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

We consider the Cauchy problem for the damped wave equation utt − u+ ut = |u|ρ−1u, (t, x) ∈ R+ ×RN and the heat equation φt − φ = |φ|ρ−1φ, (t,x) ∈ R+ ×RN. If the data is small and slowly decays likely c1(1 + |x|)−kN, 0 < k 1, then the critical exponent is ρc(k) = 1 + 2 kN 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 RN, whose asymptotic profile is given by Φ0(t, x) = RN e−|x−y|2 4t (4πt)N/2 c1 (1 + |y|2)kN/2 dy provided that the data φ0 satisfies lim|x|→∞ x kNφ0(x) = c1 ( = 0). Even in the semilinear damped wave equation in the supercritical case a time global solution u with the data (u, ut )(0, x) = (u0,u1)(x) is shown in low dimensional spaces RN, N = 1, 2, 3, to have the same asymptotic profile Φ0(t, x) provided that lim|x|→∞ x kN (u0 + u1)(x) = c1 ( = 0). Those proofs are given by elementary estimates on the explicit formulas of solutions. © 2007 Elsevier Inc. All rights reserved.