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A multi-species chemotaxis system: Lyapunov functionals, duality, critical mass

Part of: Inequalities Elliptic equations and systems Physiological, cellular and medical topics Parabolic equations and systems

Published online by Cambridge University Press:  09 October 2017

N. I. KAVALLARIS ,
T. RICCIARDI  and
G. ZECCA
N. I. KAVALLARIS
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, University of Chester, Thornton Science Park, Chester CH2 4NU, UK email: [email protected]
T. RICCIARDI
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Cintia, Monte S. Angelo, 80126 Napoli, Italy email: [email protected], [email protected]
G. ZECCA
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Cintia, Monte S. Angelo, 80126 Napoli, Italy email: [email protected], [email protected]
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Abstract

We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted versus repelled by a single chemical substance. The production versus destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model, we investigate the variational structures, in particular, we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy–Littlewood–Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.

Keywords

Multi-species chemotaxis modelsLyapunov functionalsdualitylogarithmic Hardy–Littlewood–Sobolev inequalityMoser–Trudinger inequality

MSC classification

Primary: 35K51: Initial-boundary value problems for second-order parabolic systems
Secondary: 35J20: Variational methods for second-order elliptic equations 26D15: Inequalities for sums, series and integrals 92C17: Cell movement (chemotaxis, etc.)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 3 , June 2018 , pp. 515 - 542
Copyright
Copyright © Cambridge University Press 2017 

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