ASYMPTOTIC BEHAVIOUR OF SOLUTIONS FOR THE NONLINEAR DISSIPATIVE WAVE EQUATION

We investigate the asymptotic behaviour of solutions to the initial-boundary value problem for the nonlinear dissipative wave equation in the whole space or the exterior domain outside a star-shaped obstacle. We shall treat the nonlinear dissipative term like a₁(1 + |x|)^-δ|u_t|^βu_t (a₁, β, δ > 0) and prove that the energy does not in general decay. Further, we can deduce that the classical solution is asymptotically free and the local energy decays at a certain rate as t goes to infinity.