Global Existence and Blow-up for Semilinear Wave Equations with Variable Coefficients

Abstract

The authors consider the critical exponent problem for the variable coefficients wave equation with a space dependent potential and source term. For sufficiently small data with compact support, if the power of nonlinearity is larger than the expected exponent, it is proved that there exists a global solution. Furthermore, the precise decay estimates for the energy, L ₂ and L ^p+1 norms of solutions are also established. In addition, the blow-up of the solutions is proved for arbitrary initial data with compact support when the power of nonlinearity is less than some constant.

Keywords

Semilinear wave equations / Global existence / Energy decay / L ₂ and L ^p+1 estimates / Blow up

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