Let $M$ be a smooth compact manifold and $\pi : E \to M$ a smooth vector bundle. Let $\Gamma(E)$ denote the Frechet space of smooth (i.e. $C^{\infty}$) sections of $E$ and let $\mathcal{U}\subset \Gamma(E)$ be an open subset. Suppose we have a second order differential operator $\mathrm{D} : \mathcal{U} \to \Gamma(E)$ having the property that its linearisation $\mathrm{D}^{\prime}_f$ around any $f \in \mathcal{U}$ is elliptic (in the sense that the real parts of the eigenvalues of the principal symbol $\sigma_{\xi}(\mathrm{D}^{\prime}_f)$ are strictly positive for all $\xi\neq 0$). In Hamilton's famous paper on the Ricci flow he states -- without proof -- that under these assumptions the parabolic equation \begin{align*} \partial_t f&=\mathrm{D}(f),\\ f(\cdot,0)&=f_0 \end{align*} admits a unique smooth solution $f : M \times [0,\varepsilon) \to E$ with $\varepsilon>0$ (possibly) depending on the initial condition $f_0 \in \mathcal{U}$. More precisely, Hamilton assumes that $\mathcal{U}$ consists of those smooth sections of $E$ that take values in some open subset $U \subset E$.
Two questions: Is the statement correct as phrased here? Is a proof written down anywhere (preferably in the language of vector bundles and manifolds)?