Damped wave equation with a critical nonlinearity

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity {∂_t²u+∂_tu-△u+λu ^1+2/n = 0, x ∈ Rⁿ, t>0, u(0,x) = εu ₀ (x), ∂_tu(0,x) = εu₁ (x), x ∈ Rⁿ, where ε > 0, and space dimensions n = 1,2,3, Assume that the initial data u₀ ∈ H^δ,0 ∩ H ^0,δ, u₁ ∈ H^δ-1,0 ∩ H ^-1,δ, where δ>n/2, weighted Sobolev spaces are H ^l,m = {Φ ∈ L²;∥x^mi∂x ^l Φ (x)∥_L2 <∞}, 〈x〉 = √1+x². Also we suppose that λθ ^2/n>0∫u₀(x)dx>0, where θ = ∫ (u ₀ (x) + u₁ (x)) dx. Then we prove that there exists a positive ε₀ such that the Cauchy problem above has a unique global solution u ∈ C ([0, ∞); H^δ,0) satisfying the time decay property ∥u(t)-εθG(t,x)e^-ℓ(t)∥ _Lp ≤Cε^1+2/ng^-1-n/2(t) 〈t〉^-n/2(1-1/p) for all t > 0, 1 ≤ p ≤ ∞ where ε ∈ (0, ε₀].