AI 中文总结
该论文研究带界漂移和任意小势能的抛物型方程，证明能量解的双指数衰减下界，并构造最优例子说明该衰减尺度不可改进。
AI 中文摘要
一个有界的时间依赖漂移，加上一个任意小的实值势能，可以导致全空间$L^2$范数的双指数衰减。我们证明了在$\mathbb{R}^n$（$n\geq3$）上，对于$\partial_tu-\Delta u=vu+w\cdot\nabla u$的能量解，当$w\in L^\infty_{t,x}$且要么$v$在$L^\infty_tL^{n/2}_x$中足够小，要么$v\in L^\infty_tL^p_x$对某个$n/2<p\leq\infty$成立（无需小性假设）时，匹配的定量双时间下界成立。特别地，一个非零全局解在大时间满足双指数下界。这个衰减尺度对于这些系数类是最优的：在每个维度$n\geq3$中，我们构造了一个实值有界漂移、一个在临界空间和任何指定的次临界$L^p$空间中任意小的实值势能，以及一个具有双指数衰减的非零实值能量解。该构造通过吸收局部化误差来局部化Rowan的环面例子。
英文摘要
A bounded time-dependent drift together with an arbitrarily small real potential can produce double exponential decay of the full spatial $L^2$ norm. We prove matching quantitative two-time lower bounds for energy solutions of $\partial_tu-Δu=vu+w\cdot\nabla u$ on $\mathbb{R}^n$, for $n\geq3$, when $w\in L^\infty_{t,x}$ and either $v$ is sufficiently small in $L^\infty_tL^{n/2}_x$ or $v\in L^\infty_tL^p_x$ for some $n/2<p\leq\infty$, without a smallness assumption. In particular, a nonzero global solution satisfies a double exponential lower bound at large times. This decay scale is optimal for these coefficient classes: In every dimension $n\geq3$, we construct a real bounded drift, a real potential arbitrarily small in the critical space and in any prescribed subcritical $L^p$ space, and a nonzero real energy solution with double exponential decay. The construction localizes Rowan's torus example by absorbing the localization error.
