# A Pogorelov estimate and a Liouville type theorem to parabolic $k$-Hessian equations

## TL;DR

This paper establishes Pogorelov estimates and Liouville theorems for parabolic $k$-Hessian equations, showing that solutions with certain convexity and growth conditions are necessarily quadratic in space and linear in time.

## Contribution

It introduces new Pogorelov type estimates and Liouville theorems for solutions of parabolic $k$-Hessian equations under specific convexity and growth assumptions.

## Key findings

- Solutions are linear in time plus quadratic in space.
- Any $k+1$-convex-monotone solution with quadratic growth in initial data is of a specific polynomial form.
- The results extend classical estimates to a parabolic $k$-Hessian context.

## Abstract

We consider Pogorelov type estimates and Liouville type theorems to parabolic $k$-Hessian equations of the form $-u_t \sigma_k (D^2u) =1$ in $\mathbb{R}^n\times (-\infty, 0]$. We derive that any \textbf{$k+1$-convex-monotone} solution to $-u_t \sigma_k (D^2u) =1$ when $u(x,0)$ satisfies a quadratic growth and $0<m_1\le -u_t\le m_2$ must be a linear function of $t$ plus a quadratic polynomial of $x$.