On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion

Kenneth H.  Karlsen; Süleyman  Ulusoy; Kenneth H.  Karlsen; Süleyman  Ulusoy

doi:10.3934/nhm.2016.11.181

Networks and Heterogeneous Media

2016, Volume 11, Issue 1: 181-201. doi: 10.3934/nhm.2016.11.181

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On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion

Kenneth H. Karlsen ^{1
,},
Süleyman Ulusoy ^{2
,}

1.
Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo
2.
Department of Mathematics, Faculty of Education, Zirve University, Gaziantep, 27260

Received: 01 April 2015 Revised: 01 July 2015
Primary: 35R09, 35L60, 35D; Secondary: 92C17.

We investigate a Keller-Segel model with quorum sensing and a fractional diffusion operator. This model describes the collective cell movement due to chemical sensing with flux limitation for high cell densities and with anomalous media represented by a nonlinear, degenerate fractional diffusion operator. The purpose of this paper is to introduce and prove the existence of a properly defined entropy solution.
- Keller-Segel system,
- hyperbolic equation,
- fractional diffusion,
- non-local diffusion,
- entropy solution,
- compactness,
- existence
Citation: Kenneth H. Karlsen, Süleyman Ulusoy. On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion[J]. Networks and Heterogeneous Media, 2016, 11(1): 181-201. doi: 10.3934/nhm.2016.11.181

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Abstract

We investigate a Keller-Segel model with quorum sensing and a fractional diffusion operator. This model describes the collective cell movement due to chemical sensing with flux limitation for high cell densities and with anomalous media represented by a nonlinear, degenerate fractional diffusion operator. The purpose of this paper is to introduce and prove the existence of a properly defined entropy solution.

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