Asymptotic stability of the nonlocal diffusion equation with nonlocal delay

Abstract

This work focuses on the asymptotic stability of nonlocal diffusion equations in $N$-dimensional space with nonlocal time-delayed response term. To begin with, we prove $L^2$ and $L^{\infty }$-decay estimates for the fundamental solution of the linear time-delayed equation by Fourier transform. For the considered nonlocal diffusion equation, we show that if $l>\left(p\right)$, then the solution $u(t,x)$ converges globally to the trivial equilibrium time-exponentially. If $l=\left(p\right)$, then the solution $u(t,x)$ converges globally to the trivial equilibrium time-algebraically. Furthermore, it can be proved that when $r>\left(q\right)$, the solution $u(t,x)$ converges globally to the positive equilibrium time-exponentially, and when $r=\left(q\right)$, the solution $u(t,x)$ converges globally to the positive equilibrium time-algebraically. Here, $l,p,r$, and $q$ are the coefficients of each term contained in the linear part of the nonlinear term $f$. All convergence rates above are $L^2$ and $L^{\infty }$-decay estimates. The comparison principle and low-frequency and high-frequency analyses are significantly effective in proofs. Finally, our theoretical results are supported by numerical simulations in different situations.