Interior estimates of derivatives and a Liouville type theorem for parabolic $ k $-Hessian equations.

In this paper, we establish the gradient and Pogorelov estimates for $ k $-convex-monotone solutions to parabolic $ k $-Hessian equations of the form $ -u_t\sigma_k(\lambda(D^2u)) = \psi(x, t, u) $. We also apply such estimates to obtain a Liouville type result, which states that any $ k $-convex-monotone and $ C^{4, 2} $ solution $ u $ to $ -u_t\sigma_k(\lambda(D^2u)) = 1 $ in $ \mathbb{R}^n\times(-\infty, 0] $ must be a linear function of $ t $ plus a quadratic polynomial of $ x $, under some growth assumptions on $ u $. [ABSTRACT FROM AUTHOR]