Global asymptotics for the damped wave equation with absorption in higher dimensional space

We consider the Cauchy problem for the damped wave equation with absorption u_tt - Δu + u_t + |u|^p-1u = 0, (t,x) ∈ R₊ × R^N, (*) with N = 3,4. The behavior of u as t → ∞ is expected to be the Gauss kernel in the supercritical case ρ > ρa_c(N) := 1 + 2/N. In fact, this has been shown by Karch [12] (Studia Math., 143 (2000), 175-197) for ρ > 1 + 4/N (N = 1,2,3), Hayashi, Kaikina and Naumkin [8] (preprint (2004)) for ρ > ρ_c(N)(N = 1) and by Ikehata, Nishihara and Zhao [11] (J. Math. Anal. Appl., 313 (2006), 598-610) for ρ_c(N) < ρ ≤ 1+ 4/N(N = 1,2) and ρ_c(N) < p < 1 + 3/N(N = 3). Developing their result, we will show the behavior of solutions for ρ_c(N) < p ≤ 1 + 4/N (N = 3), ρ_c(N) < ρ < 1 + 4/N (N = 4). For the proof, both the weighted L ²-energy method with an improved weight developed in Todorova and Yordanov [22] (J. Differential Equations, 174 (2001), 464-489) and the explicit formula of solutions are still usefully used. This method seems to be not applicable for N = 5, because the semilinear term is not in C² and the second derivatives are necessary when the explicit formula of solutions is estimated.