2019/10/15 by Pecher, Hartmut
#35L70 #35Q55 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1910.06612
We consider the Klein-Gordon-Schrödinger system i ∂t ψ+ Δψamp; = ϕ2 ψ- ϕψ
(\Box +1)ϕamp; = -2|ψ|2 ϕ+ |ψ|2 with additional cubic terms and Cauchy data ψ(0) = ψ0 ∈ Hs(\mathbb Rn) , ϕ(0) = ϕ0 ∈ Hk(\mathbb Rn) , (∂t ϕ)(0) = ϕ1 ∈ Hk-1(\mathbb Rn) in space dimensions n=2 and n=3 . We prove local existence, uniqueness and continuous dependence on the data in Bourgain-Klainerman-Machedon spaces for low regularity data, e.g. for s=-(1)/(8), k=(3)/(8)+ε in the case n= 2 and s=0 , k=(1)/(2)+ε in the case n=3. Global well-posedness in energy space is also obtained as a special case. Moreover, we show "unconditional" uniqueness in the space ψ∈ C0([0,T],Hs) , ϕ∈ C0([0,T],Hs+(1)/(2)) ∩ C1([0,T],Hs-(1)/(2)), if s > (3)/(22) for n=2 and s > (1)/(2) for n=3.