Stability of a coupled wave-Klein-Gordon system with quadratic nonlinearities

Global solution to the Klein-Gordon-Zakharov equations with uniform energy bounds

“…In terms of small data global existence result for the Klein-Gordon-Zakharov equations and the asymptotic behavior of the solution, there exist a few proofs, see for instance [32,39,18,8]. So the main contribution in Theorem 1.1 is that the energy for the solution to the Klein-Gordon-Zakharov equations is shown to be uniformly bounded.…”

“…∂ α u∂ β v − ∂ β u∂ α v, α, β ∈ {0, 1, 2, 3}). Later on, more physical models governed by the wave and Klein-Gordon systems were studied, for example the Klein-Gordon-Zakharov equations [32,39,18,8], the Maxwell-Klein-Gordon equations [34,35,25,11], the Dirac-Proca equations [40], the Einstein-Klein-Gordon equations [28,29,41,16,17], and the U (1) electroweak standard model [9]. Besides, in view of pure PDE's, there also exist some other results, see for instance [18,27,19,8,5].…”

Hidden structure and sharp asymptotics for the Dirac--Klein-Gordon system in two space dimensions

“…To conclude the introduction, we remind the reader of numerous recent works concerning global existence and decay for coupled wave-Klein-Gordon equations in 1 + 3 dimensions. These include wave-Klein-Gordon equations derived from mathematical physics, such as the Dirac-Klein-Gordon model, the Dirac-Proca and U (1)-electroweak model [14,21,39,40], the Einstein-Klein-Gordon equations [20,25,27,41], the Klein-Gordon-Zakharov equations [15,33,38], the Maxwell-Klein-Gordon equations [24] and certain geometric problems derived from wave maps [1].…”

“…The nonlinear transformation allows us to cancel the wave-Klein-Gordon interaction wu at the expense of introducing null and cubic terms. Such a strategy appears in [15] and was inspired by the transformation of Tsutsumi mentioned in Type 1.…”

“…We conclude with a remark concerning a more general DKG system allowing for the full range of masses: Surprisingly, (4.11) is a very difficult system to treat due to the slowly-decaying quadratic term 2m ψ vψ. In R 1+3 one would require ideas from [15], while the analysis remains open in R 1+2 . The case m ψ = m v = 0 is similarly very difficult to treat in R 1+2 and R 1+3 .…”