A unique continuation property for a class of parabolic differential inequalities in a bounded domain
Guojie Zheng, Dihong Xu, Taige Wang · Communications on Pure & Applied Analysis · 2020
This article is concerned with a strong unique continuation property of a forward differential inequality abstracted from parabolic equations proposed on a convex domain $ \Omega $ prescribed with some regularity and growth conditions. Our results show that the value of the solutions can be determined uniquely by its value on an arbitrary open subset $ \omega $ in $ \Omega $ at any given positive time $ T $. We also derive the quantitative nature of this unique continuation, that is, the estimate of a $ L^2(\Omega) $ norm of the initial data, which is majorized by that of solution on the bounded open subset $ \omega $ at terminal moment $ t = T $.