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Title: | Boundary spike-layer solutions of the singular Keller-Segel system : existence and stability | Authors: | Carrillo, JA Li, JY Wang, ZA |
Issue Date: | Jan-2021 | Source: | Proceedings of the London Mathematical Society, Jan. 2021, v. 122, no. 1, p. 42-68 | Abstract: | We explore the existence and nonlinear stability of boundary spike-layer solutions of the Keller-Segel system with logarithmic singular sensitivity in the half space, where the physical zero-flux and Dirichlet boundary conditions are prescribed. We first prove that, under above boundary conditions, the Keller-Segel system admits a unique boundary spike-layer steady state where the first solution component (bacterial density) of the system concentrates at the boundary as a Dirac mass and the second solution component (chemical concentration) forms a 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 stable under appropriate perturbations. As far as we know, this is the first result obtained on the global well-posedness of the singular Keller-Segel system with nonlinear consumption rate. We introduce 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 anti-derivatives', that is, working at the level of the distribution function. Then, we carefully choose weight functions to prove our main results by suitable weighted energy estimates with Hardy's inequality that fully captures the dissipative structure of the system. | Keywords: | 35A01 35B40 35K57 35Q92 76D10 92C17 (primary) |
Publisher: | Wiley-Blackwell | Journal: | Proceedings of the London Mathematical Society | ISSN: | 0024-6115 | EISSN: | 1460-244X | DOI: | 10.1112/plms.12319 | Rights: | © 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). The following publication Carrillo, J.A., Li, J. and Wang, Z.-A. (2021), Boundary spike-layer solutions of the singular Keller–Segel system: existence and stability. Proc. London Math. Soc., 122: 42-68 is available at https://dx.doi.org/10.1112/plms.12319 |
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