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

## TL;DR

This paper proves that solutions to certain nonlocal nonlinear equations on  with specific stationary states tend to a positive equilibrium from non-zero initial conditions, establishing the hair-trigger effect and providing conditions for well-posedness.

## Contribution

It establishes the hair-trigger effect for a class of nonlocal nonlinear equations and provides conditions for existence, uniqueness, and comparison principles.

## Key findings

- Solutions with non-zero initial data converge to  in the long term.
- The paper identifies sufficient conditions for existence and uniqueness.
- Comparison principles are established for the considered equations.

## Abstract

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.