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

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*} $x\in \mathrm{\Omega },\mathrm{\Omega }\subset {\mathbb{R}}^n,\beta \in (0,1),\alpha \in (1,2)$, we consider for $n\ge 3$ the problem of finding a time-independent upper bound of the classical solution such that as $\theta >0,C>0$ \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 $\overline{u_0}=\frac{1}{\mid \mathrm{\Omega }\mid }{\int }_{\mathrm{\Omega }}{u}_{0}dx$. 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.