In this talk I am going to present our joint results with Prof. Vasily Zhikov (Vladimir State University) on stabilization of solutions to nonlinear parabolic equations of the p-Laplace type. The result result is stabilization in the L²(Rⁿ) norm of solutions with L²(Rⁿ) initial data. The second result, which is the central part of this talk, is a criterion of the uniform stabilization of bounded solutions. This criterion generalizes a widely known criterion for linear parabolic equations proved independently by V. Zhikov and S. Kamin in 1976. The statement of the criterion is very simple: “A bounded solution converges uniformly (w.r.t. the space variables) to zero as time goes to infinity if and only if the initial data has zero uniform mean.”