We study global existence in time of small solutions to the Cauchy problem for the nonlinear damped wave equation {(partial deriv)_t~2u + (partial deriv)_tu - Δu = N(u), x ∈ R~n, t > 0, u(0,x) = εu_0(x), (partial deriv)_tu(0,x) = εu_1(x), x ∈ R~n, (0.1) 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 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 = l. Weighted Sobolev spaces are H~(l,m) = {φ ∈ L~2; ‖~m ~lφ(x)‖_(L~2) < ∞}, where = (1+x~2)~(1/2).
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