Asymptotic profile of solutions for 1-D wave equation with time-dependent damping and absorbing semilinear term

We consider the Cauchy problem for the wave equation with time-dependent damping and absorbing semilinear term u_tt-Δu+b(t)u _t+|u|^ρ-1u=0, (t,x)∈R₊×R ^N, (u,u_t)(0,x)=(u₀,u₁)(x), x∈R^N. When b(t)=b₀(t+1)^-β with -1<β<1 and b₀>0, we want to seek for the asymptotic profile as t→∞ of the solution u to in the supercritical case ρ>ρ_F(N):=1+2/N. By the weighted energy method we can show the basic decay rates of u, which are almost the same as those to the corresponding linear parabolic equation φ_t-1/b(t)Δφ=0, (t,x)∈R₊×R^N. When N=1, the decay rates of higher order derivatives of u are obtained by the energy method, so that the solution u can be regarded as that of with source term -1/b(t)(u _tt+|u|^ρ-1u). Thus, we will show θ ₀G_B(t,x) (θ₀: suitable constant) to be an asymptotic profile of u, where G_B(t,x) is the fundamental solution of.