The global boundedness and asymptotic behavior are investigated for the solutions of a nonlocal time fractional p-Laplacian reaction-diffusion equation (NTFPLRDE) $$ \frac{ \partial^{ \alpha }u}{ \partial t^{ \alpha }}=\Delta_{ p} u+\mu u^{ 2}(1-kJ*u) -\gamma u, \qquad(x,t)\in\mathbb{ R}^{ N}\times(0,+\infty)$$ with $0<\alpha <1,\beta, \mu ,k>0,N\leq 2$ and $\Delta_{ p}u =div(\left| \bigtriangledown u \right|^{ p-2}\bigtriangledown u)$. Under appropriate assumptions on $J$ and the conditions of $1<p<2$, it is proved that for any nonnegative and bounded initial conditions, the problem has a global bounded classical solution if $k^{ *}=0$ for $N=1$ or $k^{ *}=(\mu C^{ 2}_{ GN}+1)\eta^{ -1}$ for $N=2$, where $C_{ GN}$ is the constant in Gagliardo-Nirenberg 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 $t \rightarrow \infty$, which is referred as the Allee effect in sense of Caputo derivative. Moreover, under the condition of $J \equiv 1$, it is proved that the nonlinear NTFPLRDE has a global bounded solution in any dimensional space with the nonlinear p-Laplacian diffusion terms $\Delta_{ p} u^{ m}\, (2-\frac{ 2}{ N}< m\leq 3)$.
Categories: math.AP