Finite time blow-up and global solutions for a nonlocal parabolic equation at high energy level

TL;DR

This paper investigates the conditions under which solutions to a nonlocal parabolic equation either blow up in finite time or exist globally, especially at high energy levels, and demonstrates the existence of blow-up solutions regardless of energy size.

Contribution

It establishes criteria for global existence and blow-up at high energy levels and proves the existence of blow-up solutions with negative Nehari functional regardless of energy magnitude.

Findings

Criteria for global solutions at high energy levels

Conditions leading to finite time blow-up

Existence of blow-up solutions with negative Nehari functional

Abstract

In this paper, we consider the solution of a nonlocal parabolic equation. Focusing on the solutions with initial data at high energy level, we find the criteria for global existence and finite time blow up for the corresponding solution respectively. Moreover, we prove that there always exists blow up solution with negative Nehari functional no matter how large the energy is.