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On the parabolic-elliptic Keller-Segel system with signal-dependent motilities: a paradigm for global boundedness and steady states
  • Zhian Wang
Zhian Wang
Hong Kong Polytechnic University
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Abstract

This paper is concerned with a parabolic-elliptic Keller-Segel system where both diffusive and chemotactic coefficients (motility functions) depend on the chemical signal density. This system was originally proposed by Keller and Segel in \cite{KS-1971-JTB2} to describe the aggregation phase of {\it Dictyostelium discoideum} cells in response to the secreted chemical signal cyclic adenosine monophosphate (cAMP), but the available analytical results are very limited by far. Considering system in a bounded smooth domain with Neumann boundary conditions, we establish the global boundedness of solutions in any dimensions with suitable general conditions on the signal-dependent motility functions, which are applicable to a wide class of motility functions. The existence/nonexistence of non-constant steady states is studied and abundant stationary profiles are found. Some open questions are outlined for further pursues. Our results demonstrate that the global boundedness and profile of stationary solutions to the Keller-Segel system with signal-dependent motilities depend on the decay rates of motility functions, space dimensions and the relation between the diffusive and chemotactic motilities, which makes the dynamics immensely wealthy.

Peer review status:Published

22 Jun 2020Submitted to Mathematical Methods in the Applied Sciences
26 Jun 2020Submission Checks Completed
26 Jun 2020Assigned to Editor
26 Jun 2020Reviewer(s) Assigned
25 Oct 2020Review(s) Completed, Editorial Evaluation Pending
09 Nov 2020Editorial Decision: Revise Major
11 Nov 20201st Revision Received
11 Nov 2020Submission Checks Completed
11 Nov 2020Assigned to Editor
30 Nov 2020Reviewer(s) Assigned
14 Mar 2021Review(s) Completed, Editorial Evaluation Pending
04 Apr 2021Editorial Decision: Revise Minor
05 Apr 20212nd Revision Received
05 Apr 2021Submission Checks Completed
05 Apr 2021Assigned to Editor
05 Apr 2021Editorial Decision: Accept
02 May 2021Published in Mathematical Methods in the Applied Sciences. 10.1002/mma.7455