### Abstract

We discuss the solvability and the comparison principle for the heat equation with a nonlinear boundary condition ∂tu = Δ x ∈ Ω t > 0; ∇u .v(x) = up; x ∈ ∂ Δ t > 0;(x; 0) ∝ = (x) x ∈ Ω where N ≥ 1, p > 1, is a smooth domain in RN and '(x) = O(e^{λd(x)2} ) as d(x) → 1 for some λ ≥ 0. Here, d(x) = dist (x; ∈ Ω). Furthermore, we obtain the lower estimates of the blow-up time of solutions with large initial data by use of the behavior of the initial data near the boundary ∈ Ω.

Original language | English |
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Pages (from-to) | 481-504 |

Number of pages | 24 |

Journal | Differential and Integral Equations |

Volume | 30 |

Issue number | 7-8 |

Publication status | Published - 2017 Jul 1 |

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

Ishige, K., & Sato, R. (2017). Heat equation with a nonlinear boundary condition and growing initial data.

*Differential and Integral Equations*,*30*(7-8), 481-504.