Hölder estimate for non-uniform parabolic equations in highly heterogeneous media

Uniform bound for the solutions of non-uniform parabolic equations in highly heterogeneous media is concerned. The media considered are periodic and they consist of a connected high permeability sub-region and a disconnected matrix block subset with low permeability. Parabolic equations with diffusion depending on the permeability of the media have fast diffusion in the high permeability sub-region and slow diffusion in the low permeability subset, and they form non-uniform parabolic equations. Each medium is associated with a positive number ϵϵ, denoting the size ratio of matrix blocks to the whole domain of the medium. Let the permeability ratio of the matrix block subset to the connected high permeability sub-region be of the order ϵ2τϵ2τ for τ∈(0,1]τ∈(0,1]. It is proved that the Hölder norm of the solutions of the above non-uniform parabolic equations in the connected high permeability sub-region are bounded uniformly in ϵϵ. One example also shows that the Hölder norm of the solutions in the disconnected subset may not be bounded uniformly in ϵϵ.