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^{α}u}{\partial t^{α}}+(-Δ)_{p}^{s} u=μu^{2}(1-kJ*u)-γu,&(x,t)\in\mathbb{R}^{N}\times(0,T)\\ u(x,0)=u_{0}(x),& x\in\mathbb{R}^{N} \end{cases} \end{equation*} $\mu >0,k>0,\gamma \ge 1,\alpha \in (0,1),s\in (0,1),1<p$, we consider for $N\le 2$ 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 $t\to \infty$ 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 $(-\Delta {)}_p^s{u}^m(2-\frac{2}{N}<m\le 3,1<p<\frac{4}{3})$ by using Moser iteration and fractional differential inequality.

20 pages. arXiv admin note: text overlap with arXiv:2205.11040, arXiv:2112.11143, arXiv:2202.04928; text overlap with arXiv:2103.00552 by other authors