Backward uniqueness for parabolic operators with non-Lipschitz coefficients

In this paper we study the backward uniqueness for parabolic equations with non-Lipschitz coefficients in time and space. The result presented here improves an old uniqueness theorem due to Lions and Malgrange [Math. Scand. $\mathbf{8}$ (1960), 277–286] and some more recent results of Del Santo and Prizzi [J. Math. Pures Appl. $\mathbf{84}$ (2005), 471–491; Ann. Mat. Pura Appl., to appear].