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Song, Xu; Li, Jingyu

Convergence rate of solutions towards spiky steady state for the Keller-Segel system with logarithmic sensitivity. (English) Zbl 1516.35428

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 232, Article ID 113284, 25 p. (2023).
Summary: We study the large time behaviors of solutions to the Keller-Segel system with logarithmic singular sensitivity in the half space, where biological mixed boundary conditions are prescribed. The existence and asymptotic stability of spiky steady states of this system were proved by J. A. Carrillo et al. [Proc. Lond. Math. Soc. (3) 122, No. 1, 42–68 (2021; Zbl 1464.35013)]. In this paper we obtain convergence rate of solutions towards the steady state under appropriate initial perturbations. The proofs are based on a Cole-Hopf type transformation and a weighted energy method, where the weights are artfully constructed.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
92C17 Cell movement (chemotaxis, etc.)
41A25 Rate of convergence, degree of approximation

Keywords:

chemotaxis; Keller-Segel system; spike; stability; convergence rate

Citations:

Zbl 1464.35013
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Full Text: DOI

References:

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