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

Abstract

We consider Pogorelov type estimates and Liouville type theorems to parabolic $k$-Hessian equations of the form $-u_t\ddot{I}{ƒ}_k(D^{2}u)=1$ in ${\mathbb{R}}^n\times (-\infty ,0]$. We derive that any \textbf{$k+1$-convex-monotone} solution to $-u_t\ddot{I}{ƒ}_k(D^{2}u)=1$ 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 $x$.