Boundedness of solutions of a haptotaxis model

In this paper we prove the existence of global solutions of the haptotaxis model of cancer invasion for arbitrary non-negative initial conditions. Uniform boundedness of the solutions is shown using the method of bounded invariant rectangles applied to the reformulated system of reaction-diffusion equations in divergence form with a diagonal diffusion matrix. Moreover, the analysis of the model shows how the structure of kinetics of the model is related to the growth properties of the solutions and how this growth depends on the ratio of the sensitivity function (describing the size of haptotaxis) and the diffusion coefficient. One of the implications of our analysis is that in the haptotaxis model with a logistic growth term, cell density may exceed the carrying capacity, which is impossible in the classical logistic equation and its reaction-diffusion extension.