Damped wave equation in the subcritical case

We study large time asymptotics of small solutions to the Cauchy problem for the one dimensional nonlinear damped wave equation (1) {v_tt + v_t - v_xx + v^1+σ = 0, x ∈ R, t > 0, v (0, x) = ε v₀ (x), v_t (0, x) = ε v₁ (x) in the sub critical case σ ∈ (2 - ε³, 2). We assume that the initial data v₀, (1 + ∂_x)^-1 v₁ ∈ L^∞ ∩ L^1,a, a ∈ (0, 1) where L^1,a = { ∈ L¹; ∥φ∥ _L1a, = ∥〈·〉^a φ∥ _L1 < ∞}, 〈x〉 = 1 + x². Also we suppose that the mean value of initial data ∫_R (v₀ (x) + v₁ (x)) dx > 0. Then there exists a positive value ε such that the Cauchy problem (1) has a unique global solution v (t, x) ∈ C ([0, ∞); L^∞ ∩ L^1,a), satisfying the following time decay estimate: ∥v (t)∥ _L∞ ≤ C ε 〈t〉 ^-1/σ for large t > 0, here 2 - ε³ < σ < 2.