Grow-up rates of inhomogeneous semilinear heat equations in the critical dimension
Abstract
This paper concerns the grow-up rate of solutions to a semilinear heat equation in the unit ball with the exponential nonlinearity and an inhomogeneous term f. When f=0, it is known that the large-time behavior of a solution changes at the critical dimension N=10, and the grow-up phenomenon occurs for the case N≥ 10. For the inhomogeneous case with N≥ 10, the present author and a coauthor showed in [12] that the grow-up phenomenon disappears once f exceeds a threshold. In this paper, we provide a quantitative characterization of the disappearance of the grow-up phenomenon by obtaining the grow-up rate in the critical dimension. Our result shows that a qualitatively different type of grow-up behavior occurs in the critical dimension compared with the case N≥ 11 studied in [12]. The difference is caused by a change in the outer behavior of the solution. Even in the case f=0, our result is new in that it provides a rigorous justification of the formal computation by Galaktionov and King [10].
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BibTeX
@misc{kumagai2026growup,
title = {{Grow-up rates of inhomogeneous semilinear heat equations in the critical dimension}},
author = {Kenta Kumagai},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30518},
eprint = {2608.30518},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30518},
}
RIS
TY - UNPB
TI - Grow-up rates of inhomogeneous semilinear heat equations in the critical dimension
AU - Kenta Kumagai
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30518
UR - https://arxiv.org/abs/2608.30518
AB - This paper concerns the grow-up rate of solutions to a semilinear heat equation in the unit ball with the exponential nonlinearity and an inhomogeneous term f. When f=0, it is known that the large-time behavior of a solution changes at the critical dimension N=10, and the grow-up phenomenon occurs for the case N≥ 10. For the inhomogeneous case with N≥ 10, the present author and a coauthor showed in [12] that the grow-up phenomenon disappears once f exceeds a threshold. In this paper, we provide a quantitative characterization of the disappearance of the grow-up phenomenon by obtaining the grow-up rate in the critical dimension. Our result shows that a qualitatively different type of grow-up behavior occurs in the critical dimension compared with the case N≥ 11 studied in [12]. The difference is caused by a change in the outer behavior of the solution. Even in the case f=0, our result is new in that it provides a rigorous justification of the formal computation by Galaktionov and King [10].
ER -