Global Existence and Boundedness of Solutions to a Model of Chemotaxis

Issue		Math. Model. Nat. Phenom. Volume 3, Number 7, 2008 Special issue dedicated to Glenn Webb


Page(s)		17 - 35
DOI		10.1051/mmnp:2008039
Published online		23 October 2008

Math. Model. Nat. Phenom. Vol. 3, No. 7, 2008, pp. 17-35
DOI: 10.1051/mmnp:2008039

Global Existence and Boundedness of Solutions to a Model of Chemotaxis

J. Dyson¹, R. Villella-Bressan² and G. F. Webb³

¹ Mansfield College, University of Oxford, Oxford, UK
² Dipartimento di Matematica Pura e Applicata, Universita' di Padova, Padova, Italy
³ Department of Mathematics, Vanderbilt University, Nashville, Tennessee

janet.dyson@mansfield.ox.ac.uk

Published online: 23 October 2008

Abstract
A model of chemotaxis is analyzed that prevents blow-up of solutions. The model consists of a system of nonlinear partial differential equations for the spatial population density of a species and the spatial concentration of a chemoattractant in n-dimensional space. We prove the existence of solutions, which exist globally, and are $L^{\infty}$ -bounded on finite time intervals. The hypotheses require nonlocal conditions on the species-induced production of the chemoattractant.

Mathematics Subject Classification. 92C17, 92B05, 92D25, 47D03, 47H20, 35M10

Key words: chemotaxis -- global solution -- boundedness -- nonlocal conditions -- diffusion -- analytic semigroup -- fractional power

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