A unique continuation property for a class of parabolic differential inequalities in a bounded domain

This article is concerned with the 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 result shows 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 on $\Omega$, which is majorized by that of solution on the bounded open subset $\omega$ at terminal moment $t = T$.