Stability of a wave and Klein-Gordon system with mixed coupling

Shijie Dong (1)

1 LJLL - Laboratoire Jacques-Louis Lions

We are interested in establishing stability results for a system of semilinear wave and Klein-Gordon equations with mixed coupling nonlinearities, that is, we consider all of the possible quadratic nonlinear terms of the type of wave and Klein-Gordon interactions. The main difficulties are due to the absence of derivatives on the wave component in the nonlinearities. By doing a transformation on the wave equation, we reveal a hidden null structure. Next by using the scaling vector field on the wave component only, which was generally avoided, we are able to get very good L 2-type estimates on the wave component. Then we distinguish high order and low order energies of both wave and Klein-Gordon components, which allows us to close the bootstrap argument. MSC code. 35L05, 35L52, 35L71.

Mots clés: Wave and Klein-Gordon system; global existence; sharp pointwise decay; null condition; hyperboloidal foliation method

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]

Shijie Dong. Stability of a wave and Klein-Gordon system with mixed coupling. 2020. ⟨hal-02542061⟩