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Abstract. The aim of this paper is to prove the existence of the global attractor for a semilinear strongly degenerate parabolic equation on RN with the locally Lipschitz nonlinearity satisfying a subcritical growth condition
GLOBAL ATTRACTOR FOR A SEMILINEAR STRONGLY DEGENERATE PARABOLIC EQUATION ON RN CUNG THE ANH Abstract. The aim of this paper is to prove the existence of the global attractor for a semilinear strongly degenerate parabolic equation on RN with the locally Lipschitz nonlinearity satisfying a subcritical growth condition. 1. Introduction The understanding of asymptotic behavior of dynamical systems is one of the most important problems of modern mathematical physics. One way to attack the problem for a dissipative dynamical system is to consider its global attractor. The existence of global attractors has been proved for a large class of nondegenerate partial differential equations (see e.g. [5, 20, 23] and references therein). In the last few years, a number of papers are devoted to the study of long-time behavior of solutions to degenerate parabolic equations. One of the classes of degenerate equations that has been studied widely in recent years is the class of equations involving an operator of Grushin type Gα u = ∆x u + |x|2α ∆y u, α ≥ 0. This operator was studied by Grushin in [9] when α is an integer, and by Franchi & Lanconelli in [6, 7, 8] when α is not an integer. Note that G0 = ∆ is the Laplacian operator, and Gα , when α > 0, is not elliptic in domains intersecting the surface x = 0. The long-time behavior of solutions to semilinear parabolic equations involving this operator has been studied recently in [1, 2]. We also refer the reader to some recent results on the generalized Grushin operators [10, 13, 15, 16, 17]. Recently, Thuy and Tri [21] considered a strongly degenerate operator Pα,β u = ∆x u + ∆y u + |x|2α |y|2β ∆z u, α, β ≥ 0, which is degenerate on two intersecting surfaces x = 0 and y = 0, and established some compact embedding theorems for weighted Sobolev spaces associated to this operator in bounded domains. This operator falls into the class of ∆λ operators [12], or more general, the class of X-elliptic operators [14]. In this paper we study the existence and long-time behavior of solutions to the following semilinear strongly degenerate parabolic equation ∂u − Pα,β u + λu + f (X, u) = g(X), X ∈ RN , t > 0, (1.1) ∂t u(X, 0) = u (X), X ∈ RN , 0 2010 Mathematics Subject Classification. 35D35, 35K65, 35B41. Key words and phrases. strongly degenerate; unbounded domains; mild solution; global attractor; tail-estimates method; sectorial operator; Lyapunov function. 1 2 C.T. ANH where X = (x, y, z) ∈ RN1 × RN2 × RN3 = RN , λ > 0, u0 ∈ S 1 (RN ) (see the definition of this function space below) are given, the nonlinearity f and the external force g satisfy the following conditions: (F) f : RN × R → R is a function satisfying f (X, 0) = 0, (1.2) |fu (X, u)| ≤ C a(X) + |u|ρ , 0 ≤ ρ < 4 Nα,β − 2 , (1.3) fu (X, u) ≥ − (1.4) 2 where Nα,β f (X, u)u ≥ −µu − C1 (X), (1.5) µ 2 (1.6) F (X, u) ≥ − u − C2 (X), 2 = N1 + N2 + (α + β + 1)N3 , C, are positive constants, a ∈ Nα,β s LNα,β (RN )∩L 2 (RN ) is a nonnegative function, F (X, s) = 0 f (X, τ )dτ , µ < λ, C1 , C2 ∈ L1 (RN ) are two nonnegative functions; (G) g ∈ L2 (RN ). Under similar above conditions, following the approach used in [1], Thuy and Tri [22] recently proved the existence and uniqueness of a global mild solution to problem (1.1) in a bounded domain Ω with the homogeneous Dirichlet boundary condition, and they also proved the existence of a compact global attractor for the semigroup generated by this problem (see also [13] for a more general situation). The aim of the present paper is to extend these results to the case of unbounded domains, the more complicated case due to the lack of compactness of the embedding theorems. For other results related to problem (1.1) with the nonlinearity