A proof on the invariance of attractors.

Consider the semigroup S(t) corresponding to the reaction diffusion equation

u_t - \Delta u + f(u) = g(x), u(0) = u_0\in L^2

and u is vanished on the boundary of the domain.

It is noticed that S(t) is continuous in L^2.

In the previous post, we wonder that: can the invariance of the attractor in H^1 be implied from the invariance of the attractor in L^2? In this post, we give a short proof of it. To this end, we need the following lemma.

Lemma 1. Let \{x_n\} be a sequence in H^1 such that x_n\rightarrow x_0 strongly in L^2 and x_n \rightarrow y_0 in H^1. Then x_0 = y_0.

PROOF. Since H^1 \subset L^2 continuously, we can see that x_n \rightarrow y_0 strongly in L^2. Due to the uniqueness of the limit, we conclude that x_0 = y_0.

We also state a well-known result.

Theorem 1. The semigroup S(t) has an absorbing set B_2 in L^2 and is asymptotically compact in L^2. This implies that S(t) has a global attractor A_2 in L^2 and

A_2 = \cap_{s\geq 0}\overline{\cup_{t\geq s}S(t)B_2}^{L^2}.        (1)

Now, we state our main result.

Theorem. Assume that S(t) has an absorbing set B in H^1 and S(t) is asymptotically compact in H^1. Then there exists a unique global attractor A in H^1 for S(t), and

A = \cap_{s\geq 0}\overline{\cup_{t\geq s}S(t)B}^{H^1}.         (2)

PROOF.

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

Let B_0 = B\cap B_1, then B_0 is an absorbing set for S(t) in both L^2 and H^1.

Since S(t)A_2 = A_2 for all t\geq 0. We will prove that A \equiv A_2.

Indeed, first, taking any x\in A. From definition (2) (with B_0 in place of B), there exist a sequence t_n\rightarrow +\infty and x_n\in B_0 such that S(t_n)x_n \rightarrow x strongly in H^1. Because H^1 \subset L^2 continuously, we get that S(t_n)x_n \rightarrow x strongly in L^2. Hence, x\in A_2 because of the definition of A_2. Thus, A\subset A_2.

On the other hand, let x\in A_2. Then there exist sequences t_n\rightarrow +\infty and x_n\in B_0 satisfying S(t_n)x_n \rightarrow x in the strong topology of L^2. Since S(t) is asymptotically compact in H^1, there exists a subsequence \{n_k\} of \{n\} such that S(t_{n_k})x_{n_k} \rightarrow y strongly in H^1 for some y\in A. It implies that S(t_{n_k})x_{n_k} \rightarrow y strongly in L^2. Due to the uniqueness of the limit, we obtain that x = y. Thus, A_2 \subset A.

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 entry was posted in Attractors. Bookmark the permalink.

One Response to A proof on the invariance of attractors.

  1. baotangquoc says:

    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

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