Let us consider the fully nonlinear problem $$ \begin{cases} F(x,u,Du,D^2 u) = 0 & \text{ in } \Omega \\ u=0 & \text{ in } \partial \Omega \end{cases} $$ Suppose that we know that the solution $u_n$ to $$ \begin{cases} F_n(x,u_n,Du_n,D^2 u_n) = 0 & \text{ in } \Omega \\ u_n=0 & \text{ in } \partial \Omega \end{cases} $$ satisfies $u_n \to u$ uniformly on compact subsets of $\Omega$.

In this setup, is it generally true that the solution of $$ \begin{cases} \partial_t u_n + F_n(x,u_n,Du_n,D^2 u_n) = 0 & \text{ in } \Omega\times(0,\infty) \\ u_n=0 & \text{ in } \partial \Omega\times(0,\infty) \\ u_n(\cdot,0) = g & \text{ in } \Omega \end{cases} $$ converges to the one of $$ \begin{cases} \partial_t u + F(x,u,Du,D^2 u) = 0 & \text{ in } \Omega\times(0,\infty) \\ u=0 & \text{ in } \partial \Omega\times(0,\infty) \\ u(\cdot,0) = g & \text{ in } \Omega \end{cases} $$ or not? If not, under which additional conditions is it true?