The Cauchy problem of non-local space-time reaction-diffusion equation involving fractional $p$-Laplacian

Abstract

For the non-local space-time reaction-diffusion equation involving fractional $p$-Laplacian \begin{equation*} \begin{cases} \frac{\partial^{\alpha }u}{\partial t^{\alpha }}+(-\Delta)_{p}^{s} u=\mu u^{2}(1-kJ*u)-\gamma u,&(x,t)\in\mathbb{R}^{N}\times(0,T)\\ u(x,0)=u_{0}(x),& x\in\mathbb{R}^{N} \end{cases} \end{equation*} $\mu>0 ,k>0,\gamma\geq 1,\alpha\in(0,1),s\in(0,1),1<p$, we consider for $N\leq2$ the problem of finding a global boundedness of the weak solution by virtue of Gagliardo-Nirenberg inequality and fractional Duhamel's formula. Moreover, we prove such weak solution converge to $0$ exponentially or locally uniformly as $t \rightarrow \infty$ for small $\mu$ values with the comparison principle and local Lyapunov type functional. In those cases the problem is reduced to fractional $p$-Laplacian equation in the non-local reaction-diffusion range which is treated with the symmetry and other properties of the kernel of $(-\Delta)_{p}^{s}$. Finally, a key element in our construction is a proof of global bounded weak solution with the fractional nonlinear diffusion terms $(-\Delta)_{p}^{s}u^{m}(2-\frac{2}{N}<m\leq 3,1<p<\frac{4}{3})$ by using Moser iteration and fractional differential inequality.