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A weighted semigroup approach to exponential stability in linear parabolic equations
Abstract
This paper establishes the exponential L2-stability of the unique solutions to initial-boundary value problems for linear parabolic partial differential equations with general drift and zero-order coefficients in bounded domains. The key idea lies in constructing a suitable Dirichlet form with respect to a weighted measure μ= ρ dx and identifying the corresponding sub-Markovian C0-semigroup of contractions on L2(U, μ) with the unique weak solution. Remarkably, the exponential L2-stability remains valid even when the zero-order term vanishes, and it holds robustly for all drift coefficients H ∈ Lp(U, ℝd) with p ∈ (d, ∞).
Related
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BibTeX
@misc{lee2026weighted,
title = {{A weighted semigroup approach to exponential stability in linear parabolic equations}},
author = {Haesung Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05173},
eprint = {2609.05173},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05173},
}
RIS
TY - UNPB
TI - A weighted semigroup approach to exponential stability in linear parabolic equations
AU - Haesung Lee
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05173
UR - https://arxiv.org/abs/2609.05173
AB - This paper establishes the exponential L^2-stability of the unique solutions to initial-boundary value problems for linear parabolic partial differential equations with general drift and zero-order coefficients in bounded domains. The key idea lies in constructing a suitable Dirichlet form with respect to a weighted measure μ= ρ dx and identifying the corresponding sub-Markovian C_0-semigroup of contractions on L^2(U, μ) with the unique weak solution. Remarkably, the exponential L^2-stability remains valid even when the zero-order term vanishes, and it holds robustly for all drift coefficients H ∈ L^p(U, ℝ^d) with p ∈ (d, ∞).
ER -