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 t t - Δ u + u t + | u | ρ - 1 u = 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 + 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 ) < ρ < 1 + 3 N ( N = 3 ) . Developing their result, we will show the behavior of solutions for ρ c ( N ) < ρ ≤ 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 C 2 and the second derivatives are necessary when the explicit formula of solutions is estimated.