Two Classes Of Evolution Equations With Two Nonlinear Source Terms Of Different Sign

The initial boundary problems of the nonlinear wave equation and nonlinear reaction-diffusion equation are studied. By assuming the increasing condition, the continuity and the bound, the properties of the solutions are discussed.Firstly the family of potential wells W_δand the outside sets V_δareintroduced then their properties are given. Especially the analysis properties of the depth function of the potential wells family are obtained. By using W_δand V_δwe prove invariance of some sets in flow of the problems and the isolate of solutions. Then we get the threshold result of global existence and nonexistence ofsolutions. Finally the critical initial conditions of I(u₀)≥0, E(0) = d (orJ(u₀) = d ) and postive energy problem are discussed.