Summary: We prove that the logarithmic Monge-Ampère flow with uniform 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
\[ \det D^{2}u=\exp\left\{n\left(-u+\tfrac{1}{2}\, \sum_{i=1}^{n}x_{i}\;\frac{\partial u}{\partial x_{i}} \right)\right\}, \]
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
\[ \sum_{i=1}^{n} \arctan\lambda_{i}=-u+\tfrac{1}{2}\, \sum_{i=1}^{n}x_{i}\;\frac{\partial u}{\partial x_{i}} \]
must be a quadratic polynomial, where \(\lambda_{i}\) are the eigenvalues of the Hessian \(D^{2} u\).