Criteria for finite time blow up for a system of Klein-Gordon equations

TL;DR

This paper establishes three criteria based on initial data that determine whether solutions to certain Klein-Gordon systems in three dimensions blow up in finite time, using energy and potential well methods.

Contribution

It introduces new conditions for finite-time blow-up in Klein-Gordon systems, including cases with arbitrarily large energy, and classifies initial data below ground state energy.

Findings

Negative energy initial data lead to blow-up.

Positive energy data can also cause blow-up under certain conditions.

Data below ground state energy are classified into blow-up or global solutions.

Abstract

We give three conditions on initial data for the blowing up of the corresponding solutions to some system of Klein-Gordon equations on the three dimensional Euclidean space. We first use Levine's concavity argument to show that the negativeness of energy leads to the blowing up of local solutions in finite time. For the data of positive energy, we give a sufficient condition so that the corresponding solution blows up in finite time. This condition embodies datum with arbitrarily large energy. At last we use Payne-Sattinger's potential well argument to classify the datum with energy not so large (to be exact, below the ground states) into two parts: one part consists of datum leading to blowing-up solutions in finite time, while the other part consists of datum that leads to the global solutions.