Affiliations: Faculty of Political Science and Economics, Waseda University, Tokyo 169-8050, Japan. E-mail: [email protected]
Abstract: We consider the Cauchy problem for the wave equation with time-dependent damping and absorbing semilinear term utt−Δu+b(t)ut+|u|ρ−1u=0, (t,x)∈R+×RN, (u,ut)(0,x)=(u0,u1)(x), x∈RN (*). 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>ρF(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 ϕt−1/b(t) Δϕ=0, (t,x)∈R+×RN. (**) 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) (utt+|u|ρ−1u). Thus, we will show θ0GB(t,x) (θ0: suitable constant) to be an asymptotic profile of u, where GB(t,x) is the fundamental solution of (**).
