Damped wave equation with super critical nonlinearities

We study global existence in time of small solutions to the Cauchy problem for the nonlinear damped wave equation ∂2 t u+ ∂tu−∆u = N (u) , x ∈ Rn, t > 0, u(0, x) = eu0 (x) , ∂tu(0, x) = eu1 (x) , x ∈ Rn, (1) where e > 0. The nonlinearity N (u) ∈ Ck (R) satisfies the estimate d j duj N (u) ≤ C |u|ρ−j , 0 ≤ j ≤ k ≤ ρ. The power ρ > 1+ 2 n is considered as super critical for large time. We assume that the initial data u0 ∈ H ∩ H , u1 ∈ Hα−1,0 ∩ H , where δ > n 2 , [α] ≤ ρ, α ≥ n 2 − 1 ρ−1 for n ≥ 2 and α ≥ 12 − 1 2(ρ−1) for n = 1. Weighted Sobolev spaces are H = n φ ∈ L; 〈x〉 〈i∂x〉 φ (x) L2 0 such that for any e ∈ (0, e0] there exists a unique global solution u ∈ C [0,∞) ;Hα,0 ∩H0,δ for the Cauchy problem (1) and solutions satisfy the time decay property ‖u (t)‖Lp ≤ Ct − 2 1− 1 p for all t > 0, where 2 ≤ p ≤ 2n n−2α if α n 2 .