Stability of constant states and qualitative behavior of solutions to a one dimensional hyperbolic model of chemotaxis

Francesca Romana Guarguaglini; Corrado Mascia; Roberto Natalini; Magali Ribot

doi:10.3934/dcdsb.2009.12.39

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2009, Volume 12, Issue 1: 39-76. Doi: 10.3934/dcdsb.2009.12.39

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Stability of constant states and qualitative behavior of solutions to a one dimensional hyperbolic model of chemotaxis

1.
Dipartimento di Matematica Pura e Applicata, Universitá degli Studi di L’Aquila, Via Vetoio, I–67100 Coppito (L’Aquila)
2.
Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Piazzale A. Moro, 2, I–00185 Roma
3.
Istituto per le Applicazioni del Calcolo “Mauro Picone”, Consiglio Nazionale delle Ricerche, c/o Department of Mathematics, University of Rome “Tor Vergata”, Via della Ricerca Scientifica, 1; I-00133 Roma
4.
Laboratoire J. A. Dieudonné, Université de Nice-Sophia Antipolis, Parc Valrose, F-06108 Nice Cedex 02

Received: August 31, 2008

Revised: January 31, 2009

Published: June 2009

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Abstract

We consider a general model of chemotaxis with finite speed of propagation in one space dimension. For this model we establish a general result of stability of some constant states both for the Cauchy problem on the whole real line and for the Neumann problem on a bounded interval. These results are obtained using the linearized operators and the accurate analysis of their nonlinear perturbations. Numerical schemes are proposed to approximate these equations, and the expected qualitative behavior for large times is compared to several numerical tests.

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Mathematics Subject Classification: Primary: 35L60; Secondary: 35L50, 92B05, 92C17.

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Stability of constant states and qualitative behavior of solutions to a one dimensional hyperbolic model of chemotaxis

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