The global existence and asymptotic behavior for anisotropic parabolic equation in  are considered. By using the classical Galerkin approximation and suitable conditions on the weighted function, it is obtained the global estimate and the uniformly estimates for the global weak solution. Besides, the 2 () ∩ () global attractor for anisotropic parabolic-Laplacian equation is also investigated by proving the-limit compactness for the multivalued semigroups or multivalued semiflows.