论文标题
涉及信号依赖性运动的Keller-Segel征服系统的精制规律性分析
Refined regularity analysis for a Keller-Segel-consumption system involving signal-dependent motilities
论文作者
论文摘要
我们考虑Keller-Segel-type迁移 - 消费系统，涉及信号依赖性动作，$$ \ left \ {\ begin {array} {l} u_t =δ\ big（ux（v）\ big），\\ [1mm] v_t =ΔV-uv， \ end {array} \ right。 \ qquad \ qquad $ $ $$在平滑界面$ω\ subset \ mathbb {r}^n $，$ n \ ge 1 $中。假设$ [[0，\ infty））$在$ [0，\ infty）$上是正面的，并且对于$（c^0（c^0（\overlineΩ））^\ star \ star \ star \ star \ times l^\ infty（ω）$的非负初始数据，以前的文献具有较大的态度，并且在全球范围内提供了很大的态度，并且在全球范围内提供了很大的态度，并且在全球范围内提供了很大的态度，并且在全球范围内提供了很大的态度，并且是在全球范围内提供的。相对于拓扑中的半序平衡，$（w^{1,2}（ω））^\ star \ times l^\ infty（ω）$。 The present study reveals that solutions in fact enjoy significantly stronger regularity features when $0<ϕ\in C^3([0,\infty))$ and the initial data belong to $(W^{1,\infty}(Ω))^2$: It is firstly shown, namely, that then in the case $n\le 2$ an associated no-flux initial-boundary value problem even admits a global classical solution, and that each of these解决方案在某种意义上平稳稳定，因为作为$ t \至\ infty $，我们有$$ \ begin {align*} u（\ cdot，t）\ to \ frac {1} {|ω|} \int_Ωu_0 \ qquad \ text {and} \ qquad v（\ cdot，t）\至0 \ qquad \ qquad（\ star）\ end {align*} $$即使相对于两个组件中的$ l^\ infty（ω）$中的norm。 在此情况下，当$ n \ ge 3 $（其次发现了一些真正的弱解决方案）存在于全球存在的情况下，在l^\ frac {4} {4} {3} {3} {3} _ {loc} _ {loc} _ {\overlineΩ\ times times [0，\ infty）; \ mathbbbbbb in l^\ frac {4} {3} {3} {3} {3} {3} {3}在特定的三维环境中，任何此类解决方案都可以最终变得光滑和满足（$ \ star $）。
We consider the Keller-Segel-type migration-consumption system involving signal-dependent motilities, $$\left\{ \begin{array}{l} u_t = Δ\big(uϕ(v)\big), \\[1mm] v_t = Δv-uv, \end{array} \right. \qquad \qquad$$ in smoothly bounded domains $Ω\subset\mathbb{R}^n$, $n\ge 1$. Under the assumption that $ϕ\in C^1([0,\infty))$ is positive on $[0,\infty)$, and for nonnegative initial data from $(C^0(\overlineΩ))^\star \times L^\infty(Ω)$, previous literature has provided results on global existence of certain very weak solutions with possibly quite poor regularity properties, and on large time stabilization toward semitrivial equilibria with respect to the topology in $(W^{1,2}(Ω))^\star \times L^\infty(Ω)$. The present study reveals that solutions in fact enjoy significantly stronger regularity features when $0<ϕ\in C^3([0,\infty))$ and the initial data belong to $(W^{1,\infty}(Ω))^2$: It is firstly shown, namely, that then in the case $n\le 2$ an associated no-flux initial-boundary value problem even admits a global classical solution, and that each of these solutions smoothly stabilizes in the sense that as $t\to\infty$ we have $$ \begin{align*} u(\cdot,t) \to \frac{1}{|Ω|}\int_Ωu_0 \qquad \text{ and } \qquad v(\cdot,t)\to 0 \qquad \qquad (\star) \end{align*}$$ even with respect to the norm in $L^\infty(Ω)$ in both components. In the case when $n\ge 3$, secondly, some genuine weak solutions are found to exist globally, inter alia satisfying $\nabla u\in L^\frac{4}{3}_{loc}(\overlineΩ\times [0,\infty);\mathbb{R}^n)$. In the particular three-dimensional setting, any such solution is seen to become eventually smooth and to satisfy ($\star$).
