Stable and Unstable Sets for the Cauchy Problem for a Nonlinear Wave Equation with Nonlinear Damping and Source Terms

For the Cauchy problem for the nonlinear wave equation with nonlinear damping and source terms, we define stable and unstable sets for the initial data. We prove that if during the evolution the solution enters into the stable set, the solution is global and we are able to estimate the decay rate of the energy. If during the evolution the solution enters into the unstable set, the solution blows up in finite time.