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October  2010, 14(3): 1055-1080. doi: 10.3934/dcdsb.2010.14.1055

Mathematical analysis of a kinetic model for cell movement in network tissues

Thomas Hillen 1, , Peter Hinow 2, and  Zhi-An Wang 3,

1. 

Department of Mathematical and Statistical Sciences, Centre for Mathematical Biology, University of Alberta, Edmonton, T6G 2G1
2. 

Department of Mathematical Sciences, University of Wisconsin – Milwaukee, P.O. Box 413, Milwaukee, WI 53201-0413, United States
3. 

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, United States

Received  March 2009 Revised  March 2010 Published  July 2010

Mesenchymal motion describes the movement of cells in biological tissues formed by fibre networks. An important example is the migration of tumour cells through collagen networks during the process of metastasis formation. We investigate the mesenchymal motion model proposed by T. Hillen in [14] in higher dimensions. We formulate the problem as an evolution equation in a Banach space of measure-valued functions and use methods from semigroup theory to show the global existence of mild and classical solutions. We investigate steady states of the model and show that patterns of network type exist as steady states. For the case of constant fibre distribution, we find an explicit solution and we prove the convergence to the parabolic limit.
Keywords: pointwise steady states., parabolic limits, Mesenchymal motion, kinetic theory.
Mathematics Subject Classification: Primary: 35L03; Secondary: 92C1.
Citation: Thomas Hillen, Peter Hinow, Zhi-An Wang. Mathematical analysis of a kinetic model for cell movement in network tissues. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1055-1080. doi: 10.3934/dcdsb.2010.14.1055
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