Abstract In this paper, we proved two types of uniqueness results for solutions to parabolic equations whose traces at time T > 0 T>0, for large spatial variables satisfy exponential decay assumption. For solutions to parabolic equations in unbounded domains, we can state the main results as follows. The first uniqueness asserts that if | u (T, x) | ≤ C e - | x | 2 4 T |u (T, x) | Ce^{-|x|^{24T}} as | x | → ∞ |x| with some T > 0 T>0 and a potential satisfies some restrictions, then u (t, x) = 0 u (t, x) =0 for 0 t T 0 x ∈ ℝ n {x^{n}. The second result asserts that if | u (t, x) | + | ∇ u <
Imanuvilov et al. (Thu,) studied this question.