Remark on the strong unique continuation property for parabolic operators

We consider solutions u = u ( x , t ) u = u(x,t) , in a neighbourhood of ( x , t ) = ( 0 , 0 ) (x,t) =(0,0) , to a parabolic differential equation with variable coefficients depending on space and time variables. We assume that the coefficients in the principal part are Lipschitz continuous and that those in the lower order terms are bounded. We prove that, if u ( ⋅ , 0 ) u( \cdot ,0) vanishes of infinite order at x = 0 x=0 , then u ( ⋅ , 0 ) ≡ 0 u( \cdot ,0) \equiv 0 .