## Abstract

We show the non-uniform bound for a solution to the Cauchy problem of a drift-diffusion equation of a parabolic-elliptic type in higher space dimensions. If an initial data satisfies a certain condition involving the entropy functional, then the corresponding solution to the equation does not remain uniformly bounded in a scaling critical space. In other words, the solution grows up at ∞ in the critical space or blows up in a finite time. Our presenting results correspond to the finite time blowing up result for the two-dimensional case. The proof relies on the logarithmic entropy functional and a generalized version of the Shannon inequality. We also give the sharp constant of the Shannon inequality.

Original language | English |
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Pages (from-to) | 145-183 |

Number of pages | 39 |

Journal | Analysis and Applications |

Volume | 14 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2016 Jan 1 |

## Keywords

- Critical space
- Drift-diffusion
- Finite time blow up
- Generalized Shannon's inequality
- Unbounded solution
- Virial laws

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics