Mathematical analysis of a model describing the invasion of bacteria in burn wounds

doi:10.1016/j.na.2006.01.009

Nonlinear Analysis: Theory, Methods & Applications

Volume 66, Issue 5, 1 March 2007, Pages 1118-1140

https://doi.org/10.1016/j.na.2006.01.009 Get rights and content

Abstract

We investigate a reaction–diffusion system for a parabolic equation coupled with an ordinary differential equation on an unbounded space domain. This system arises as a model for host tissue degradation by bacteria and involves a parameter describing the degradation rate that is typically very large. We prove the existence and uniqueness of solutions to this system and the convergence to a Stefan-like free boundary problem as the degradation rate tends to infinity.

Section snippets

Introduction and main results

The development of alternative treatments for bacterial infections has become a major issue in recent years. The formulation and analysis of suitable mathematical models can serve as a valuable tool in understanding the basic mechanisms underlying such infections and in providing insight into possible means of fighting the spread of bacteria in host tissue. We take our cue here from [12], where a mathematical model of host tissue degradation by extracellular bacteria was introduced and analysed

Existence of solutions for the RD system

In this section we prove Theorem 1. Instead of working in the upper half-space, it is convenient to consider the problem on the whole space. We extend the initial data to the whole of $R^{n}$ by ${\bar{u}}_{0} (x_{1}, \dots, x_{n}) ≔ {\bar{u}}_{0} (x_{1}, \dots, - x_{n}), {\bar{w}}_{0} (x_{1}, \dots, x_{n}) ≔ {\bar{w}}_{0} (x_{1}, \dots, - x_{n}),$ for $x = (x_{1}, \dots, x_{n})$ with $x_{n} < 0$ . Then, Theorem 1 will follow from the corresponding result for the whole space and from the uniqueness of solutions.

Moreover, in addition to (1.3), (1.4) we first assume that both initial data are continuous and have compact

The fast degradation rate limit

In this section we prove the convergence of the solutions $(u_{k}, w_{k})$ of Problem $(P_{k})$ to solutions of $(P_{\infty})$ . We show the uniqueness of solutions and give the classical formulation of the limit problem.

Proof of Theorem 2

The solutions $u_{k}, w_{k}$ of Problem $(P_{k})$ with initial data ${\bar{u}}_{0}, {\bar{w}}_{0}$ are bounded in $L^{2} (Q_{T})$ uniformly in $k \in N$ according to the estimates (2.7), (2.8), (1.13). Recalling the estimates for time and space differences (2.53), (2.54), (2.55), and invoking again the Theorem of Fréchet–Kolmogorov–Riesz (see for

Acknowledgements

The authors gratefully acknowledge funding from the EU Research Training Network FRONTS-SINGULARITIES, HPRN-CT-2002-00274. The third author thanks the Laboratoire de Mathématiques, Universitè de Paris-Sud and the School of Mathematical Sciences, University of Nottingham for their great hospitality during his visits there.

References (16)

E.C.M. Crooks et al.
Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions
Nonlinear Anal. Real World Appl.
(2004)
D. Hilhorst et al.
The fast reaction limit for a reaction-diffusion system
J. Math. Anal. Appl.
(1996)
D. Hilhorst et al.
Nonlinear diffusion in the presence of fast reaction
Nonlinear Anal.
(2000)
K. Anguige et al.
Modelling antibiotic- and anti-quorum sensing treatment of a spatially-structured pseudomonas aeruginosa population
J. Math. Biol.
(2005)
E.N. Dancer et al.
Spatial segregation limit of a competition-diffusion system
European J. Appl. Math.
(1999)
N. Dunford et al.
Linear operators. Part I
(1988)
R. Eymard et al.
A reaction-diffusion system approximation of a one-phase stefan problem
L.C. Evans
Partial differential equations

There are more references available in the full text version of this article.

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Mathematical analysis of a model describing the invasion of bacteria in burn wounds

Abstract

Section snippets

Introduction and main results

Existence of solutions for the RD system

The fast degradation rate limit

Acknowledgements

Nonlinear Anal. Real World Appl.

J. Math. Anal. Appl.

Nonlinear Anal.

Modelling antibiotic- and anti-quorum sensing treatment of a spatially-structured pseudomonas aeruginosa population

J. Math. Biol.

Spatial segregation limit of a competition-diffusion system

European J. Appl. Math.

Linear operators. Part I

A reaction-diffusion system approximation of a one-phase stefan problem

Partial differential equations