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In this paper, an equivalence relation between theω-limit set of ini-tial values and theω-limit set of solutions is established for the Cauchy prob-lem of evolution p-Laplacian equation in the unbounded space Yσ(R N ). To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions, we establish the propagation estimate and the growth estimate for the solutions. It will be demonstrated that the equivalence relation can be used to study the asymptotic behavior of solutions.