۱- Introduction

۲- Local existence of classical solutions

۳- Global boundedness

References

Introduction

Cancer invasion is associated with the degradation of the extracellular matrix (ECM), which is degraded by matrix degrading enzymes (MDEs) such as the urokinase-type plasminogen activator (uPA) secreted by cancer cells. The degradation provides space for cancer cells to proliferate and invade into the surrounding tissue. In addition to random diffusion, the migration of cancer cells is biased towards a gradient of the diffusible MDEs by chemotaxis which as a significant mechanism of directional migration of cells is the movement of cells in response to concentration gradients of a chemical signal emitted by the cells themselves in many biological process, and towards a gradient of the nondiffusible ECM through detecting the ECM material vitronectin VN adhered therein by haptotaxis (see [1]). In addition, the cancer cells undergo birth and death in a logistic manner, competing for space with the ECM. The MDE is assumed to be produced by cancer cells, and to diffuse and decay, whereas the ECM is assumed to be degraded up contact with MDE. The combination of these two cell migration mechanisms forms the core of the modeling approach pursued by Chaplain and Lolas to describe cancer cell invasion into surrounding healthy tissue. Except for the model [2], a variety of mathematical models describing the different stages of cancer invasion and metastasis have been developed before (see [1–۱۰]).