Remark on a weakly coupled system of nonlinear damped wave equations

We study global existence of small solutions to the Cauchy problem for a weakly coupled nonlinear damped wave equation{(∂t2+∂t−Δ)u=N1(v),(∂t2+∂t−Δ)v=N2(u),x∈Rn,t>0u(0,x)=εu0(x),∂tu(0,x)=εu1(x),v(0,x)=εv0(x),∂tv(0,x)=εv1(x),x∈Rn, with super-critical nonlinearities Nk(ϕ)=|ϕ|ρkNk(ϕ)=|ϕ|ρk, k=1,2k=1,2, where ε>0ε>0, the space dimension n≥4n≥4. Our purpose is to remove the exponential decay condition on the data and the lower bound for ρ1ρ1 which was assumed in [4] when proving the global existence of solutions in the case of higher space dimensions.