Approximate two-sphere one-cylinder inequality in parabolic periodic homogenization with suitable lower-order terms

Abstract

We continue the study of approximate propagation of smallness in parabolic periodic homogenization in [SIAM J. Math. Anal., 53(5):5835–5852, 2021], where by using the asymptotic behaviors of fundamental solutions and the Lagrange interpolation technique, we obtained the approximate two-sphere one-cylinder inequality in parabolic homogenization. In this paper, we consider the parabolic equations with suitable lower-order terms in homogenization, and the approximate two-sphere one-cylinder inequality continues to hold. The difficulty is to handle the more “worse" junior coefficients in parabolic homogenization. The results obtained in the paper can be easily extended to the elliptic case, which are also new in elliptic homogenization.