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首页> 外文期刊>Discrete and continuous dynamical systems▼hSeries S >BOUNDEDNESS IN A QUASILINEAR FULLY PARABOLIC KELLER-SEGEL SYSTEM VIA MAXIMAL SOBOLEV REGULARITY
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BOUNDEDNESS IN A QUASILINEAR FULLY PARABOLIC KELLER-SEGEL SYSTEM VIA MAXIMAL SOBOLEV REGULARITY

机译：通过最大SOBOLEV规律性的Quasilinear全抛物型Keller-Segel系统中的有界性

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摘要

This paper deals with the quasilinear Keller-Segel system {u_t=▽·(D(u)▽u)-▽·(S(u)▽υ),x∈Ω,t＞0,υ_t=Δυ-υ+u,x∈Ω,t＞0 in Ω=R~N or in a smoothly bounded domain Ω⊂R~N, with nonnegative initial data u_0∈L~1(Ω)∩L~∞(Ω), and υ_0∈L~1(Ω)∩W~(1,∞)(Ω);in the case that is bounded, it is supplemented with homogeneous Neumann boundary condition. The diffusivity D(u) and the sensitivity S(u) are assumed to fulfill D(u) ≥ u~(m-1) (m≥1) and S(u) ≤ u~(q-1) (q≥2), respectively. This paper derives uniform-in-time boundedness of nonnegative solutions to the system when q ＜ m+2/N. In the case Ω=R~N the result says boundedness which was not attained in a previous paper (J. Differential Equations 2012; 252:1421-1440). The proof is based on the maximal Sobolev regularity for the second equation. This also simplifies a previous proof given by Tao-Winkler (J. Differential Equations 2012; 252:692-715) in the case of bounded domains.
机译：本文涉及Quasilinear Keller-Segel系统{U_T =▽·（D（U）▽U） - ▽·（s（u）▽υ），x∈ω，t> 0，υ_t=Δυ-υ+ u ，x∈ω，t> 0inω= r〜n或在平滑的界域ωνr〜n中，具有非负初始数据u_0∈l〜1（ω）∩1°（ω），和υ_0∈l 〜1（ω）∩W〜（1，∞）（ω）;在界定的情况下，它补充了均匀的neumann边界条件。假设扩散性D（U）和敏感性S（U）满足D（u）≥U〜（m≥1）和s（u）≤u〜（q-1）（q≥ 2）分别。本文在Q

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