## Why study “pullback attractors”?

Pullback attractors for PDE systems was my first research project. Consider a non-autonomous (partial) differential equation, with $t> \tau \in \mathbb R$,

$u'(t) = A(t)u(t) \qquad u(\tau) = u_{\tau}$

with $u_\tau \in X$ for a Banach space $X$. If the evolution equation has a unique solution in $X$ for each $u_\tau \in X$, the it generates an evolution process $U(t,\tau): X \rightarrow X$ where $U(t,\tau)u_{\tau}$ is the solution of the equation at the time $t$ with initial data $u_\tau$ at the time $\tau$.

The pullback attractor for $\{U(t,\tau)\}$ is a set of compact sets $\{A(t)\}_{t\in\mathbb R}$ which is invariant, i.e. $U(t,\tau)A(\tau) = A(t)$ and is pullback attracting all bounded sets, that is for any fixed time $t\in \mathbb R$,

$\lim_{\tau \rightarrow -\infty}d_{H}(U(t,\tau)B, A(t)) = 0$

for any bounded set $B$.

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 $t$ to infinity, we send the initial time $\tau$ 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

$u' = -u + t$ and initial data $u(\tau) = u_{\tau} \in \mathbb R$.

By solving the equation we easily get that

$u(t) = e^{-(t-\tau)}(u_{\tau} - \tau + 1) + t - 1$.

It is obvious that $\lim_{t\rightarrow+\infty}u(t) = +\infty$ and there is no hope for a compact (or even bounded) attracting set as $t\rightarrow+\infty$.

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

$(u - v)' = -(u-v)$ or equivalently $(u-v)(t) = e^{-(t-\tau)}(u_\tau - v_{\tau})$.

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.