A standard result in the theory of dynamical systems states that for a continuous system $(*)$ defined by $\dot{x}(t) = F(x)$, where $ F : \mathbb{R}^n \rightarrow \mathbb{R}^n $ locally Lipschitz, if one finds a Lyapunov function, i.e. a continuously differentiable function $L : \mathbb{R}^n \rightarrow \mathbb{R}$ that is everywhere non-negative and has compact sublevel sets $ \mathcal{L}_K = \{ x \in \mathbb{R}^n \mid L(x) \leq K \}, \forall K > 0 $ and is non-increasing along trajectories $x(t)$, then one can conclude that the said system has bounded solutions, i.e. for each initial condition $x_0$, there exists some constant $B>0$ such that $ \| x(t) \| \leq B $, where $x(t)$ is a solution of the system $(*)$ with $x(0) = x_0$.
I am interested in this result in the case of gradient flows; that is, when there exists some lower bounded function $\phi : \mathbb{R}^n \rightarrow \mathbb{R}$ such that $F = - \nabla \phi$ and I would like to relax the assumption that the sublevel sets are bounded.
For example, one might look at the simple function $\phi(x_1, x_2) = x_1^2 $ which gives us the system $ \begin{bmatrix} \dot{x}_1(t) \\ \dot{x}_2(t) \end{bmatrix} = \begin{bmatrix} -2 x_1(t) \\ 0 \end{bmatrix} $ which has the solution $x_1(t) = x_1(0) e^{-2t}, x_2(t) = x_2(0)$ where $(x_1(0), x_2(0))$ are the initial conditions. We see that in this case $x_1(t) \to 0 $ as $t \to \infty$ and it's clear that whatever initial conditions we choose, we can bound the solution. Yet the sublevel sets of the Lyapunov function of the system (which in this case is the function itself) are not bounded, since for example $\mathcal{L}_1 = \{ (x_1, x_2) \in \mathbb{R}^2 \mid x_1^2 \leq 1 \} = [-1, 1] \times \mathbb{R} $.
Of course, what I showed here is a quite special instance, since the function $\phi$ is convex. But one can easily imagine some non-convex functions where this still holds.
So my question is: for gradient flows, if one drops the assumption of compact sublevel sets, what other weaker, suitable assumptions could one impose to still obtain that the solutions remain bounded for any initial condition?
Any thoughts and references are welcome! Thank you!