I am reading this paper https://projecteuclid.org/journals/differential-and-integral-equations/volume-26/issue-5_2f_6/Convergence-to-equilibrium-for-discretizations-of-gradient-like-flows-on/die/1363266079.short . In it they introduce a definition and say a function $F:\mathbb{R}^n\to \mathbb{R}$ satisfies a Lojasiewicz inequality at a point $x$ there exists $\beta,\sigma >0$ and $\nu\in(0,\frac{1}{2}]$ such that
$$ \|F(x)-F(y)\|^{1-\nu}\leq \beta \|\nabla F(y)\|~~~\forall y\in B_\sigma(x) $$ ( in the paper they state this definition in a more general setting).
I know this property is meant to be useful for proving convergence of a numerical scheme to a stationary solution, when $F$ is the Lyapunov function measuring stability. However this is the first time I have seen the Lojasiewicz inequality and have very little intuition for it. Could anyone give a reason why this kind of property might be useful, or provide some examples of types of functions $F$ which satisfy this property.