Backward uniqueness for solutions of linear parabolic equations

Abstract

We address the backward uniqueness property for the equation u t - Δu = w j ∂ j u + vu in R x (T 0 ,0]. We show that under rather general conditions on v and w, u| t=0 = 0 implies that u vanishes to infinite order for all points (x,0). It follows that the backward uniqueness holds if w = 0 and v ∈ L∞([0,T 0 ], L p (R n )) when p > n/2. The borderline case p = n/2 is also covered with an additional continuity and smallness assumption.