Asymptotically self-similar solutions of the damped wave equation

We consider the damped hyperbolic equation equation(1)εuττ+uτ=(a(ξ)uξ)ξ−|u|p−1u,(ξ,τ)∈R×R+, where ε>0,a(ξ)→1ε>0,a(ξ)→1 as |ξ|→+∞|ξ|→+∞ and 1