I have read the paper "Some application of the Lojasiewicz gradient inequality" of Alain Haraux in the last days, and I found it really difficult for me from some first lines. For example, what is the definition for Lyapunov function that matches things referred in the article, since I've found several definitions for it on the internet. Here is the first page of the paper:
I want to ask for some foundation knowledge to study first in order to prepare for reading this article.
Thank you very much.
In the context of this paper, Haraux calls $F$ a Liapunov function simply because for any solution $t\mapsto u(t)$ for (1.1), the function $t\mapsto F(u(t))$ is non-increasing.
In fact, $F\circ u$ is strictly decreasing for any non-stationary solution of (1.1), and this precludes the possibility of periodic behavior and many kinds of recurrent behavior such as chaotic attractors. What is tricky to prove (and not true in general for smooth systems) is that every solution must approach some single equilibrium state.
I don't have handy access to Haraux's article, but I believe it reviews recent work concerning sufficient conditions for such convergence, for rather general PDE systems analogous to (1.1) but with $u$ belonging to an infinite-dimensional state space.
In different contexts, when the stability of some single equilibrium state is under consideration, the term `Liapunov function' is sometimes reserved for a function $F$ which decreases along any solution in a neighborhood of the given equilibrium. In that case the existence of such a function with a strict minimum at the equilibrium establishes its stability.