## Abstract

We consider the isotropic initial boundary value problem for the heat equation on open regions with noncompact boundary and construct differential inequalities for a generalized heat flow measure defined over a spherical cross section. Under suitable assumptions, integration of the differential inequality leads to spatial growth and decay rate estimates for mean-square cross-sectional measures of the time-weighted temperature spatial gradient. The estimates are then used to obtain similar results for the time-weighted temperature. In particular, when the base heat flow measure is positive, the time-weighted temperature becomes pointwise unbounded at large spatial distance.

Original language | English |
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Pages (from-to) | 1049-1076 |

Number of pages | 28 |

Journal | Studies in Applied Mathematics |

Volume | 152 |

Issue number | 4 |

Early online date | 16 Jan 2024 |

DOIs | |

Publication status | Published - May 2024 |

## Keywords

- heat equation
- noncompact region
- spatial growth and decay rate estimates

## ASJC Scopus subject areas

- Applied Mathematics