uniqueness of solutions in the future

Question

I'm asked to proof this: If we have $$f:\mathbb R^d\rightarrow\mathbb R^d, f \in C^1$$ verifying:$$\ f(0)=0 \ $$ $$\ \langle p,f(p)\rangle \leq0 ,\ \forall p \in \mathbb R^d$$ then the initial value problem $$ \left \{ \begin{matrix} x'=f(x)\\ x(0)=0\\ \end{matrix} \right .\}$$ has a unique solution in the future. Any answer would be appreciated.

Answer 1

Since $f$ is $C^1$, existence and uniqueness of a local solution is guaranteed by the Existence and Uniqueness Theorem (a.k.a. Picard's theorem,...) What you are asked to prove is that the solution is defined on $[0,\infty)$. For this it is enough to prove that the solution does not blow-up, that is, it is bounded on any finite interval $[0,T]$ (with a bound that may depend on $T$.) Let $x(t)$ be the solutuion. Then $$ \frac{d}{dt}\|x(t)\|^2=\frac{d}{dt}\langle x(t),x(t)\rangle=2\,\langle x(t), x'(t)\rangle=2\,\langle x(t), f(x(t))\rangle\le0. $$ Thus, $\|x(t)\|^2$ is decreasing, and $\|x(t)\|\le\|x(0)\|$ for all $t$ for which the solution is defined.