Consider a system of first-order ODE $$\dot{x}=\Phi(x).$$
$\Phi(x)$ is locally Lipschitz. By the theorem, given any initial condition $x(t_0)=x_0$, the solution $x(t)$ is unique and continuous on some interval $[t_0, T)$, where $T\in \mathbb{R}\geq t_0$.
Somehow, $x(t)$ has the property $$\|x(t)\|\leq te^t.$$ Thus, $x(t)$ is unique and continuous in the interval $[t_0, \infty)$.
My question is, can $x(t)$ be bounded on $[t_0, \infty)$ with what we have so far? If not, then what do we need more to ensure boundedness of the solution?