Asymptotic Behavior of Solutions to the Semilinear Wave Equation with Time-dependent Damping

We consider the Cauchy problem for the semilinear wave equation with time-dependent damping {u(tt) - Delta u + b(t)u(t) = f (u), (t, x) epsilon R+ x R-N (u, u(t))(0, x) = (u(0), u(1))(x), x epsilon R-N. (*) When b(t) = (t + 1)(-beta) with 0 <= beta < 1, the damping is effective and the solution u to (*) behaves as that to the corresponding parabolic problem. When f (u) = O(vertical bar u vertical bar(rho)) as u -> 0 with 1 < rho < N+2/[N-2](+) (the Sobolev exponent), our main aim is to show the time-global existence of solutions for small data in the supercritical exponent rho > rho(F) (N) := 1 + 2/N. We also obtain some blow-up results on the solution within a finite time, so that the smallness of the data is essential to get global existence in the supercritical exponent case.