Qnique Continuation for

Let u be a solution of the heat equation (1.1) {partial_derivative}u/{partial_derivative}t = {triangle}u + {sigma}{sup n}{sub i=l} b{sub i}(x,t) {center_dot} {partial_derivative}u/{partial_derivative}x{sub i} + c(x,t)u for x {epsilon} R{sup n} and t > 0. Let c and b = b{sub 1},b{sub 2},...,b{sub n} be uniformly bounded functions, i.e., (1.2) sup{sub x{epsilon}R{sup n},t>0} {parallel}c(x,t){parallel} + {parallel}b(x,t){parallel} {le} M for some constant M > 0. We will show Theorem 1.1. Suppose that both u and {triangledown}u are uniformly bounded. If u vanishes of infinite order in space-time at any point in R{sup n} X (0,{infinity}), then u is identically zero.