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

Abstract

We consider the Cauchy problem for the damped wave equation with absorption
u_tt-Δu+u_t+|u|^ρ-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>ρ_c(N):=1+2/N. In fact, this has been shown by Karch [12] (Studia Math., 143 (2000), 175-197) for ρ>1+$¥frac{4}{N}$ (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+$¥frac{4}{N}$ (N=1,2) and ρ_c(N)<ρ<1+$¥frac{3}{N}$ (N=3). Developing their result, we will show the behavior of solutions for ρ_c(N)<ρ≤1+$¥frac{4}{N}$ (N=3), ρ_c(N)<ρ<1+$¥frac{4}{N}$ (N=4). For the proof, both the weighted L²-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.