  The authors prove that the logarithmic Monge-Amp\`{e}re flow with uniformly
bound and convex initial data satisfies uniform decay estimates away from time
$t=0$. Then applying the decay estimates, we conclude that every entire
classical strictly convex solution of the equation {equation*} \det
D^{2}u=\exp\{n(-u+1/2\sum_{i=1}^{n}x_{i}\frac{\partial u}{\partial x_{i}})\},
{equation*} should be a quadratic polynomial if the inferior limit of the
smallest eigenvalue of the function $|x|^{2}D^{2}u$ at infinity has an uniform
positive lower bound larger than $2(1-1/n)$. Using a similar method, we can
prove that every classical convex or concave solution of the equation
{equation*}
\sum_{i=1}^{n}\arctan\lambda_{i}=-u+1/2\sum_{i=1}^{n}x_{i}\frac{\partial
u}{\partial x_{i}}. {equation*} must be a quadratic polynomial, where
$\lambda_{i}$ are the eigenvalues of the Hessian $D^{2}u$.
