differential equations – \$G:mathbb{R}^dtomathbb{R}^d\$ locally lipschitz

There is a function $$G:mathbb{R}^dtomathbb{R}^d$$ locally lipschitz, suppose there exists $$R>0$$ such that $$xcdot G(x)leq 0 , forall xin mathbb{R}^d$$ with $$||x||geq R$$. Show that every maximal solution to $$x’=G(x)$$ is defined in an interval of the form $$(a,infty)$$.

This problem seems complicated to solve, any help is appreciated. The existence of this solution is guaranteed by the Picard Theorem. But I’m confused because the differential equation is $$x’=G(x)$$ so it means that $$x’in mathbb{R}^d$$ but at the same time $$x$$ should be defined on some $$(a,infty)$$? So the proper statement should be $$x'(t)=G(x(t))$$ so $$x:Isubseteqmathbb{R}tomathbb{R}^d$$