，克服了所涉及ODE的非正则化特性，得到了弱解的存在性结果。此外，我们基于有限差分格式开展了数值模拟，结果表明该系统存在行波解，并且我们通过数值方法研究了当ECM降解率趋于无穷大时系统的渐近行为。
英文摘要：
We study the existence of weak solutions for a model of cell invasion into the extracellular matrix (ECM), which consists of a non-linear partial differential equation for the density of cells, coupled with an ordinary differential equation (ODE) describing the ECM density. The model contains cross-species density-dependent diffusion and proliferation terms that capture the role of the ECM in providing structural support for the cells during invasion while also preventing growth via volume-filling effects. Furthermore, the model includes ECM degradation by the cells. We present an existence result for weak solutions which is based on carefully exploiting the partial gradient flow structure of the problem which allows us to overcome the non-regularising nature of the ODE involved. In addition, we present simulations based on a finite difference scheme that illustrate that the system exhibits travelling wave solutions, and we investigate numerically the asymptotic behaviour as the ECM degradation rate tends to infinity.
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