Quantitative Decay for Linear Parabolic Equations

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/