Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations

TL;DR

This paper establishes strong blow-up instability for standing wave solutions in a quadratic nonlinear Klein-Gordon system, filling a gap in the understanding of stability properties for such coupled equations.

Contribution

It proves the first known strong blow-up instability results for standing waves in the quadratic nonlinear Klein-Gordon system, using techniques adapted from previous studies.

Findings

Proves strong blow-up instability under mass resonance conditions

Extends instability analysis from single equations to coupled systems

Uses techniques from Ohta and Todorova to establish results

Abstract

This paper is concerned with strong blow-up instability (Definition 1.3) for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. In the single case, namely the nonlinear Klein-Gordon equation with power type nonlinearity, stability and instability for standing wave solutions have been extensively studied. On the other hand, in the case of our system, there are no results concerning the stability and instability as far as we know. In this paper, we prove strong blow-up instability for the standing wave to our system. The proof is based on the techniques in Ohta and Todorova [25]. It turns out that we need the mass resonance condition in two or three space dimensions whose cases are the mass-subcritical case.