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 { ∂ 2 t u + ∂ t u - Δu + λu 1+2 n = 0, x ∈ R n , t > 0, u(0, x) = eu 0 (x), ∂ t u(0, x) = eu 1 (x), x ∈ R n , where e > 0, and space dimensions n = 1, 2,3. Assume that the initial data uo ∈ H δ,0 n H 0,δ , u 1 e H δ-1,0 n H -1,δ , where δ > n 2, weighted Sobolev spaces are H l,m = {Φ e L2; m l Φ(x)∥ L 2 = √1 + x 2 . Also we suppose that λθ 2/n > 0, ∫u 0 (x) dx > 0, where Then we prove that there exists a positive e 0 such that the Cauchy problem above has a unique global solution u ∈ C ([0, oo); H δ,0 ) satisfying the time decay property ∥u(t)-eθG(t,x)e -φ(t) ∥ Lp ≤ Ce 1+2 n g -1-n 2 (t) -n 2(1-1/p) for all t > 0, 1 ≤ p ≤ ∞, where e ∈ (0, e 0 ].