“…Waurick showed in [11] that as N → ∞, the sequence of solutions (u N ) N ∈N converges weakly in L 2 loc ([0, 1] × R) to u, the solution to the limit equation 1 2 ∂ 2 t u(ξ, t) + ∂ t u(ξ, t) + 1 2 u(ξ, t) − 2∂ 2 ξ u(ξ, t) = f (ξ, t) + ∂ t f (ξ, t), t ∈ R, subject to zero initial conditions and Neumann boundary on both ends. Furthermore, he showed that this limit admitted exponentially stable solutions in the sense of [10]. However, when the elliptic part, u N (ξ, t) − ∂ 2 ξ u N (ξ, t) = f (ξ, t), is replaced with the corresponding parabolic part, ∂ t u N (ξ, t) − ∂ 2 ξ u N (ξ, t) = f (ξ, t), the limit equation becomes 1 2 ∂ 2 t u(ξ, t) + ∂ t u(ξ, t) − 2∂ 2 ξ u(ξ, t) = f (ξ, t) + ∂ t f (ξ, t), t ∈ R, subject again to zero initial conditions and Neumann boundary.…”