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Let $(S_i)_{i\in\mathcal I}$ be a collection of subsets of a domain $\mathcal D\subseteq\mathbb R^n$ with non-empty interior. Let $A_i$ be linear maps on $S_i$ respectively. I don't want to assume that they are patched together continuously. Does a unique solution to \begin{equation} \dot x=\sum_{i}\mathbb I_{x\in S_i}A_ix \end{equation} exist? If so, under which conditions? Are there results or references?

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    As you can formulate the standard examples $\dot x = -sign(x)$ and $\ddot x +x =-sign(\dot x)$ (use a component with derivative zero, initial value $1$) in that way, you will get situations where the existence of solutions ends at the boundaries. – 2017-02-06
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    Thanks Lutz. Of course I'd love to have existence across the boundaries. So i wonder if there are conditions when I can have it. – 2017-02-06

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You might be interested in these notes of Jorge Cortes and the references therein. The notes describe several notions of existence for solutions to ODE with discontinuous vector fields. You might be particularly interested in the case of Fillipov solutions, which are a generalized notion of solution for ODE of the general kind you describe. See in particular Proposition 6, which discusses the particular case of piecewise continuous vector fields.