Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption

Abstract We consider the Cauchy problem for the damped wave equation with absorption (∗) u t t − Δ u + u t + | u | ρ − 1 u = 0 , ( t , x ) ∈ R + × R N . The behavior of u as t → ∞ is expected to be same as that for the corresponding heat equation ϕ t − Δ ϕ + | ϕ | ρ − 1 ϕ = 0 , ( t , x ) ∈ R + × R N . In the subcritical case 1 ρ ρ c ( N ) : = 1 + 2 / N there exists a similarity solution w b ( t , x ) with the form t − 1 / ( ρ − 1 ) f ( x / t ) depending on b = lim | x | → ∞ | x | 2 / ( ρ − 1 ) f ( | x | ) ⩾ 0 . Our first aim is to show the decay rates (∗∗) ( ‖ u ( t ) ‖ L 2 , ‖ u ( t ) ‖ L ρ + 1 , ‖ ∇ u ( t ) ‖ L 2 ) = O ( t − 1 ρ − 1 + N 4 , t − 1 ρ − 1 + N 2 ( ρ + 1 ) , t − 1 ρ − 1 − 1 2 + N 4 ) provided that the initial data without initial data size restriction spatially decays with reasonable polynomial order. The decay rates (∗∗) are sharp in the sense that they are same as those of the similarity solution. The second aim is to show that the Gauss kernel is the asymptotic profile in the supercritical case, which has been shown in case of one-dimensional space by Hayashi, Kaikina and Naumkin [N. Hayashi, E.I. Kaikina, P.I. Naumkin, Asymptotics for nonlinear damped wave equations with large initial data, preprint, 2004]. We show this assertion in two- and three-dimensional space. To prove our results, both the weighted L 2 -energy method and the explicit formula of solutions will be employed. The weight is an improved one originally developed in [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464–489].