Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign

We study the initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms of different sign. By introducing a family of potential wells W-delta and corresponding outside sets V-delta or W-delta we first obtain the invariant sets and vacuum isolating of solutions. Then we get the threshold result of global existence and nonexistence of solutions. Finally we prove the global existence of solutions for the problem with critical initial conditions I(u(0)) >= 0, E(0) = d (or J(u(0)) = d).