Definition. Let $X$ be a manifold. A smooth map $f:X\to \mathbb R$ is said to satisfy the Palais-Smale condition over $y\in \mathbb R$ if whenever $y_n\to y$, any sequence $x_n\in f^{-1}(y_n)$ such that $\nabla f(x_n)\to 0$ has a convergent subsequence.
If $x_{n_k}\to x_0$ then $\nabla f(x_{n_k})\to \nabla f(x_0)=0$ meaning the gradient field actually has a critical point.
I have read an interpretation of the failure of the Palais-Smale condition as "integral curves of the gradient field disappearing at infinity".
Question 1. What does this mean? The only reasonable meaning I can think of is that integral curves are defined for all time, but do not intersect all fibers of $f$ that it should. Is this the intended meaning?
I want to understand the geometry of the Palais-Smale condition with an example.
Consider the Reeb foliation of the plane, defined by the surjective submersion $f:\mathbb R^2\to \mathbb R$ given by $(x,y)\to (x^2-1)e^y$. The gradient is $\nabla f(x,y)=(2xe^y,(x^2-1)e^y)$ and its integral curves are shown below.
The submersion $f$ does not satisfy the Palais-Smale condition above zero, since we have the sequence $a_n=(0,-n)$ on which the gradient and the function tend to zero, but there's no convergent subsequence.
I would like to understand the picture here. The integral curves certainly slow down as $y\to -\infty $, but it seems there's only one problematic curve: the $y$-axis intersects the negative fibers orthogonally, so there's an integral curve of the gradient field heading downwards along it. There's no reason for this curve to be defined only for a finite time, so it seems to be defined for all time (including backwards). Evidently it never meets a non-negative fiber, consistent with the suggested interpretation above.
I am confused by this. Any connection on $f$ is a parametric vector field on $\mathbb R^2$ (with real parameter) whose flow lifts the parametric radial field $(x,y)\mapsto x+ty$ whenever it exists. Taking the orthogonal connection seems to give the gradient field, so its integral curves should lift straight paths. Consequently, a gradient field curve defined for all time should lift an infinite straight path in $\mathbb R$, and therefore intersect all fibers of $f$. But this contradicts reality: the vertical gradient field curve along the $y$-axis is defined for all time but does not intersect any non-negative fibers!
Question 2. What am I missing here?
Looking at the submersion again, there are many more Palais-Smale sequences. Consider for instance $a_n=(-n,-n)$, on which the gradient and the function tend to zero. Let us restrict our submersion to e.g $ \left\{ x<-1 \right\}$. Then despite not meeting the Palais-Smale condition, this seems to be a fiber bundle - we can "comb the fibers downwards". There don't seem to be any problematic integral curves for the gradient field.
Question 3. What is the geometric interpretation of the failure of the Palais-Smale condition for $f|_ {\left\{ x<-1 \right\}}$?
Question 4. Why does the sequence $(0,-n)$ "see" a problem that $(-n,-n)$ does not?
Question 5. Why did it matter at all the the derivative tends to zero on the above sequences? It seems the same pathology with the vertical integral curve could occur even if it sped up as $y\to -\infty$...
I don't think there's too much going on here. You understand this example well, in particular you understand that it fails to satisfy the Palais-Smale condition.
That phrase about "integral curves of the gradient field disappearing at infinity" is intended to be a spark to your understanding and your intuition. Sometimes sparks fizzle; even so, you may understand it anyway on your own terms.
For this vector field, the $y$-axis is an integral curve of the gradient field, and that integral curve slows way down as $y \to -\infty$, as you notice. If the Palais-Smale condition were true, that integral curve would have to approach a finite limit. But it doesn't: it goes off to infinity; it "disappears at infinity". That's all. I think this addresses Questions 1,2,3.
For Question 4, regarding the sequence $x_n = (-n,+n)$, that sequence does see a problem. The intuition is that if the Palais-Smale condition were true then (some subsequence of) $x_n$ would approach some kind of saddle singularity. So in this example, the sequence of integral curves through the points $x_n$ should be getting closer and closer to some kind of saddle singularity. But they are not: they are instead "disappearing at infinity".
I don't know what to tell you about Question 5. The Palais-Smale condition has consequences. If an example satisfies the Palais-Smale condition, then you know those consequences are true. So the real answer to Question 5 is to read a theorem in which the Palais-Smale condition is a hypothesis, and to see how that condition is used in the proof.