Travelling-wave analysis of a model describing tissue degradation by bacteria
Danielle Hilhorst, John R. King, and Matthias Röger
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Submission date: 28. Feb. 2007
published in: European journal of applied mathematics, 18 (2007) 5, p. 583-605
DOI number (of the published article): 10.1017/S0956792507007139
MSC-Numbers: 35K57, 92E20, 35B25, 80A22
Keywords and phrases: Travelling waves, reaction-diffusion system, Stefan problem
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We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the `degradation-rate parameter' solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave-speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range, nonlinear selection of the minimal speed occurs.