AI 中文总结
本文采用加权半群方法，证明了带一般漂移项和零阶系数的线性抛物型方程初边值问题解的指数L²稳定性，该结果对漂移项具有鲁棒性。
AI 中文摘要
本文针对有界域内带有一般漂移项和零阶系数的线性抛物型偏微分方程的初边值问题，证明了其唯一解的指数L²稳定性。核心思路是构造关于加权测度μ=ρdx的合适狄利克雷形式，并将L²(U,μ)上对应的次马尔可夫C₀压缩半群与唯一弱解对应起来。值得注意的是，即使零阶项消失，指数L²稳定性仍然成立，且对所有满足p∈(d,∞)的漂移项H∈Lᵖ(U,ℝᵈ)均具有鲁棒性。
英文摘要
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 $\mathbf{H} \in L^p(U, \mathbb{R}^d)$ with $p \in (d, \infty)$.
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