A remark on reaction-diffusion equations in unbounded domains

We prove the existence of a compact L^2-H^1 attractor for a reaction-diffusion equation in R^n. This improves a previous result of B. Wang concerning the existence of a compact L^2-L^2 attractor for the same equation.


Introduction
In this note we consider the reaction-diffusion equation We make the following assumptions: |f ′ (s)| ≤ C(1 + |s| β ) for all s ∈ R, (1.4) where C is some positive constant and 0 ≤ β if N ≤ 2; 0 ≤ β ≤ min{(2 * /2) − 1, 4/N } if N ≥ 3. (1.5) In the sequel, we write L 2 and H 1 instead of L 2 (R N ) and H 1 (R N ). Moreover, we denote by ·, · the scalar product of L 2 , by · the norm of L 2 and by · E the norm of any other space E. It is well known (see e.g. [2]) that the Cauchy problem for equation (1.1) is well posed for initial data u(0, x) = u 0 (x) (1.6) in the space L 2 . Actually, equation (1.1) generates a global semiflow π in L 2 and in H 1 . The following lemma summarises some well known a-priori estimates for the semiflow π (see [9], Lemmas 1 and 2): there exists a constant K > 0 and for every R > 0 there exists T (R) > 0 such that, if u ∈ L 2 and u ≤ R, then π(t, u) It is a classical result that, if R N is replaced by a bounded domain Ω ⊂ R N , equation (1.1) possesses a compact (L 2 − H 1 ) attractor (see e.g. [1], [4] and [5]). The proof of this fact relies essentially on the compactness of the Sobolev embedding H 1 (Ω) ⊂ L 2 (Ω). If Ω is unbounded, the Sobolev embedding may be no longer compact and new techniques are needed. In [2] Babin and Vishik proved the existence of a compact attractor for equation (1.1) in an unbounded domain, but they needed to introduce weighted spaces in order to overcome the difficulties arising from the lack of compactness. The choice of weighted spaces, however, imposes some severe conditions on the forcing term g and on the initial data. In [9], B. Wang obtained for the first time the existence of a compact attractor for equation (1.1) in the 'natural' space L 2 . The crucial step in his proof is the following Theorem 1.2 (Wang '98). Let (u n ) n∈N be a bounded sequence in L 2 and let (t n ) n∈N be a sequence of positive numbers, t n → ∞ as n → ∞. Then there exists a strictly increasing sequence of natural numbers (n k ) k∈N and a function u ∈ L 2 such that π(t n k , u n k ) → u in L 2 as k → ∞. In other words, π is asymptotically compact in the strong L 2 topology.
Combining Lemma 1.1 and Theorem 1.2 with the abstract results of [4], [6] and [8], one easily obtains On the other hand, the 'natural' energy associated to equation (1.1) is given by where F ′ (s) = f (s), s ∈ R. The gradient structure of equation (1.1) implies that, along any given nonconstant solution of (1.1), the energy E decays and asymptotically approaches some limit level. The question then arises, whether the ω-limit of a given orbit lies at the energy level asymptotically approached by the orbit itself. Since E is continuous on H 1 , the problem becomes that of proving the asymptotic compactness of π in the strong H 1 topology. However, it seems definitely not trivial to obtain such a result by means of estimates involving the energy E.
Very recently Efendiev and Zelik in [3] considered the more general problem in three spatial dimensions. Using energy estimates and comparison arguments they proved, among other things, the existence of a compact (H α − H α ) attractor for (1.8), where H α is some fractional space between H 2 and H 1 . Their technique, however, exploits the Sobolev imbedding H 2 ⊂ L ∞ , which is no longer true in higher space dimensions.
We have found more convenient to follow a different way: in this note we show that the asymptotic compactness in H 1 can be recovered from the asymptotic compactness in L 2 by a simple continuity argument (Theorem 2.3). As a consequence, we deduce that the (L 2 − L 2 ) attractor, whose existence was established by B. Wang, is actually an (L 2 − H 1 ) attractor, like in the case of bounded domains (Theorem 2.4). The proof is very simple and is based on Henry's theory of abstract parabolic equations (see [5]).
Finally, we would like to mention the recent results obtained by Zelik in [10], concerning the existence and the entropy of locally compact attractors for equations like (1.1). See also [11], where similar results have been obtained for a dampded wave equation in R N .
Following Henry [5], the Cauchy problem (1.1)-(1.6), for initial data u 0 ∈ H 1 , can be formulated as an abstract parabolic initial value problem u + Au = −λu +f (u) + g u(0) = u 0 (2.5) It is well known that equation (2.5) is equivalent to the integral equation We have the following crucial Lemma 2.2. Let (u n ) n∈N be a sequence in H 1 , let u ∈ H 1 , and assume that u n ⇀ u in H 1 , u n → u in L 2 . Then π(t, u n ) → π(t, u) in H 1 , uniformly on [t 0 , t 1 ], for all t 1 > t 0 > 0.
It follows that, for t ∈ [0, t 1 ], By the singular Gronwall's Lemma (see [5]), there is a constant Finally, a simple computation shows that there exists a constant C 2 = C 2 (λ, L, M, a, t 1 ) such that, for t ∈ [0, t 1 ], This completes the proof.
Remark. It seems that such a result cannot be obtained as a consequence of a-priori estimates like the ones in [9]. In a different context, a similar observation was made also in [7].
The next theorem shows that, thanks to Lemma 2.2, the asymptotic compactness of π in L 2 implies the asymptotic compactness of π in H 1 . Theorem 2.3. Let (u n ) n∈N be a bounded sequence in H 1 and let (t n ) n∈N be a sequence of positive numbers, t n → ∞ as n → ∞. Then there exists a strictly increasing sequence of natural numbers (n k ) k∈N and a function u ∈ H 1 such that π(t n k , u n k ) → u in H 1 as k → ∞. In other words, π is asymptotically compact in the strong H 1 topology.
Proof. Fix any positive T . Since t n → ∞ as n → ∞, we have t n > T for all sufficiently large n, say n ≥ n 0 . Since the sequence (u n ) n∈N is bounded in H 1 , by Lemma 1.1 also the sequence π(t n − T, u n ) n≥n 0 is bounded in H 1 . Then there exists a strictly increasing sequence of natural numbers (n k ) k∈N , n k ≥ n 0 for all k ∈ N, and a functionū ∈ H 1 such that π(t n k − T, u n k ) ⇀ū in H 1 as k → ∞. On the other hand, by Theorem 1.2, we can choose the sequence (n k ) k∈N in such a way that π(t n k − T, u n k ) →ū in L 2 as k → ∞. Then, by Lemma 2.2, we have π(t n k , u n k ) = π(T, π(t n k − T, u n k )) → π(T,ū) in H 1 as k → ∞.
The proof is complete.
Finally, we can state and prove Theorem 2.4. The semiflow π possesses a compact (L 2 − H 1 ) attractor.
Proof. By Lemma 1.1 there exists a bounded set B in H 1 and for any bounded set B in L 2 there exists T (B) > 0 such that π(t, B) ⊂ B for all t ≥ T (B). On the other hand, by Theorem 2.3 π is asymptotically compact in H 1 . The conclusion follows from the abstract results of [4], [6] and [8].
Remark. Of course, by Lemma 1.1, the (L 2 −L 2 ) attractor and the (L 2 −H 1 ) attractor coincide. Moreover, by the continuity result of Lemma 2.2, the Hausdorff (or the fractal) dimension of the attractor is the same in L 2 and in H 1 .