Subsolutions and Derivative Estimates for a Class of Parabolic Hessian Quotient Type Equations

In this paper, we consider the Dirichlet problem for a type of parabolic Hessian quotient equations $-u_{t} \frac{\sigma_{k}(U[u])}{\sigma_{l}(U[u])} = f(x,t,u,Du)$ in a cylinder domain $Q = \Omega \times (0,T]$, where $U[u] = (\Delta u)I - D^{2}u$. If $k \lt n$, we construct an admissible subsolution without imposing any curvature restriction on $\partial \Omega$. If $k = n$, we construct such a subsolution as long as the mean curvature of $\partial \Omega$ is positive. Moreover, this result holds true also for the elliptic case. The existence of such a subsolution represents a new feature of the Hessian quotient operator involving the matrix $U[u]$, compared to that involving the classic $D^{2}u$. Furthermore, we use the subsolution and appropriate auxiliary functions to establish the global estimates for first-order derivatives $Du$ and $u_{t}$ respectively.