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
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
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.