Now I want to consider the following pde
$u_t(x,t)=\sigma(x,t)(1+|D_xu(x)|^2)^{1/2}$, with initial condition $u(x,0)=g(x)$ which is analytic, and on domain $D\times \mathbf{R}^{+}$, $D\subset \mathbf{R}^n$ is a bounded open domain. (If we replace $\sigma(x,t)$ by the mean curvature, this would be the mean curvature flow, but now $\sigma(x,t)$ is prescribed on $D\times [0,\epsilon]$, for $\epsilon$ small enough)
My original assumption on regularity of $\sigma(x,t)$ is analytic w.r.t $x$ and Lipschitz w.r.t $t$, and jointly continuous. I want to know whether there is a standard method to say the local existence, uniqueness and regularity of the solution $u(x,t)$.
If the regularity on $\sigma(x,t)$ can be assumed to be good enough, what can we say about the above questions?
