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In this paper we prove decay estimates for solutions to a nonlocal p−Laplacian evolution problem with mixed boundary conditions, that is, for (x, t) ∈ Ω × R + and u(x, t) = 0 in Ω0 × R + . The proof of these estimates is based on bounds for the associated first eigenvalue. Mathematics Subject Classification (2010). 35B40, 35P15, 45K05.doi:10.3934/dcds.2015.35.1469 fatcat:qeufsmg6nfbbdkztq63jjcpp4m