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

Abstract

We consider the Cauchy problem for the damped wave equation u t t − Δ u + u t = | u | ρ − 1 u , ( t , x ) ∈ R + × R N and the heat equation ϕ t − Δ ϕ = | ϕ | ρ − 1 ϕ , ( t , x ) ∈ R + × R N . If the data is small and slowly decays likely c 1 ( 1 + | x | ) − k N , 0 < k ⩽ 1 , then the critical exponent is ρ c ( k ) = 1 + 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 by Φ 0 ( t , x ) = ∫ R N e − | x − y | 2 4 t ( 4 π t ) N / 2 c 1 ( 1 + | y | 2 ) k N / 2 d y provided that the data ϕ 0 satisfies lim | x | → ∞ 〈 x 〉 k N ϕ 0 ( x ) = c 1 ( ≠ 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 ) = ( u 0 , u 1 ) ( x ) is shown in low dimensional spaces R N , N = 1 , 2 , 3 , to have the same asymptotic profile Φ 0 ( t , x ) provided that lim | x | → ∞ 〈 x 〉 k N ( u 0 + u 1 ) ( x ) = c 1 ( ≠ 0 ) . Those proofs are given by elementary estimates on the explicit formulas of solutions.