# American Institute of Mathematical Sciences

January  2007, 7(1): 171-189. doi: 10.3934/dcdsb.2007.7.171

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

 1 Department of Applied Mathematics, Harbin Engineering University, Harbin 150001, China, China

Received  October 2005 Revised  June 2006 Published  October 2006

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 $V_\delta$ of $W_\delta$ 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 $I(u_0)\ge 0$, $E(0)=d$ (or $J(u_0)=d$).
Citation: Yacheng Liu, Runzhang Xu. Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 171-189. doi: 10.3934/dcdsb.2007.7.171
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