Optimal three cylinder inequality at the boundary for solutions to parabolic equations and unique continuation properties

Let G be a portion of a C1,a boundary of an n-dimensional domain D. Let u be a solution to a second order parabolic equation in D x (-T, T) and assume that u = 0 on G x (-T, T), 0 G. We prove that u satisfies a three cylinder inequality near G x (-T, T). As a consequence of the previous result we prove that if u (x, t) = O (xk) for every t (-T, T) and every k N, then u is identically equal to zero.