AI 中文总结
本文建立了线性抛物方程能量解全空间L²范数的尖锐时间下界，证明在次临界势下二次指数最优，并构造局部化Meshkov例子验证尖锐性。
AI 中文摘要
我们在$\mathbb{R}^n$（$n\geq3$）上建立了$\partial_tu-\Delta u=vu$的能量解的全空间$L^2$范数的尖锐时间下界。对于全局解，当$v$在尺度不变空间$L^infty_tL^{n/2}_x$中很小时，在单个时刻非消失意味着下界$ce^{-Ct}$；而在次临界范围$v\in L^infty_tL^p_x$（$p>n/2$）中，无需小性条件，下界为$ce^{-Ct^2}$。线性时间指数对于自由热方程已经是最优的，而我们证明二次指数在$\mathbb{R}^3$上的有界复势情形下是最优的。势不需要额外的正则性。证明使用了一个凸化的混合Carleman估计，其范数与最终时间截断一起设计。对空间中心积分去除了热核权重，并给出了全解范数的定量比较。为了证明欧几里得能量类中的二次尖锐性，我们构造了一个依赖于时间的函数$g$，它局部化了Meshkov的周期抛物例子，同时使诱导势在周期剖面的零点附近保持有界。
英文摘要
We establish sharp temporal lower bounds for the full spatial $L^2$-norm of energy solutions to $\partial_tu-Δu=vu$ on $\mathbb{R}^n$, with $n\geq3$. For global solutions, nonvanishing at a single time implies the lower bound $ce^{-Ct}$ when $v$ is small in the scale-invariant space $L^\infty_tL^{n/2}_x$, and $ce^{-Ct^2}$ throughout the subcritical range $v\in L^\infty_tL^p_x$, with $p>n/2$, without smallness. The linear time exponent is optimal already for the free heat equation, while we show that the quadratic exponent is optimal for bounded complex potentials on $\mathbb{R}^3$. No additional regularity of the potential is required. The proof uses a convexified mixed Carleman estimate whose norms are designed together with the final time truncation. Integration over spatial centers removes the heat-kernel weight and yields quantitative comparisons of the full solution norms. To prove quadratic sharpness in the Euclidean energy class, we construct a time-dependent function $g$ that localizes Meshkov's periodic parabolic example while keeping the induced potential bounded across the zeros of the periodic profile.
