PDE problems arising in mathematical biology

Avner Friedman; Avner Friedman

doi:10.3934/nhm.2012.7.691

Networks and Heterogeneous Media

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: 01 March 2012 Revised: 01 September 2012
Primary: 49J10, 35Q92, 35R35, 92C50; Secondary: 35J47, 35K57, 35L40, 35M30.

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.
- Free boundary problems,
- reaction-hyperbolic equations,
- reaction-diffusion equations,
- tumor growth,
- wound healing
Citation: Avner Friedman. PDE problems arising in mathematical biology[J]. Networks and Heterogeneous Media, 2012, 7(4): 691-703. doi: 10.3934/nhm.2012.7.691

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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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