Affiliations: Department of Mathematics, State University of Maringá, 87020-900, Av. Colombo 5790, Maringá, PR, Brazil. E-mails: [email protected], [email protected]
Abstract: The following coupled damped Klein–Gordon–Schrödinger equations are considered
iψt+Δψ+iαb(x)(|ψ|2+1)ψ=ϕψχΩRin Rn×(0,∞)(α>0),ϕtt−Δϕ+ϕ+a(x)ϕt=|ψ|2χΩRin Rn×(0,∞),
where ΩR=Rn∖BR={|x|⩾R} and χΩR represents the characteristic function of ΩR. Assuming that a,b∈W1,∞(Rn) are nonnegative functions such that a(x)⩾a0>0 in ΩR and b(x)⩾b0>0 in ΩR, the exponential decay rate is proved for every regular solution of the above system. Our result generalizes substantially the previous ones given by Cavalcanti et al. in the references (NoDEA, Nonlinear Differ. Equ. Appl. 15 (2008) 91–113), (NoDEA, Nonlinear Differ. Equ. Appl. 7 (2000) 285–307), (Communications on Pure and Applied Analysis 17 (2018) 2039–2061) and (Evolution Equations and Control Theory 8 (2019) 847–865).
