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 u_tt - Δu + u_t + u ^p-1u = 0, (t,x) ∈ R₊ × R^N. The behavior of u as t → ∞ is expected to be the same as that for the corresponding heat equation φ_t - Δφ + φ ^p-1φ = 0, (t,x) ∈ R₊ × R^N, which has the similarity solution w_a(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 p_c(N) := 1+2/N. In this paper, as a first step, if 1 < p < p_c(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 w_a (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].