Backward uniqueness for general parabolic operators in the whole space

Abstract

We prove the backward uniqueness for general parabolic operators of second order in the whole space under assumptions that the leading coefficients of the operator are Lipschitz and their gradients satisfy certain decay conditions. The point is that the decay rate is related to the exponential growth rate of the solution, which is quite different from the case of the half-space (Wu and Zhang in Commun Contemp Math 18(1):1550011, 2016). This result extends in some ways a classical result of Lions and Malgrange (Math Scand 8:277–286, 1960 ) and a recent result of Wu and Zhang (2016).