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+ ∂tu−∆u+ λu 2 n = 0, x ∈ Rn, t > 0, u(0, x) = εu0 (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 ∩H, u1 ∈ Hδ−1,0 ∩H−1,δ, where δ > n 2 , weighted Sobolev spaces are H = { φ ∈ L; ∥∥∥〈x〉m 〈i∂x〉 φ (x)∥∥∥ L2 < ∞ } , 〈x〉 = √ 1 + x2. Also we suppose that λθ 2 n > 0, ∫ u0 (x) dx > 0, where θ = ∫ (u0 (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,∞) ;Hδ,0 ) satisfying the time decay property ∥∥∥u (t)− εθG (t, x) e−φ(t)∥∥∥ Lp ≤ Cε 2 n g−1− n 2 (t) 〈t〉 n 2 ( 1− 1 p ) for all t > 0, 1 ≤ p ≤ ∞, where ε ∈ (0, ε0] .