Application of an Adams type inequality to a two-chemical substances chemotaxis system
Application of an Adams type inequality to a two-chemical substances chemotaxis system
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DOI：
10.1016/j.jde.2017.02.031
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发表时间：
2017-07
影响因子：
2.4
通讯作者：
Kentarou Fujie;T. Senba
中科院分区：
文献类型：
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作者：
Kentarou Fujie;T. Senba
研究了全抛物型方程组{ut = Δ u− χ <$u（u <$v）in Ω×（0，∞），τ 1 vt = Δ v− v+ w in Ω×（0，∞），τ 2 wt = Δ w− w+ u in Ω×（0，∞）在齐次Neumann边界条件或混合边界条件下的正解（无通量和Dirichlet条件）在光滑有界区域Ω <$Rn（n≤ 4）上，具有正参数τ 1，τ 2，χ> 0和非负光滑初值（u 0，v0，w 0）.在低维情形下（n≤ 3），证明了对于所有合理的初值，系统的解在时间上全局存在且保持有界.在n= 4的情形下，证明了在径向对称的情形下，该方程组的Neumann边值问题解在时间上整体存在且有界，如果满足<$u0 <$L1（Ω）<（8 π）2/χ;对于混合边值问题，如果满足<$u0 <$L1（Ω）<（8 π）2/χ，在无径向对称的情形下，我们将证明解的整体存在性和有界性.关键成分是一个李雅普诺夫功能和亚当斯型不等式。构造了上述问题的一个李雅普诺夫泛函，并由亚当斯型不等式中的临界常数导出了常数（8 π）2/χ.这一结果被认为是Keller-Segel系统中著名的8π问题向高维的推广。
This paper deals with positive solutions of the fully parabolic system,{u t= Δ u− χ∇⋅(u∇ v) in Ω×(0,∞), τ 1 v t= Δ v− v+ w in Ω×(0,∞), τ 2 w t= Δ w− w+ u in Ω×(0,∞), under homogeneous Neumann boundary conditions or mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded domain Ω⊂ R n (n≤ 4) with positive parameters τ 1, τ 2, χ> 0 and nonnegative smooth initial data (u 0, v 0, w 0). In the lower dimensional case (n≤ 3), it is proved that for all reasonable initial data solutions of the system exist globally in time and remain bounded. In the case n= 4, it is shown that in the radially symmetric setting solutions to the Neumann boundary value problem of the system exist globally in time and remain bounded if‖ u 0‖ L 1 (Ω)<(8 π) 2/χ; as to the mixed boundary value problem, we will establish global existence and boundedness of solutions if‖ u 0‖ L 1 (Ω)<(8 π) 2/χ without radial symmetry. The key ingredients are a Lyapunov functional and an Adams type inequality. A Lyapunov functional of the above problems will be constructed and the constant (8 π) 2/χ is deduced from the critical constant in the Adams type inequality. This result is regarded as a generalization of the well-known 8π problem in the Keller–Segel system to higher dimensions.