非齐次半线性热方程的大时间行为与成长速率
Large-time behavior and grow-up rates of inhomogeneous semilinear heat equations
AI总结：
该研究针对带非齐次项的半线性热方程，揭示了非齐次项阈值对分岔结构及成长现象的影响，定量给出高维精确成长速率并发现N=11时的特殊对数-对数修正现象。
AI中文摘要：
我们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。已知当$f=0$时，定常问题的分岔结构在临界维数$N=10$处发生定性变化，这种变化会影响热方程解的大时间行为，尤其当$N\ge 10$时会出现成长（grow-up）现象。\n 本文证明，一旦$f$超过某个阈值，分岔结构会转变为$f=0$情形下不存在的类型，分岔结构的变化会导致阈值以上成长现象消失。此外，我们通过确定$N\ge 11$时的精确成长速率，对这一转变给出了定量刻画。特别地，我们在阈值情形中发现了一种新的依赖维数的现象：仅当$N=11$时，成长速率中会出现对数-对数型修正项。
英文摘要：
We consider the semilinear heat equation in the unit ball with the exponential nonlinearity and an inhomogeneous term $f$. When $f=0$, it is known that the bifurcation structure of the stationary problem undergoes a qualitative change at the critical dimension $N=10$. This change affects the large-time behavior of solutions to the heat equation, and in particular, the grow-up phenomenon occurs for $N\ge 10$. In this paper, we show that once $f$ exceeds a threshold, the bifurcation structure changes to a type that does not appear in the case $f=0$. The change in the bifurcation structure leads to the disappearance of the grow-up phenomenon beyond the threshold. Moreover, we provide a quantitative characterization of this transition by determining the sharp grow-up rates for $N\ge 11$. In particular, we identify a new dimension-specific phenomenon in the threshold case: a log-log type correction term emerges in the grow-up rate only for $N=11$.
