Quantitative Parabolic Decay With Bounded Drift

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 ${\mathrm{\partial }}_{t}u-\mathrm{\Delta }u=vu+w\cdot \mathrm{\nabla }u$ on ${\mathbb{R}}^n$, for $n\ge 3$, when $w\in L_{t,x}^{\mathrm{\infty }}$ and either $v$ is sufficiently small in $L_t^{\mathrm{\infty }}{L}_x^{n\mathrm{/}2}$ or $v\in L_t^{\mathrm{\infty }}{L}_x^p$ for some $n\mathrm{/}2<p\le \mathrm{\infty }$, 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 $n\ge 3$, 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.