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

TL;DR

This paper investigates the global boundedness and long-term behavior of solutions to a nonlocal time fractional p-Laplacian reaction-diffusion equation, revealing conditions for boundedness, decay, and the Allee effect across different dimensions.

Contribution

It establishes new conditions for global boundedness and decay of solutions to a nonlocal fractional p-Laplacian PDE, including the Allee effect and solutions in higher dimensions.

Findings

Solutions are globally bounded under specified conditions.

For small parameters, solutions decay exponentially or locally uniformly.

The model exhibits the Allee effect in the fractional Caputo derivative context.

Abstract

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 $\Delta_{p} u^{m}\, (2-\frac{2}{N}< m\leq 3)$.