How far does small chemotactic interaction perturb the Fisher–KPP dynamics?
How far does small chemotactic interaction perturb the Fisher–KPP dynamics?
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DOI：
10.1016/j.jmaa.2017.03.005
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发表时间：
2016-10
影响因子：
1.3
通讯作者：
J. Lankeit;M. Mizukami
中科院分区：
文献类型：
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作者：
J. Lankeit;M. Mizukami
研究了具有光滑边界的有界凸区域Ω <$Rn（n≥ 1）中具有正小参数ε> 0的完全抛物型趋化增长系统{（u ε）t= Δ u ε− ε <$u ε（u ε <$v ε）+ μ u ε（1− u ε），x ∈ Ω，t> 0，（v ε）t= Δ v ε− v ε+ u ε，x∈ Ω，t> 0的Neumann初边值问题的非负解.当ε→ 0时，解收敛于Fisher-KPP方程的解u。证明了对所有μ> 0和任意适当正则的非负初值（u i n i t，v i n i t），存在常数ε 0> 0和C> 0，使得对于所有ε∈（0，ε 0），supt> 0 <$u ε（n，t）− u（n，t）<$L∞（Ω）≤ C ε.
This paper deals with nonnegative solutions of the Neumann initial–boundary value problem for the fully parabolic chemotaxis-growth system,{(u ε) t= Δ u ε− ε∇⋅(u ε∇ v ε)+ μ u ε (1− u ε), x∈ Ω, t> 0,(v ε) t= Δ v ε− v ε+ u ε, x∈ Ω, t> 0, with positive small parameter ε> 0 in a bounded convex domain Ω⊂ R n (n≥ 1) with smooth boundary. The solutions converge to the solution u to the Fisher–KPP equation as ε→ 0. It is shown that for all μ> 0 and any suitably regular nonnegative initial data (u i n i t, v i n i t) there are some constants ε 0> 0 and C> 0 such that sup t> 0‖ u ε (⋅, t)− u (⋅, t)‖ L∞(Ω)≤ C ε for all ε∈(0, ε 0).