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In this paper we consider a nonlinear wave equation with nonlinear boundary damping and source terms, subject to a localized viscoelastic damping and localized frictional damping. We obtain the global existence of solution via potential well method. Furthermore, by introducing suitable Lyapunov functionals, using multiplier method and constructing convex differential inequality, we compare the effect of locally viscoelastic term and frictional damping and establish an explicit and general decay rate result. This stability result allows the localized viscoelastic damping and localized frictional damping acting simultaneously or complementarily in the domain, which gives the competing relationship of the viscoelastic and frictional damping. [ABSTRACT FROM AUTHOR]