I want to learn how to apply certain ODE theory to PDE. If we have a Banach space ODE $$x'(t) = f(t, x(t), p),$$ $$x(0) = x_0$$ where the equation is over same domain $t \in (a,b)$, then via the implicit function theorem, there is a unique solution around neighborhoods of the initial value and the parameter, and the solution depends continuously on the initial value and the parameter $p$. This is according to Zeidler's book in Section 4.11

Is there a way this ODE theory can be directly applied to PDE theory for some parabolic equation to get continuous dependence? Or does one just adapt the proof?