The necessary and sufficient conditions of Hyers-Ulam stability for a class of parabolic equation

The aim of this paper is to consider the Hyers-Ulam stability of a class of parabolic equation { ∂u ∂t − a 2∆u+ b · ∇u+ cu = 0, (x, t) ∈ Rn × (0,+∞), u(x, 0) = φ(x), x ∈ Rn. We conclude that (i) it is Hyers-Ulam stable on any finite interval; (ii) if c 6= 0, it is Hyers-Ulam stable on the semi-infinite interval; (iii) if c = 0, it is not Hyers-Ulam stable on the semi-infinite interval by using Fourier transformation. Furthermore, our results can be applied to the mean square Hyers-Ulam stability of parabolic equations driven by an ndimensional Brownian motion.