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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

Yacheng Liu 1, and  Runzhang Xu 1,

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 $J(u_0)=d$).
Keywords: global existence, semilinear wave equations, potential wells, vacuum isolating., reaction-diffusion equations, nonexistence.
Mathematics Subject Classification: Primary: 35C75, 35G20; Secondary: 35G3.
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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