论文标题
时间空间分数抛物线型凯勒 - 塞格模型的有限性和指数稳定
Boundedness and exponential stabilization for time-space fractional parabolic-elliptic Keller-Segel model in higher dimensions
论文作者
论文摘要
对于时间空间分数简并凯勒 - 塞格方程\ begin {qore*} \ begin {case} \ partial _ {t}^{β} u = - （ - δ）^{\fracα{2}}}（ρ（v）u），＆t> 0 \\（ - δ）^{\fracα{2}}} \end{cases} \end{equation*} $x\inΩ, Ω\subset \mathbb{R}^{n}, β\in (0,1),α\in (1,2)$, we consider for $n\geq 3$ the problem of finding a time-independent upper bound of the classical solution such that as $θ>0,C>0$ \ begin {equation*} \ left \ | u（\ cdot，t） - \ edline {u_ {0}} \ right \ | _ {l^{\ infty}（ω）}+\ lest \ | | v（\ cdot，t） - \ edline {u_ {0}} \ right \ | _ {w^{1，\ infty}（ω）} \ leq ce^{（ - θ）^{1/β{1/β{1/β{1/β} t}， \ end {equation*}其中$ \ +叠加{u_ {0}} = \ frac {1} {\ left | ω\ right |} \ int _ {ω} u_ {0} dx $。我们在浓度的特殊情况下发现了这种解决方案，该解决方案是通过Alikakos-Moser迭代和分数差异不等式的。在这种情况下，问题将减少为时间空间的分数抛物线 - 椭圆方程，该方程用Lyapunov功能方法处理。我们构造中的一个关键要素是通过使用分数Duhamel型积分方程来证明对恒定稳态的指数稳定。
For the time-space fractional degenerate Keller-Segel equation \begin{equation*} \begin{cases} \partial _{t}^{β}u=-(-Δ)^{\fracα{2}}(ρ(v)u),& t>0\\ (-Δ)^{\fracα{2}} v+v=u,& t>0 \end{cases} \end{equation*} $x\inΩ, Ω\subset \mathbb{R}^{n}, β\in (0,1),α\in (1,2)$, we consider for $n\geq 3$ the problem of finding a time-independent upper bound of the classical solution such that as $θ>0,C>0$ \begin{equation*} \left \| u(\cdot ,t)-\overline{u_{0}} \right \|_{L^{\infty }(Ω)}+\left \| v(\cdot ,t)-\overline{u_{0}} \right \|_{W^{1,\infty }(Ω)}\leq Ce^{(-θ)^{1/β}t}, \end{equation*} where $\overline{u_{0}}=\frac{1}{\left | Ω\right |}\int _{Ω}u_{0}dx$. We find such solution in the special cases of time-independent upper bound of the concentration with Alikakos-Moser iteration and fractional differential inequality. In those cases the problem is reduced to a time-space fractional parabolic-elliptic equation which is treated with Lyapunov functional methods. A key element in our construction is a proof of the exponential stabilization toward the constant steady states by using fractional Duhamel type integral equation.
