### Abstract

We consider the initial-boundary value problem (p) {∂/∂tu = Δu - V(|x|)u in Ω_{L} × (0,∞), μu + (1 - μ)∂/∂vu = 0 on ∂Ω_{L} × (0, ∞), u(·, 0) = ø(·) ∈ L^{p}(Ω_{L}), p ≥ 1, where Ω_{L} = {x ∈^{N} : |x| > L}, N ≥ 2, L > 0, 0 ≤ μ ≤ 1, v is the outer unit normal vector to ∂Ω_{L}, and V is a nonnegative smooth function such that V(r) = O(r^{-2}) as r → ∞. In this paper, we study the decay rates of the derivatives ▽_{x}^{j}u of the solution u to (P) as t → ∞.

Original language | English |
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Pages (from-to) | 861-898 |

Number of pages | 38 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 59 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2007 Jul 1 |

### Keywords

- Decay rates estimate
- Linear parabolic equation
- Radial solutions

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Ishige, K., & Kabeya, Y. (2007). Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball.

*Journal of the Mathematical Society of Japan*,*59*(3), 861-898. https://doi.org/10.2969/jmsj/05930861