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Would someone please explain to me the concept of converting a parabolic PDE to an ODE on Banach space?

If I have a PDE, say $$u_t = f(u_{xx}, u_x, u, p)$$ where $p$ is a parameter and the solution $u$ should lie in some Banach space $X$ (so $X = C^{2,1}(S^1 \times [0,T]$ for example), can I write this as an ODE?

I ask because I want the continuous dependence on parameter in the solution and it is easier to get this from and ODE than from a PDE I think. And I think this must be possible since I read papers that say for a PDE solution that "the solution depends smoothly on the parameter by the implicit function theorem", and they reference Zeidler which contains a theorem about smooth dependence for a Banach space ODE.

Anyway, my understanding is bad on this and I can't find any books. I appreciate your help.

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