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Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign. (English) Zbl 1121.35085

The paper deals with solutions to the equation \(u_{tt}-\triangle u=f(u)\) satisfying Dirichlet boundary condition, where \(f(u)\) is a sum of the nonlinearities \(b_k| u| ^{p-1}u\). The authors introduce two families of the potential wells and deduce their properties. Then they prove global existence and finite time blow up of the solutions by use of the family of the potential wells. A threshold results of global existence and nonexistence of the solution are also given.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35G30 Boundary value problems for nonlinear higher-order PDEs
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