Boundary spike-layer solutions of the singular Keller-Segel system: existence and stability
File(s)
Published version
Author(s)
Carrillo de la Plata, Jose Antonio
Li, Jingyu
Wang, Zhian
Type
Journal Article
Abstract
We explore the existence and nonlinear stability of boundary spike/layer solutionsof the Keller-Segel system with logarithmic singular sensitivity in the half space, where thephysical zero-flux and Dirichlet boundary conditions are prescribed. We first prove that, underabove boundary conditions, the Keller-Segel system admits a unique boundary spike-layer steadystate where the first solution component (bacterial density) of the system concentrates at theboundary as a Dirac mass and the second solution component (chemical concentration) formsa boundary layer profile near the boundary as the chemical diffusion coefficient tends to zero.Then we show that this boundary spike-layer steady state is asymptotically nonlinearly stableunder appropriate perturbations. As far as we know, this is the first result obtained on theglobal well-posedness of the singular Keller-Segel system with nonlinear consumption rate. Weintroduce a novel strategy of relegating the singularity, via a Cole-Hopf type transformation,to a nonlinear nonlocality which is resolved by the technique of “taking antiderivatives”, i.e.working at the level of the distribution function. Then, we carefully choose weight functions toprove our main results by suitable weighted energy estimates with Hardy’s inequality that fullycaptures the dissipative structure of the system.
Date Issued
2021-01-01
Date Acceptance
2019-12-13
Citation
Proceedings of the London Mathematical Society, 2021, 122 (1), pp.42-68
ISSN
0024-6115
Publisher
London Mathematical Society
Start Page
42
End Page
68
Journal / Book Title
Proceedings of the London Mathematical Society
Volume
122
Issue
1
Copyright Statement
© 2020 The Authors. Proceedings of the London Mathematical Society is copyright © London Mathematical Society.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
License URL
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/P031587/1
Subjects
Science & Technology
Physical Sciences
Mathematics
35A01
35B40
35K57
35Q92
76D10
92C17 (primary)
HYPERBOLIC-PARABOLIC SYSTEM
ZERO DIFFUSION LIMIT
TRAVELING-WAVES
CONSERVATION-LAWS
NONLINEAR STABILITY
CHEMOTAXIS MODEL
GLOBAL DYNAMICS
WELL-POSEDNESS
CONVERGENCE
math.AP
math.AP
0101 Pure Mathematics
0104 Statistics
Publication Status
Published
Date Publish Online
2020-05-01