Asymptotic behavior at infinity and existence of solutions to the Lagrangian mean curvature flow equation in $\mathbb R^{n+1}_-$

Abstract: This paper investigates the asymptotic behavior at infinity of ancient solutions to the Lagrangian mean curvature flow equation. Under conditions that admit Liouville type rigidity theorems, we prove that every classical solution converges exponentially to a quadratic polynomial of form $\tau t+\frac{1}{2}x'Ax+bx+c$ at infinity, with an explicitly derived convergence rate. Furthermore, we investigate Dirichlet type problems with prescribed asymptotic behavior, featuring two key innovations: applicability to all dimensions $n\geq 2$, and no requirement for the matrix $A$ to be positive definite or close to a scalar multiple of the identity matrix. These results establish the relationship among Liouville type rigidity, asymptotic analysis at infinity, and boundary value problems (including Dirichlet type and initial value type).