Given an ODE for a function $u \in C^1(I,V)$, where $V$ is some locally convex TVS (topological vector space) and $I \subset \mathbb{R}$, i.e.

$\frac{d}{dt} u = f(t,u)$

for some function $f: I \times V \to V$. Are there results concerning the uniqueness of the initial value problem? Can someone give me some references or outline the idea of how to prove uniqueness? What is the suitable condition on $f$ that replaces Lipschitz continuity for ODE's with values in Banach spaces?

An explicit example: When solving the heat equation $ \frac{\partial u}{\partial t} - \Delta u = 0$ in the class $C^\infty(\mathbb{R}^+, S'(\mathbb{R}^n)$ using the Fourier transform ($S'$ denotes the tempered distributions), one gets an ODE

$\frac{d}{dt} u = -|\cdot|^2 u$.

Does the initial value problem have a unique solution?