Tohoku Mathematical Journal
2024
December
SECOND SERIES VOL. 76, NO. 4

Tohoku Math. J.
76 (2024), 609-628

Title STABILITY OF A WAVE AND KLEIN-GORDON SYSTEM WITH MIXED COUPLING

Author Shijie Dong

(Received December 16, 2022, revised April 7, 2023)
Abstract. We are interested in establishing global 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 nonlinear transformation on the wave equation, we reveal a hidden null structure. Next, by using the scaling vector field on the wave component only, which is generally avoided, we are able to get good $L^2$-type estimates on the wave component. Then we distinguish high-order and low-order energy of both wave and Klein-Gordon components, which allows us to close the bootstrap argument.

Mathematics Subject Classification. Primary 35L05; Secondary 35L52, 35L71.
Key words and phrases. Wave and Klein-Gordon system, global existence, sharp time decay.

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