Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion (Q2878989)

The authors have proved the global existence of weak solution to the model of tumor cells multiscale invasion through surrounding tissue, presented in the form of a PDE-ODE system in dimensionless variables in their preprint [\textit{G. Meral} et al., On a multiscale model involving cell contractivity and its effects on tumor invasion. Kaiserslautern: TU Kaiserslautern] NEWLINE\[NEWLINE \begin{cases} \partial _t c =\nabla \cdot \left(\frac{\kappa}{1+cv}\nabla c\right)-\nabla \cdot \left(\frac{\kappa v}{1+v}c\nabla v\right) -\nabla \cdot \left(\frac{c}{1+cl}\nabla l\right)+\mu_c c(1-c-\eta_1 v), \; x\in\Omega, \;t>0, \\ \partial _t v = \mu_vv(1 - v) - \lambda cv, \; x\in\Omega, \;t>0, \\ \partial _t l =\Delta l - l + cv, \; x\in\Omega, \; t>0, \\ \partial _t y_1 = k_1(1 - y_1 - y_2)v - k_{-1}y_1, \; x\in\Omega, \;t>0, \\ \partial _t y_2 = k_2(1 - y_1 - y_2)l-k_{-2}y_2, \; x\in\Omega,\; t>0, \\ \partial _t \kappa = -\kappa + \frac{My_1(\cdot,t-\tau)}{1+y_2(\cdot,t-\tau)}, \; x\in\Omega, \;t>0, \end{cases}NEWLINE\]NEWLINE where \(C(x.t)\) is the cancer cells density; \(v(x,t)\) -- the density of tissue fibers in the extracellular matrix (ECM); \(l(x,t)\) -- the concentration of chemoattractant (i.e., proteolytic rests resulting from degradation of ECM fibers by matrix degrading enzymes produced by the tumor cells); \(y_1(x, t) \;(y_2(x, t))\) -- the concentration of integrins bound to ECM fibers; (concentration of integrins bound to proteolytic residuals); \(\kappa(x, t)\) is the contractivity function; \(\mu_c\) -- the proliferation rate of tumor cells; \(\mu_v\) -- the production/reestablishment rate of ECM fibers; \(\delta\) -- the degradation rate of ECM fibers due to the action of matrix degrading onzymes; \(\eta_1, \eta_2\) are interaction rates allowing crowding to be taken into account; \(k_{-1}, k_1, k_{-2},k_2\) are reaction rates in the binding of integrins to insoluble (ECM fibers) and soluble (proteolytic rests) ligands in the peritumoral environment; with abbreviatere \(\lambda=\delta+\mu_v \eta_2\).NEWLINENEWLINEThis model takes into account the macroscopic level for the cell and tissue densities evolution with subcellular level such as the binding of integrins to soluble and insoluble components in the peritumoral region and also the interaction between them, contains the temporal delay stiplated by chemotactic and haptotactic cross diffusion.