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Boundedness and stabilization in a multi-dimensional chemotaxis—haptotaxis model

Published online by Cambridge University Press:  03 October 2014

Youshan Tao  and
Michael Winkler
Youshan Tao
Affiliation:
Department of Applied Mathematics, Dong Hua University, Shanghai 200051, People's Republic of China, ([email protected])
Michael Winkler
Affiliation:
Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany, ([email protected])
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Abstract

This paper deals with the coupled chemotaxis-haptotaxis model of cancer invasion given by

where χ, ξ and μ are positive parameters and Ω ⊂ ℝn (n ≥ 1) is a bounded domain with smooth boundary. Under zero-flux boundary conditions, it is shown that, for any μ > χ and any sufficiently smooth initial data (u0, w0) satisfying u0 ≥ 0 and w0 > 0, the associated initial–boundary-value problem possesses a unique global smooth solution that is uniformly bounded. Moreover, we analyse the stability and attractivity properties of the non-trivial homogeneous equilibrium (u, v, w) ≡ (1,1, 0) and establish a quantitative result relating the domain of attraction of this steady state to the size of μ. In particular, this will imply that whenever u0 > 0 and 0 < w0 < 1 in there exists a positive constant μ* depending only on χ, ξ, Ω, u0 and w0 such that for any μ < μ* the above global solution (u, v, w) approaches the spatially uniform state (1, 1, 0) as time goes to infinity.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 5 , October 2014 , pp. 1067 - 1084
Copyright
Copyright © Royal Society of Edinburgh 2014 

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