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

θ₀G_B(t,x)  (θ₀: suitable constant)

to be an asymptotic profile of u, where G_B(t,x) is the fundamental solution of (**).