We derive the total energy decay E(t) ≤ I0(1 + t)-1 and L2 boundedness ∥u(t)∥2 ≤ CIo for the solutions to the initial boundary value problem for the wave equation in an exterior domain Ω: utt - Δu + a(x)ut = 0 in Ω × (0, ∞) with u(x, 0) = u0(x), ut(x, 0) = u1(x) and u|∂Ω = 0, where I0 = ∥u0∥H1 + ∥u1∥2 and a(x) is a nonnegative function which is positive near some part of the boundary ∂Ω and near infinity. We apply these estimates to prove the global existence of decaying solutions for semilinear wave equations with nonlinearity f(u) like |u|αu, α > 0. We note that no geometrical condition is imposed on the boundary ∂Ω.
ASJC Scopus subject areas
- 数学 (全般)