Regular Approximation for Degenerate Parabolic PDE

Question

Let $T>0$ and $\Omega \subset \mathbb{R}^N$ be a bounded domain with smooth boundary. I'm considering a problem of porous medium equation $$ \partial_t u -d \Delta u^m = F(u) \ in \ (0,T) \times \Omega. $$ with homogeneous Dirichlet boundary condition. We assume $m \geq 1, \ d>0$ be a constant, $F: \mathbb{R} \rightarrow \mathbb{R}$ be Lip.continuous. Since I want to construct a "Regular Approximation" of the above equation, I considered an approximate problem $$ \partial_t u^\epsilon -d\nabla \cdot [D_\epsilon(u^\epsilon) \nabla u^\epsilon]=F(u^\epsilon) $$ where $D_\epsilon\in C^2([0,\infty);\mathbb{R}) (\epsilon>0)$ and $D_\epsilon(s)(s \in [0,\infty))$ is an approximation of $s^{m-1}$ and satisfies the following properties: $$ D_\epsilon(s) \geq \epsilon \ and\ s^{m-1} \leq D_\epsilon(s) \leq s^{m-1}+2\epsilon \ \ \forall s,\epsilon. $$ We can construct a local in time regular solution provided that initial data is regular. Then, is the boundedness of $F$, i.e. $\exists M>0 \ s.t. \ |F(x)| \leq M \ \forall x \in \mathbb{R}$ a sufficient condition for global existence of regular solution $u_\epsilon$?