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 absorptionutt−Δu+ut+|u|p−1u=0,(t,x)∈R+×RN. The behavior of u   as t→∞t→∞ is expected to be the same as that for the corresponding heat equationϕt−Δϕ+|ϕ|p−1ϕ=0,(t,x)∈R+×RN, which has the similarity solution wa(t,x)wa(t,x) with the form t−1/(p−1)f(x/t) depending on a=lim|x|→∞|x|2/(p−1)f(x)⩾0a=lim|x|→∞|x|2/(p−1)f(x)⩾0 provided that p   is less than the Fujita exponent pc(N):=1+2/Npc(N):=1+2/N. In this paper, as a first step, if 1