Well-posedness results for a generalized Klein-Gordon-Schrödinger system

Abstract

We consider the Klein-Gordon-Schrödinger system i∂tψ + Δψ = ϕ2ψ − ϕψ, (□ + 1)ϕ = −2|ψ|2ϕ + |ψ|2 with additional cubic terms and Cauchy data ψ(0)=ψ0∈Hs(Rn),ϕ(0)=ϕ0∈Hk(Rn),and (∂tϕ)(0)=ϕ1∈Hk−1(Rn) in space dimensions n = 2 and n = 3. We prove the local existence, uniqueness, and continuous dependence on the data in Bourgain-Klainerman-Machedon spaces for low regularity data, e.g., for s=−18 and k=38+ϵ in the case n = 2 and s = 0 and for k=12+ϵ in the case n = 3. Global well-posedness in energy space is also obtained as a special case. Moreover, we show the “unconditional” uniqueness in the space ψ ∈ C0([0, T], Hs), ϕ∈C0([0,T],Hs+12)∩C1([0,T],Hs−12), if s>322 for n = 2 and s>12 for n = 3.