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This paper is concerned with the study of the uniform decay rates of the energy associated with mixed problems involving the wave equation with nonlinear localized damping. The domain is an unbounded open set of R2 with finite measure and has an unbounded smooth boundary Γ = ΓN ∪ ΓD such that ΓN ∩ ΓD ≠ ∅. On ΓD and ΓN we place the homogeneous Dirichlet and Neumann boundary conditions, respectively. Due to lack of local regularity, we used the elliptic decomposition of the solution as did Grisvard [J. Math. Pures Appl. (9), 68 (1989), pp. 215-259] combined with the use of appropriated cutoff functions. [ABSTRACT FROM AUTHOR]