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**Global solution for a chemotactic-haptotactic model of cancer invasion.**
*(English)*
Zbl 1160.35431

Tumour growth is a highly complex process. It includes the initial avascular phase of growth, angiogenesis - the formation of new blood vessels, and the vascular phase of growth. Tumour growth has been extensively studied, but tumor invasion is relatively a new area for mathematical modeling. It is associated with the degradation of the extracellular matrix (ECM), which is degraded by the proteolytic enzymes secreted by tumour cells. The authors analyse a mathematical model of cancer invasion of tissue consisting of a reaction-diffusion-taxis partial differential equation (PDE) describing the evolution of tumour cell density, a reaction-diffusion PDE governing the evolution of the proteolytic enzyme concentration and an ordinary differential equation modeling the proteolysis of the ECM. In addition to random motion, the tumour cells are directed not only by haptotaxis, but also by chemotaxis. The global existence and uniqueness of a classical solution of this combined chemotactic-haptotactic model is proved in one space dimension. The global existence is proved in two and three space dimensions under a certain restriction \(\frac{\chi}{\mu}\) small, a necessary condition for the global existence of a solution of the full model.

Reviewer: J. M. Tchuenche (Dar es Salaam)

### MSC:

35K45 | Initial value problems for second-order parabolic systems |

35G25 | Initial value problems for nonlinear higher-order PDEs |

92C17 | Cell movement (chemotaxis, etc.) |

35K57 | Reaction-diffusion equations |