Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms

Tong Li, Anthony Suen, Michael Winkler, Chuan Xue

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Abstract

We study non-negative solutions to the chemotaxis system {equation presented} under no-flux boundary conditions in a bounded planar convex domain with smooth boundary, where f and S are given parameter functions on × [0,)2 with values in [0,) and R2×2, respectively, which are assumed to satisfy certain regularity assumptions and growth restrictions. Systems of type (∗), in the special case {equation presented} reducing to a version of the standard Keller-Segel system with signal consumption, have recently been proposed as a model for swimming bacteria near a surface, with the sensitivity tensor then given by S{equation presented}, reflecting rotational chemotactic motion. It is shown that for any choice of suitably regular initial data (u0, v0) fulfilling a smallness condition on the norm of v0 in L(), the corresponding initial-boundary value problem associated with (∗) possesses a globally defined classical solution which is bounded. This result is achieved through the derivation of a series of a priori estimates involving an interpolation inequality of Gagliardo-Nirenberg type which appears to be new in this context. It is next proved that all corresponding solutions approach a spatially homogeneous steady state of the form (u, v) (μ, κ) in the large time limit, with μ:= fu0 and some κ ≥ 0. A mild additional assumption on the positivity of f is shown to guarantee that κ = 0. Finally, numerical solutions are presented which suggest the occurrence of wave-like solution behavior.

Original languageEnglish (US)
Pages (from-to)721-746
Number of pages26
JournalMathematical Models and Methods in Applied Sciences
Volume25
Issue number4
DOIs
StatePublished - Apr 22 2015
Externally publishedYes

Bibliographical note

Funding Information:
The authors would like to thank the referees for their comments and suggestions on the improvement of the paper. C.X. is supported by the National Science Foundation in the United States through Grant DMS 1312966. C.X. is also supported by the Mathematical Biosciences Institute at the Ohio State University as a long-term visitor.

Publisher Copyright:
© 2015 World Scientific Publishing Company.

Keywords

  • Global existence
  • Keller-Segel model
  • Rotational flux
  • Symptotic behavior.

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