کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
8900214 1631557 2018 29 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?
ترجمه فارسی عنوان
چقدر محکم لجستیک می تواند مانع از نفوذ سیستم حداقل سیستم کلر-سگل شود؟
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات آنالیز ریاضی
چکیده انگلیسی
We study nonnegative solutions of parabolic-parabolic Keller-Segel minimal-chemotaxis-growth systems with prototype given by{ut=∇⋅(d1∇u−χu∇v)+κu−μu2,x∈Ω,t>0,vt=d2Δv−βv+αu,x∈Ω,t>0 in a smooth bounded smooth but not necessarily convex domain Ω⊂Rn (n≥3) with nonnegative initial data u0,v0 and homogeneous Neumann boundary data, where d1,d2,α,β,μ>0, χ,κ∈R. We provide quantitative and qualitative descriptions of the competition between logistic damping and other ingredients, especially, chemotactic aggregation to guarantee boundedness and convergence. Specifically, we first obtain an explicit formula μ0=μ0(n,d1,d2,α,χ) for the logistic damping rate μ such that the system has no blow-ups whenever μ>μ0. In particular, for Ω⊂R3, we get a clean formula for μ0:μ0(3,d1,d2,α,χ)={34d1αχ,if d1=d2,χ>0 and Ω is convex,310−2(1d1+2d2)α|χ|, otherwise. This offers a quantized effect of the logistic source on the prevention of blow-ups. Our result extends the fundamental boundedness principle by Winkler [42] with d1=1,d2=α=β:=1/τ, Ω being convex and sufficiently large values of μ beyond a certain number not explicitly known (except the simple case τ=1 and χ>0) and quantizes the qualitative result of Yang et al. [52]. Besides, in non-convex domains, since μ0(3,1,1,1,χ)=(7.743416⋯)χ, the recent boundedness result, μ>20χ, of Mu and Lin [25] is greatly improved. Then we derive another explicit formula:μ1=μ1(d1,d2,α,β,κ,χ)=α|χ|4κ+d1d2β for the logistic damping rate so that convergence of bounded solutions is ensured and the respective convergence rates are explicitly calculated out whenever μ>μ1. Recent convergence results of He and Zheng [9] are therefore complemented and refined.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Mathematical Analysis and Applications - Volume 459, Issue 2, 15 March 2018, Pages 1172-1200
نویسندگان
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