Global Existence of Classical Solutions to a Cancer Invasion Model

Khadijeh Baghaei,Mahmoud Hesaaraki,Mohammad Bagher Ghaemi

doi:10.4236/am.2012.34059

Khadijeh Baghaei, Mahmoud Hesaaraki + Show 1 more

Open Access

https://doi.org/10.4236/am.2012.34059

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Journal: Applied mathematics	Publication Date: Jan 1, 2012
Citations: 15	License type: CC BY 4.0

Affiliation: Sharif University of Technology

Abstract

This paper deals with a chemotaxis-haptotaxis model of cancer invasion of tissue. The model consists of three reaction- diffusion- taxis partial differential equations describing interactions between cancer cells, matrix degrading enzymes, and the host tissue. The equation for cell density includes two bounded nonlinear density-dependent chemotactic and haptotactic sensitivity functions. In the absence of logistic damping, we prove the global existence of a unique classical solution to this model by some delicate a priori estimate techniques

Highlights

This paper deals with a chemotaxis-haptotaxis model of cancer invasion of tissue
In the absence of logistic damping, we prove the global existence of a unique classical solution to this model by some delicate a priori estimate techniques
Cancer invasion is associated with the degradation of the extra cellular matrix (ECM), which is degraded by matrix degrading enzymes (MDEs) secreted by tumor cells

Summary

Introduction

Cancer invasion is associated with the degradation of the extra cellular matrix (ECM), which is degraded by matrix degrading enzymes (MDEs) secreted by tumor cells. Chaplain and Lolas [1] proposed a PDE model of cancer invasion of tissue, which considers the competition between the following several biological mechanisms: random diffusion, chemotaxis, haptotaxis and logistic growth. Walker and Webb [8] proved the global existence solutions to the Chaplain and Anderson’s model [9]. In addition to global existence and uniqueness, the uniform-in-time boundedness of solutions to a simplified parabolic-ODE-elliptic-chemotaxis-haptotaxis system has been proved for 0 in two space dimensions and for large in three space dimensions (see [17]). This paper extends Chaplain and Lolas’ model to a parabolic-parabolic-parabolic chemotaxis-haptotaxis system, and we study the global existence and boundedness of solutions to this model.

Mathematical Model

Local Existence and Uniqueness

A Priori Estimates and Global Existence