Stabilization of solutions of parabolic equations with growing leading coefficients

Precise sufficient conditions are obtained for the coefficients of a second-order parabolic equation to ensure that the solutions of the Cauchy problem with polynomially growing initial functions stabilize to zero on compact sets. It is shown, by means of an example, that these sufficient conditions cannot be improved. In the case of bounded initial functions, we find conditions on the coefficients that guarantee that the solutions of the Cauchy problem stabilize to zero at a power rate and this stabilization is uniform in the spatial variables on compact sets.