Asymptotic behavior of solutions for the damped wave equation with slowly decaying data

We consider the Cauchy problem for the damped wave equationu_{t t} - Δ u + u_t = | u |^{ρ - 1} u, (t, x) ∈ R₊ × R^N and the heat equationφ{symbol}_t - Δ φ{symbol} = | φ{symbol} |^{ρ - 1} φ{symbol}, (t, x) ∈ R₊ × R^N . If the data is small and slowly decays likely c₁ (1 + | x |)^{- k N}, 0 < k ≤ 1, then the critical exponent is ρ_c (k) = 1 + frac(2, k N) for the semilinear heat equation. In this paper it is shown that in the supercritical case there exists a unique time global solution to the Cauchy problem for the semilinear heat equation in any dimensional space R^N, whose asymptotic profile is given byΦ₀ (t, x) = under(∫, R^N) frac(e^{- frac(| x - y |2, 4 t)}, (4 π t)^{N / 2}) frac(c₁, (1 + | y |²)^{k N / 2}) d y provided that the data φ{symbol}₀ satisfies lim_{| x | → ∞} 〈 x 〉^{k N} φ{symbol}₀ (x) = c₁(≠ 0) . Even in the semilinear damped wave equation in the supercritical case a time global solution u with the data (u, u_t) (0, x) = (u₀, u₁) (x) is shown in low dimensional spaces R^N, N = 1, 2, 3, to have the same asymptotic profile Φ₀ (t, x) provided that lim_{| x | → ∞} 〈 x 〉^{k N} (u₀ + u₁) (x) = c₁(≠ 0) . Those proofs are given by elementary estimates on the explicit formulas of solutions.