The hair-trigger effect for a class of nonlocal nonlinear equations

We prove the hair-trigger effect for a class of nonlocal nonlinear evolution equations on $\mathbb{R}^d$ which have only two constant stationary solutions, $0$ and $\theta>0$. The effect consists in that the solution with an initial condition non identical to zero converges (when time goes to $\infty$) to $\theta$ locally uniformly in $\mathbb{R}^d$. We find also sufficient conditions for existence, uniqueness and comparison principle in the considered equations.