We consider the Cauchy problem for the damped wave equation with absorption utt - Δu + ut + |u|p-1u = 0, (t,x) ∈ R+ × RN, (*) with N = 3,4. The behavior of u as t → ∞ is expected to be the Gauss kernel in the supercritical case ρ > ρac(N) := 1 + 2/N. In fact, this has been shown by Karch  (Studia Math., 143 (2000), 175-197) for ρ > 1 + 4/N (N = 1,2,3), Hayashi, Kaikina and Naumkin  (preprint (2004)) for ρ > ρc(N)(N = 1) and by Ikehata, Nishihara and Zhao  (J. Math. Anal. Appl., 313 (2006), 598-610) for ρc(N) < ρ ≤ 1+ 4/N(N = 1,2) and ρc(N) < p < 1 + 3/N(N = 3). Developing their result, we will show the behavior of solutions for ρc(N) < p ≤ 1 + 4/N (N = 3), ρc(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  (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.
ASJC Scopus subject areas
- 数学 (全般)