*Pullback attractors* for PDE systems was my first research project. Consider a non-autonomous (partial) differential equation, with ,

with for a Banach space . If the evolution equation has a unique solution in for each , the it generates an evolution process where is the solution of the equation at the time with initial data at the time .

The *pullback attractor* for is a set of compact sets which is invariant, i.e. and is pullback attracting all bounded sets, that is for any fixed time ,

for any bounded set .

The main difference between the pullback attractor and the usual global attractor is that the pullback attractor attracts bounded set in a “pullback” sense, that is instead of sending the final time to infinity, we send the initial time to minus infinity. In other words, instead of starting at a initial time and going to the future, what we do with the pullback attractor is to fix the current time and go back to history.

I have been asked many times the question “*What does that mean that you go back to the history, while the main idea of attractors to investigate the large time behavior or equivalently the future?*“. To be honest, I did not know a convincing answer.

Luckily, after attending a talk of Stefanie Sonner today I now have in mind a good answer for that question.

Consider the simple non-autonomous ODE equation

and initial data .

By solving the equation we easily get that

.

It is obvious that and there is no hope for a compact (or even bounded) attracting set as .

However, if we take the differences between two solutions and (with different initial data) then we have

or equivalently .

This suggests that there is still an asymptotic behavior of the equation that needs to be studied. And that is where the pullback attractor theory comes in.