Boundedness and exponential stabilization for time-space fractional parabolic-elliptic Keller-Segel model in higher dimensions

Abstract

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.