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

TL;DR

This paper derives interior derivative estimates and proves a Liouville type theorem for solutions of parabolic $k$-Hessian equations, showing that under certain conditions, solutions must be affine in time and quadratic in space.

Contribution

It provides new gradient and Pogorelov estimates for $k$-convex solutions and establishes a Liouville theorem characterizing entire solutions.

Findings

Gradient and Pogorelov estimates for solutions.

Liouville theorem for entire solutions in $R^n imes (- abla,0]$.

Solutions are affine in time and quadratic in space under growth conditions.

Abstract

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 $u$.