AI 中文总结
本文研究带指数非线性项和非齐次项的半线性热方程在临界维下的解成长速率，定量刻画非齐次项导致成长现象消失的机制，发现临界维与N≥11情形的成长行为存在定性差异，并严格证明了相关形式化计算结果。
AI 中文摘要
本文研究单位球内带指数非线性项和非齐次项$f$的半线性热方程解的成长速率。当$f=0$时，已知解的大时间行为在临界维$N=10$处发生变化，且当$N\ge 10$时会出现成长现象。对于$N\ge 10$的非齐次情形，本文作者与合作者曾在文献[12]中证明，一旦$f$超过某一阈值，成长现象就会消失。\n 本文通过获取临界维下的成长速率，对成长现象的消失给出了定量刻画。结果表明，与文献[12]中研究的$N\ge 11$情形相比，临界维下会出现性质不同的成长行为，这种差异由解的外部行为的变化所导致。即使在$f=0$的情形下，本文结果也具有创新性，它为Galaktionov和King[10]的形式化计算提供了严格证明。
英文摘要
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 $N\ge 10$. For the inhomogeneous case with $N\ge 10$, 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 $N\ge 11$ 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].
Comments24 pages
