# Vacuum isolating, blow up threshold and asymptotic behavior of solutions for a nonlocal parabolic equation

## TL;DR

This paper studies a nonlocal parabolic equation, analyzing conditions for solution blow-up or decay, and introduces a threshold for global existence versus blow-up based on initial energy and potential wells.

## Contribution

It establishes a new threshold criterion for global existence and blow-up, and explores the asymptotic behavior of solutions in relation to initial energy and potential well structure.

## Key findings

- Solutions with initial energy below a threshold exist globally.
- Solutions with critical initial energy may blow up or exist globally depending on initial conditions.
- Global solutions decay exponentially, while blow-up solutions grow exponentially.

## Abstract

In this paper, we consider a nonlocal parabolic equation associated with initial and Dirichlet boundary conditions. Firstly, we discuss the vacuum isolating behavior of solutions with the help of a family of potential wells. Then we obtain a threshold of global existence and blow up for solutions with critical initial energy. Furthermore, for those solutions satisfy $J(u_0)\leq d$ and $I(u_0)\neq 0$, we show that global solutions decay to zero exponentially as time tends to infinity and the norm of blow-up solutions increase exponentially.