Quantitative Decay for Linear Parabolic Equations

Abstract

We establish sharp temporal lower bounds for the full spatial $L^2$-norm of energy solutions to $\partial_tu-\Delta u=vu$ 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.