Let there be given a parabolic differential operator

where A is a second order linear elliptic (<0) differential operator in an open set Ω ⊂ Rn, having coefficients depending on x ∈ Ω and t ∈ [0, ∞]. Recently, Protter (1) investigated the asymptotic behaviour of functions u(x, t) that satisfy the differential inequality

(1.1)

Under suitable restrictions on the functions ci(t) and the coefficients of A, he proved that any solution of (1.1), subject to certain homogeneous boundary conditions, that vanishes sufficiently fast, as t → ∞, must be identically zero in Ω × [0, ∞). For example, conditions are given under which no solution of (1.1) can vanish faster than e-λt, ∀ λ > 0, unless identically zero.