DAMPED WAVE EQUATION WITH SUPER CRITICAL NONLINEARITIES NAKAO HAYASHI, ELENA I. KAIKINA, AND PAVEL I. NAUMKIN Abstract. We study global existence in time of small solutions to the Cauchy problem for the nonlinear damped wave equation ∂ 2 t u + ∂t u - Δu = N (u) ,x ∈ R n ,t> 0, u(0,x)= εu 0 (x) ,∂t u(0,x)= εu 1 (x) ,x ∈ R n , (1) where ε> 0. The nonlinearity N (u) ∈ C k (R) satisfies the estimate d j du j 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 u 0 ∈ H α,0 ∩ H 0,δ ,u 1 ∈ H α-1,0 ∩ H 0,δ , where δ> n 2 , [α] ≤ ρ, α ≥ n 2 - 1 ρ-1 for n ≥ 2 and α ≥ 1 2 - 1 2(ρ-1) for n =1. Weighted Sobolev spaces are H l,m = φ ∈ L 2 ; 〈x〉 m 〈i∂x〉 l φ (x) L 2 < ∞ , where 〈x〉 = √ 1+ x 2 . Then we prove that there exists a small ε 0 > 0 such that for any ε ∈ (0,ε 0 ] there exists a unique global solution u ∈ C [0, ∞); H α,0 ∩ H 0,δ for the Cauchy problem (1) and solutions satisfy the time decay property ‖u (t)‖ L p ≤ Ct - n 2 1- 1 p for all t> 0, where 2 ≤ p ≤ 2n n-2α if α< n 2 , 2 ≤ p< ∞ if α = n 2 , and 2 ≤ p ≤ ∞ if α> n 2 . 1. Introduction We study the global existence in time of small solutions to the Cauchy problem for the nonlinear damped wave equation  Lu = N (u) ,x ∈ R n ,t> 0 u (0,x)= εu 0 (x) ,∂ t u (0,x)= εu 1 (x) ,x ∈ R n , (2) where L = ∂ 2 t + ∂ t - Δ,ε> 0. The nonlinearity N (u) ∈ C k (R) satisfies the estimate     d j du j N (u)     ≤ C |u| ρ-j , 0 ≤ j ≤ k ≤ ρ. The power ρ> 1+ 2 n is considered as super critical for large time since it is known that solutions of (2) blow up when the data are positive, N (u)= |u| ρ ,ε> 0, 1 < ρ ≤ 1+ 2 n , see [5], [9], [12]. From the previous works [4], [6] we know that under the condition 〈ξ〉 δ  u 0 (ξ) , 〈ξ〉 δ-1  u 1 (ξ ) ∈ L 2 with δ> n 2 , the Fourier transform of a solution to the linearized problem corresponding to (2) decays exponentially in 1