### Abstract

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 ∈, 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 ∈ ^{2τ} for 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 . One example also shows that the Hölder norm of the solutions in the disconnected subset may not be bounded uniformly in ∈.

Original language | English |
---|---|

Pages (from-to) | 3723-3745 |

Number of pages | 23 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 75 |

Issue number | 9 |

DOIs | |

State | Published - 1 Jun 2012 |

### Keywords

- Highly heterogeneous media
- Infinitesimal generator
- Numerical range
- Paramatrix
- Pseudo-differential operator
- Strict solution