Quantitative Decay for Linear Parabolic Equations
Abstract
We establish sharp temporal lower bounds for the full spatial L2-norm of energy solutions to ∂tu−Δu=vu on ℝn, with n≥3. For global solutions, nonvanishing at a single time implies the lower bound ce−Ct when v is small in the scale-invariant space L∞tLn/2x, and ce−Ct2 throughout the subcritical range v∈ L∞tLpx, 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 ℝ3. 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.
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BibTeX
@misc{kukavica2026quantitative,
title = {{Quantitative Decay for Linear Parabolic Equations}},
author = {Igor Kukavica and Qi Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30630},
eprint = {2609.30630},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30630},
}
RIS
TY - UNPB
TI - Quantitative Decay for Linear Parabolic Equations
AU - Igor Kukavica
AU - Qi Xu
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30630
UR - https://arxiv.org/abs/2609.30630
AB - We establish sharp temporal lower bounds for the full spatial L^2-norm of energy solutions to ∂_tu−Δu=vu on ℝ^n, with n≥3. For global solutions, nonvanishing at a single time implies the lower bound ce^−Ct when v is small in the scale-invariant space L^∞_tL^n/2_x, and ce^−Ct^2 throughout the subcritical range v∈ L^∞_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 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.
ER -