Conservation laws in mathematical biology

Avner Friedman

doi:10.3934/dcds.2012.32.3081

Article Contents

2012, Volume 32, Issue 9: 3081-3097. Doi: 10.3934/dcds.2012.32.3081

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Conservation laws in mathematical biology

Avner Friedman^1,

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

Received: September 2011

Revised: March 2012

Published: September 2012

Abstract / Introduction Related Papers Cited by

Abstract

Many mathematical models in biology can be described by conservation laws of the form \begin{equation}\tag{0.1} \frac{\partial{\bf{u}}}{\partial t} + \rm{div}(V{\bf{u}})=F(t,{\bf{x}}, {\bf{u}})\quad ({\bf{x}}=(x_1,\dots, x_n)) \end{equation} where ${\bf{u}}={\bf{u}}(t,{\bf{x}})$ is a vector $(u_1,\dots,u_k)$, ${\bf{F}}$ is a vector $(F_1,\dots,F_k)$, $V$ is a matrix with elements $V_{ij}(t,{\bf{x}},{\bf{u}})$, and $F_i(t,{\bf{x}}, {\bf{u}})$, $V_{ij}(t,{\bf{x}}, {\bf{u}})$ are nonlinear and/or non-local functions of ${\bf{u}}$. From a mathematical point of view one would like to establish, first of all, the existence and uniqueness of solutions under some prescribed initial (and possibly also boundary) conditions. However, the more interesting questions relate to establishing properties of the solutions that are of biological interest.
In this article we give examples of biological processes whose mathematical models are represented in the form (0.1). We describe results and present open problems.

Keywords:

Mathematics Subject Classification: 35K57, 35L60, 35L65, 35R35, 92C17, 92C37, 92C50.

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