Global boundedness and asymptotic behavior of time-space fractional nonlocal reaction-diffusion equation

Abstract

The global boundedness and asymptotic behavior are investigate for the solution of time-space fractional non-local reaction-diffusion equation (TSFNRDE) $$\frac{{\partial }^{\alpha }u}{\partial t^{\alpha }}=-(-\Delta {)}^{s}u+\mu u^2(1-kJ\ast u)-\gamma u,(x,t)\in {\mathbb{R}}^N\times (0,+\infty ),$$ where $s\in (0,1),\alpha \in (0,1),N\le 2$. 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 $t\to \infty$. 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 $-(-\Delta {)}^s{u}^m(2-\frac{2}{N}<m<1)$.

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