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This paper is concerned with strong blow-up instability (Definition 1.3) forstanding wave solutions to the system of the quadratic nonlinear Klein-Gordonequations. In the single case, namely the nonlinear Klein-Gordon equation withpower type nonlinearity, stability and instability for standing wave solutionshave 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 weknow. In this paper, we prove strong blow-up instability for the standing waveto 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 spacedimensions whose cases are the mass-subcritical case.

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