Backward uniqueness for parabolic operators with variable coefficients in a half space

It is shown that a function u satisfying vertical bar partial derivative(t)u+ Sigma(i,j) partial derivative(i)(a(ij)partial derivative(j)u)vertical bar <= N(vertical bar u vertical bar+vertical bar del u vertical bar), vertical bar u( x, t)vertical bar <= Ne-N| x| 2 in R-+(n) x [0, T] and u(x, 0) = 0 in R-+(n) under certain conditions on {a(ij)} must vanish identically in R-+(n) x [0, T]. The main point of the result is that the conditions imposed on {a(ij)} are of this type: {a(ij)} are Lipschitz and vertical bar del(x)a(ij) (x, t)vertical bar <= E/vertical bar x vertical bar, where E is less than a given number, and the conditions are optimal in some sense.