Boundedness of Solutions of a Haptotaxis Model

Abstract

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-di®usion equations in divergence form with a diagonal di®usion 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 di®usion coe±cient. 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-di®usion extension.

Boundedness in the higher-dimensional chemotaxis–haptotaxis model with nonlinear diffusion

“…Global existence and asymptotic behavior of solution were studied in [12,13,27]. Compared with the chemotaxis-only system and the haptotaxis-only system, the coupled chemotaxis-haptotaxis system (1.1) is much less understood.…”

Boundedness in a quasilinear chemotaxis–haptotaxis system with logistic source

“…In addition, in the case that g(w) = w and C S2 is a continuous and positive function of w, Marciniak-Czochra and Ptashnyk [22] proved the solutions of (1.10) globally exist and are uniformly bounded.…”

Global Weak Solutions in a PDE-ODE System Modeling Multiscale Cancer Cell Invasion

“…Here the lack of diffusibility of v is reflected in the absence of any regularizing effect on v during evolution. Correspondingly, the mathematical literature on (1.3) is comparatively thin, the main contributions on haptotaxis systems concentrating on particular choices of f [6,38] or on modified variants involving additional dampening effects such as logistic growth inhibition [21,17]. More recently, certain combined chemotaxis-haptotaxis models with linear cell diffusion and standard cross-diffusive terms as in (1.2) and (1.3), have been investigated, and some results on global existence and also on asymptotic solution behavior could be gained for various special choices of f in the respective ODE ∂ t v = f (c, v, l) [29,33,35,36,12].…”