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+N/4, t-1/p-1+N/2(p+1), t-1/p-1-1/2+N/4), (*) 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].
ASJC Scopus subject areas
- Applied Mathematics