Assume $F:R^n\supset U \rightarrow R^{n+1}$ is a hyperplane . Evolving $F$ as $$ \partial_tF (x,t)=h(t)\nu(t) \\ F(\cdot, 0)=F_0(U) \\ F(\cdot,t)|_{\partial U}= F_0|_{\partial U} \\ h(t)=\frac{\int_{M_t}Hd\mu}{\int_{M_t}d\mu} \\ H=g^{ij}h_{ij} $$ $h_{ij}$ is second fundamental form. $\nu(t)$ is outward normal vector. Then , how to know the system has short time solution ?
What I try : this question is from the volume preserving mean curvature flow: $$ \partial_tF(x,t)=(h(t)-H(x,t))\cdot\nu(x,t) ~~~x\in U, t\ge 0 \\ F(\cdot, 0 )=F_0 \\ h(t)=\frac{\int_{M_t}Hd\mu}{\int_{M_t}d\mu} $$ I want to show it has short time solution. Because the $\nu$ is 1-order term of $F$ , and $h(t)$ is a function about $t$ (I don't know whether it needs to consider it function of $H$ and $F$), so the keypoint it to show $$ \partial_tF=-H\nu + \text{lower order term} $$ is strict parabolic. Although it is not , but by way of De Turck , we can get the volume preserving mean curvature flow has short time solution.
Then , we can do same thing for usual mean curvature flow. And know it has short time solution. So, we know the firt system in this question has short time solution. But are there any general theorem about it ?
Not sure exactly what you're expecting as an answer here, but yes; there are general short-time existence theorems for parabolic equations. As you mentioned, a lot of geometric evolution equations need a DeTurck trick first. In the case of the volume-preserving flow you can't just treat $h$ as a given function of $t$, since it is defined in terms of the solution $F$; but you can just use the theory to produce a solution of the un-normalized flow and do a time-dependent rescaling to obtain a solution of the normalized flow.
If you want a reference, Chapter 3 of Charlie Baker's thesis proves a fairly general existence theorem and applies it to mean curvature flow.