For a convex lower-semicontinuous functional on a Hilbert space $I\colon H\rightarrow\mathbb{R}$, it is shown in Evans' PDE that the Hilbert-space-valued ODE $$\begin{cases}\mathbf{u}'(t)\in-\partial I[\mathbf{u}(t)]\\\mathbf{u}(0)=u\in H\end{cases}$$ is well-posed for all time, where $\partial I[\cdot]$ denotes the subdifferential. The intuition makes perfect sense, and I am wondering if there are similar results by relaxing convexity. In particular, is there a well-posedness result if $I$ is say simply bounded from below, or coercive? Intuitively it seems plausible, but I haven't been able to find a nice resource on this matter. Perhaps even a nice text on variational methods might have what I'm looking for?