I'm looking for general existence of a PDE of the form

$$ f: U \times [0, \delta) \to \mathbb{R}$$

$$\frac{\partial f}{\partial t}(p,t) = F(f(p,t))$$

where $f(p,0)$ is prescribed and $F$ is non-linear but it is known that

$$|F(f) - F(g)| \leq K ||f(\cdot, t) - g(\cdot, t)||_{C^1(U)}$$

I'm interested in how one would prove short time existence, and also if I can allow for $|F(f) - F(g)| \leq K ||f(\cdot, t) - g(\cdot, t)||_{C^k(U)}$ with $k \geq 2$ and still get the same existence. I'm struggling to find analogies to previous existence theorems, e.g. those for parabolic systems or using Picard-iterates (which would require a different lipschitz bound)

My PDE is similar in spirit to the following: solve $(\Delta - 1) u = 0$ on $\Omega \subseteq \mathbb{R}^n$ a smooth convex set with boundary and $u \Big|_{\partial \Omega} = F(\nabla^2 f, \nabla f)$ with $f: \partial \Omega \to \mathbb{R}$. Then

$$\frac{\partial f}{\partial t}(p) = \frac{\partial u}{\partial \nu}(p) $$