  For the time-space fractional degenerate Keller-Segel equation
\begin{equation*}
  \begin{cases}
  \partial _{t}^{\beta }u=-(-\Delta )^{\frac{\alpha}{2}}(\rho (v)u),& t>0\\
(-\Delta )^{\frac{\alpha}{2}} v+v=u,& t>0
  \end{cases} \end{equation*} $x\in\Omega, \Omega \subset \mathbb{R}^{n},
\beta\in (0,1),\alpha\in (1,2)$, we consider for $n\geq 3$ the problem of
finding a time-independent upper bound of the classical solution such that as
$\theta>0,C>0$
  \begin{equation*}
  \left \| u(\cdot ,t)-\overline{u_{0}} \right \|_{L^{\infty }(\Omega )}+\left
\| v(\cdot ,t)-\overline{u_{0}} \right \|_{W^{1,\infty }(\Omega )}\leq
Ce^{(-\theta)^{1/\beta}t},
  \end{equation*} where $\overline{u_{0}}=\frac{1}{\left | \Omega \right |}\int
_{\Omega }u_{0}dx$. 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.
