Question

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?

Answer 1

There is a detailed survey on this subject:

Lobanov, S.G.; Smolyanov, O.G. Ordinary differential equations in locally convex spaces. Russ. Math. Surv. 49, No.3, 97-175 (1994); translation from Usp. Mat. Nauk 49, No.3(297), 93-168 (1994).

Answer 2

Hi.

By using functionals one can reduces uniqueness to the one-dimensional case: If $u$ is a solution to $\frac{d}{d}t u = f(t,u)$ then $\lambda u$ is a solution of $\frac{d}{dt} (\lambda u) = (\lambda f)(t,u)$ for all $\lambda\in V'$. In your case all the functions $\lambda f$ satisfy a Lipschitz-condition so Picard-Lindelöf gives you the uniqueness of $\lambda u$. Since the values $\lambda u$ determine $u$ uniquely in a LCS, this means $u$ is unique. A lot a other problems can be handled completely analogously.

The more difficult question is the existence of $u$. I'm not sure, but I think one could use Schauder's fixed-point theorem after one has replace the ODE with an integral equation. (Using Pettis-integrals for example).