A weighted semigroup approach to exponential stability in linear parabolic equations

Abstract

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 $\mu = \rho\,dx$ and identifying the corresponding sub-Markovian $C_0$-semigroup of contractions on $L^2(U, \mu)$ 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 $p \in (d, \infty)$.