I am looking for a reference treating parabolic equations on vector bundles. In particular, I look for precise conditions which guarantee short time existence. I found it quoted at different places in the literature as standard fact for parabolic PDEs.
The concrete PDE in which I am interested has the following shape: $$\partial_t h = - ( \Delta_A h + f(h, d_A h) ),\quad h_0 = 1$$ Here $h \in \Omega^0(\Sigma, \text{End}(E))$ is a section the endomorphism bundle of a vector bundle $E$ over a Riemann surface $\Sigma$ and $A$ some fixed connection on $E$. The term $f$ is a non-linear function in $h$ and its first order derivatives.
The RHS is an (uniformly) elliptic operator on $\Omega^0(\Sigma, E)$. Is ellipticity on the RHS in general enough to guarantee short time existence?
