Quantitative uniqueness of solutions to parabolic equations

We investigate the quantitative uniqueness of solutions to parabolic equations with lower order terms on compact smooth Riemannian manifolds. Quantitative uniqueness is a quantitative form of strong unique continuation property. We characterize quantitative uniqueness by the rate of vanishing. We can obtain the sharp vanishing order for solutions in term of the C1,1 norm of the potential functions, as well as the L∞ norm of the coefficient functions. Some new quantitative Carleman estimates and three cylinder inequalities are established.