Consider the semigroup corresponding to the reaction diffusion equation

and is vanished on the boundary of the domain.

It is noticed that is continuous in .

In the previous post, we wonder that: **can the invariance of the attractor in be implied from the invariance of the attractor in ?** In this post, we give a short proof of it. To this end, we need the following lemma.

**Lemma 1.** Let be a sequence in such that strongly in and in . Then .

**PROOF.** Since continuously, we can see that strongly in . Due to the uniqueness of the limit, we conclude that .

We also state a well-known result.

**Theorem 1.** The semigroup has an absorbing set in and is asymptotically compact in . This implies that has a global attractor in and

. (1)

Now, we state our main result.

**Theorem.** Assume that has an absorbing set in and is asymptotically compact in . Then there exists a unique global attractor in for , and

. (2)

**PROOF.**

The compactness and the attracting property of are implied from the asymptotic compactness of and definition of . Thus, we only have to prove the invariance of under .

Let , then is an absorbing set for in both and .

Since for all . We will prove that .

Indeed, first, taking any . From definition (2) (with in place of ), there exist a sequence and such that strongly in . Because continuously, we get that strongly in . Hence, because of the definition of . Thus, .

On the other hand, let . Then there exist sequences and satisfying in the strong topology of . Since is asymptotically compact in , there exists a subsequence of such that strongly in for some . It implies that strongly in . Due to the uniqueness of the limit, we obtain that . Thus, .

This completes the proof.

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## About baotangquoc

Lecturer
School of Applied Mathematics and Informatics
Hanoi University of Science and Technology
No 1, Dai Co Viet Street, Hanoi

This proof is true and has proved similarly in a paper of Li et. al.(See Theorem 2.8)

http://www.sciencedirect.com/science/article/pii/S0362546X11006055