The equation:
Let $\Omega = (0,1) \times (0, T) \subset \mathbb R \times \mathbb R.\\$
I would like to find a solution $u(t,x)$ to the following Dirichlet problem:
$$ u_t = h(x, u) \frac{u_{xx}}{1+(u_x)^2} \hspace{5mm}\textrm{ on } \Omega, $$ and $$ u(0,x) =u_0(x); \quad u(t,0) = u(t,1)= 0. $$
This equation is of the form: $$ u_t = a(t,x,u,u_x) u_{xx}, $$ where $$ a(t,x,u,u_x) = \frac{h(x,u)}{1+(u_x)^2}. $$ So it is quasi-linear. It is also (non-uniformly) parabolic, since $h>0$ on $\Omega$ (but goes to $0$ at $x= 0,1$).
Info about $h$: $$0 < h(t,x,u) \leq 1 \quad \textrm{for }\quad x\in (0,1). $$ and $$ h(0,u) = h(1,u) = 0 \quad \textrm{for} \quad t\in [0,T). $$
Observations: I have come across this equation in my work, but I don't have much experience with parabolic PDE. So far, looking at the literature on quasi-linear parabolic pde, I am unable to find much theory that I can apply. The main issue seems to be that the equation is not uniformly parabolic, as $a(t,x,u,u_x)$ goes to $0$ at the endpoints, and so I am unable to use any of the results in [LSU].
It may be relevant that $u_t = \frac{u_{xx}}{1+(u_x)^2}$ is the equation for graphical mean curvature flow (MCF), so this equation could be considered a modified curvature flow. Since the MCF can be solved for all time, my hope was that because $h$ is fairly well behaved (bounded, smooth, non-negative, etc), this equation might be too.
My question:
Does any existing theory cover this type of problem, or are there standard ways to get around this degeneracy at the endpoints?