PDE problems arising in mathematical biology

Avner Friedman

doi:10.3934/nhm.2012.7.691

Article Contents

2012, Volume 7, Issue 4: 691-703. Doi: 10.3934/nhm.2012.7.691

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PDE problems arising in mathematical biology

Avner Friedman^1,

1.
The Ohio State University, Department of Mathematics, Columbus, OH 43210

Received: February 29, 2012

Revised: August 31, 2012

Published: November 2012

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Abstract

This article reviews biological processes that can be modeled by PDEs, it describes mathematical results, and suggests open problems. The first topic deals with tumor growth. This is modeled as a free boundary problem for a coupled system of elliptic, hyperbolic and parabolic equations. Existence theorems, stability of radially symmetric stationary solutions, and symmetry-breaking bifurcation results are stated. Next, a free boundary problem for wound healing is described, again involving a coupled system of PDEs. Other topics include movement of molecules in a neuron, modeled as a system of reaction-hyperbolic equations, and competition for resources, modeled as a system of reaction-diffusion equations.

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Mathematics Subject Classification: Primary: 49J10, 35Q92, 35R35, 92C50; Secondary: 35J47, 35K57, 35L40, 35M30.

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