Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
7222646 | Nonlinear Analysis: Theory, Methods & Applications | 2018 | 19 Pages |
Abstract
The Keller-Segel system ut=DÎuâDÏââ
(uvâv),xâΩ,t>0,vt=DÎvâv+u,xâΩ,t>0,is considered in a bounded domain ΩâRn, nâ¥2, with smooth boundary, where Ï>0 and D>0. The main results identify a condition on the parameters Ï<2n and D>0, essentially reducing to the assumption that Ï2D be suitably small, under which for all reasonably regular and positive initial data the corresponding classical solution of an associated Neumann initial-boundary value problem, known to exist globally according to previous findings, approaches the homogeneous steady state (u¯0,u¯0) at an exponential rate with respect to the norm in (Lâ(Ω))2 as tââ, where u¯0â1|Ω|â«Î©u(â
,0). As a particular consequence, this entails a convergence statement of the above flavor in the normalized system with D=1 and fixed Ï<2n, provided that Ω satisfies a certain smallness condition.
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Authors
Michael Winkler, Tomomi Yokota,