Recently, the long time behavior of stochastic reaction diffusion equations (RDE) is considered by many mathematicians. The equation writes

(1)

in a bounded domain , where is a two-sided real-valued Weiner process on a probability space.

By putting in which

we get a new deterministic equation with random terms

(2)

The existence of a random attractor for RDE in is well known, in is solved recently by many authors. However, the regularity of the random attractor is not well understood. Compare to deterministic case (that says ), where many results show that the attractor in fact lies in or even , that should be should be something different in here.

Now, look back to the usual method of obtaining the regularity of attractors in deterministic case. One **crucial step **is usually to have

or roughly speaking, the time derivative of is bounded (in some sense). In order to do this, because of high growth of the nonlinearity, one (almost) **have to **take the derivative in time of the equation [I said almost, since there may be a way around, but up to the best of my knowledge, I don’t know]. This **couldn’t be done in stochastic case** because of the presence of random noise.

Therefore, we have to adapt another method to deal with stochastic RDE. Fortunately, there is a hope coming from some papers of Chengkui Zhong, in which he showed that

a set in a Hilbert space is pre-compact if,

for any you can find a – dimensional sub-space such that

i) is bounded in ; and

ii)

where is the canonical projection.

We will apply the above result to the RDE equation to prove that the random attractor is actually compact in . Of course, following the abstract result we set and the Hilbert space .

Motivated by the abstract result, inorder to prove that the random attractor is compact in , we prove that

for every there exists , such that for all .

Transform to the random dynamical system corresponding to the equation (2), we want to prove that

for every , every , there exist and such that

for all .

(Here is a norm).

**How to do that? **We will use eigenvalues of in that says and its eigenfunctions .

Denote by and is the canonical projector from to .

For any , we decompose where and .

Now we only have to prove that

for every , every , there exist and such that

for all .

To do that, we rewrite here equation (2)

(2)

Multiply (2) by , we arrive

Using Cauchy’s inequality and , after standard calculations, we have

.

Using Gronwall’s lemma and replace by , we get

(3)

The first term and the second on the right hand side of (3) are easy to proved that they tend to zero as .

The only thing (and also the technical thing) that we have to deal with is the last term on the right hand side of (3).

This can be implied from

(4)

But how can we get (4)?

This is done in a work of W. Zhao and Y. Li

random attractors for stochastic reaction diffusion equation on unbounded domains