Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption

We consider the Cauchy problem for the damped wave equation with absorption utt −Δu+ ut + |u|p−1u = 0, (t,x)∈ R+ ×RN. The behavior of u as t→∞is expected to be the same as that for the corresponding heat equation φt −Δφ + |φ|p−1φ = 0, (t,x)∈ R+ ×RN, which has the similarity solution wa(t, x) with the form t−1/(p−1)f (x/√t) depending on a = lim|x|→∞ |x|2/(p−1)f (x) 0 provided that p is less than the Fujita exponent pc(N) := 1 + 2/N. In this paper, as a first step, if 1 < p < pc(N) and the data (u0,u1)(x) decays exponentially as |x| → ∞ without smallness condition, the solution is shown to decay with rates as t→∞, u(t) L2 , u(t) Lp+1 , ∇u(t) L2 = O t− 1 p−1+N4 , t− 1 p−1+ N 2(p+1) , t− 1 p−1−12 +N4 , (∗) those of which seem to be reasonable, because the similarity solution wa(t, x) has the same decay rates as (∗). For the proof, the weighted L2-energy method will be employed with suitable weight, similar to that in Todorova and Yordanov [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464–489].  2005 Elsevier Inc. All rights reserved.