Global boundedness and Allee effect for a nonlocal time fractional reaction-diffusion equation

Abstract

The global boundedness and asymptotic behavior are investigated for the solutions of a nonlocal time fractional reaction-diffusion equation (NTFRDE) $$\frac{{\partial }^{\alpha }u}{\partial t^{\alpha }}=\Delta u+\mu u^2(1-kJ\ast u)-\gamma u,(x,t)\in {\mathbb{R}}^N\times (0,+\infty )$$ with $0<\alpha <1,\beta ,\mu ,k>0,N\le 2$ and $u(x,0)=u_0(x)$. Under appropriate assumptions on $J$ and the property of time fractional derivative, it is proved that for any nonnegative and bounded initial conditions, the problem has a global bounded classical solution if $k^{\ast }=0$ for $N=1$ or $k^{\ast }=(\mu C_{GN}^2+1){\eta }^{-1}$ 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 $t\to \infty$, 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 NTFRDE has a global bounded solution in any dimensional space with the nonlinear diffusion terms $\Delta u^m(2-\frac{2}{N}<m\le 3)$.

45 pages