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 $\mathrm{\infty }$) to $\theta$ locally uniformly in ${\mathbb{R}}^d$. We find also sufficient conditions for existence, uniqueness and comparison principle in the considered equations.