  The global boundedness and asymptotic behavior are investigate for the
solution of time-space fractional non-local reaction-diffusion equation
(TSFNRDE)
  $$ \frac{\partial^{\alpha }u}{\partial t^{\alpha }}=-(-\Delta)^{s} u+\mu
u^{2}(1-kJ*u)-\gamma u, \qquad(x,t)\in\mathbb{R}^{N}\times(0,+\infty),$$ where
$s\in(0,1),\alpha\in(0,1), N \leq 2$. The operator $\partial_{t}^{\alpha }$ is
the Caputo fractional derivative, which $-(-\Delta)^{s}$ is the fractional
Laplacian operator. For appropriate assumptions on $J$, it is proved that for
homogeneous Dirichlet boundary condition, this problem admits a global bounded
weak solution for $N=1$, while for $N=2$, global bounded weak solution exists
for large $k$ values by Gagliardo-Nirenberg inequality and fractional
differential inequality. With further assumptions on the initial datum, for
small $\mu$ values, the solution is shown to converge to $0$ exponentially or
locally uniformly as $t \rightarrow \infty$. Furthermore, under the condition
of $J \equiv 1$, it is proved that the nonlinear TSFNRDE has a unique weak
solution which is global bounded in fractional Sobolev space with the nonlinear
fractional diffusion terms $-(-\Delta)^{s} u^{m}\, (2-\frac{2}{N}<m<1)$.