of polynomial type, we refer the reader to some very recent works [3, 4]. In order to study problem (1.1), we use some weighted Sobolev spaces. We use the space S 1 (RN ) defined as the completion of C0∞ (RN ) in the norm u 2 S 1 (RN ) |u|2 + |∇x u|2 + |∇y u|2 + |x|2α |y|2β |∇z u|2 dX. = RN Then S 1 (RN ) is a Hilbert space with the inner product uv + ∇x u∇x v + ∇y u∇y v + |x|2α |y|2β ∇z u∇z v dX. (u, v)S 1 (RN ) = RN We also use the space S 2 (RN ) defined as the completion of C0∞ (RN ) in the norm u 2 S 2 (RN ) |u|2 + |Pα,β u|2 dX. = RN In Lemma 2.1 below, we will establish some embedding results related to these function spaces. Denote A = −Pα,β + λu, the positive and self-adjoint unbounded linear operator with domain of the definition D(A) = {u ∈ S 1 (RN ) : Au ∈ L2 (RN )} = S 2 (RN ), and define the corresponding Nemytski map f by fˆ(u)(X) = f (X, u(X)), u ∈ S 1 (RN ). A SEMILINEAR STRONGLY DEGENERATE PARABOLIC EQUATIONS 3 Then, (1.1) can be formulated as an abstract evolutionary equation du + Au + fˆ(u) = g, dt u(0) = u0 . The aim of this paper is to study the existence of a global mild solution and of a global attractor for the semigroup generated by problem (1.1). One basic feature of (1.1) is that it admits the following (strict) natural Lyapunov function 1 |∇x u|2 + |∇y u|2 + |x|2α |y|2β |∇z u|2 dX + λ Φ(u) = u2 dX 2 RN N R (1.7) + (F (X, u) − gu)dX. RN One can check that d Φ(u(t)) = − ut 2L2 (RN ) dt for any solution u of problem (1.1). This Lyapunov function plays an essential role in proving the global existence of solutions and it implies the structure of the global attractor obtained later. The structure of the paper is as follows. In Section 2, we prove the existence of a global mild solution to problem (1.1) by using the existence theorem for abstract parabolic equations and the Lyapunov function Φ. Then we can construct the semigroup S(t) generated by problem (1.1). In Section 3, we prove the existence of a compact global attractor for S(t) by showing the existence of a bounded absorbing set and the asymptotic compactness of the semigroup S(t). The existence of bounded absorbing sets can be quite easily deduced by using a priori estimates of solutions, while the proof of the asymptotic compactness is much more involved. To prove the asymptotic compactness of S(t) in L2 (RN ) we exploit the tail-estimates method introduced by B. Wang [24] to overcome the difficulty arising due to the lack of compactness of the embedding theorems, while the asymptotic compactness of S(t) in S 1 (RN ) is proved by combining the asymptotic compactness in L2 (RN ), the singular Gronwall inequality and the arguments introduced by Prizzi and Rybokowski in [18, 19]. It is noticed that the results obtained in the paper are also true for problem (1.1) in an arbitrary (bounded or unbounded) domain Ω in RN , not necessary the whole space RN , with the homogeneous Dirichlet boundary condition. Then, instead of S 1 (RN ) and S 2 (RN ), we use the spaces S01 (Ω) and S02 (Ω), defined as the completions of C0∞ (Ω) in the corresponding norms. 2. Existence and uniqueness of a global mild solution First, we prove the following embedding results. Lemma 2.1. The following embeddings are continuous: i) S 1 (RN ) → Lp (RN ) for all 2 ≤ p ≤ 2∗α,β := 2Nα,β , where Nα,β = Nα,β − 2 N1 + N2 + (α + β + 1)N3 ; ii) S 2 (RN ) → S 1 (RN ). Proof. i) We follow the ideas in the case of bounded domains (see the proofs of Theorem 3.3 and Proposition 3.2 in [12]). More precisely, we first embed S 1 (RN ) 4 C.T. ANH into an anisotropic Sobolev-type space, and then use an embedding theorem for classical anisotropic Sobolev-type spaces of fractional orders. Because the proof is very similar to the case of bounded domains [12], so we omit it here. ii) For all u ∈ C0∞ (RN ), we have u 2 S 1 (RN ) (|u|2 + |∇x u|2 + |∇y u|2 + |x|2α |y|2β |∇z u|2 )dX = RN |u|2 dX u(u − Pα,β u)dX ≤ C = RN RN =C u 1/2 L2 (RN ) u (|u|2 + |Pα,β u|2 )dX 1/2 RN S 2 (RN ) . Noting that u L2 (RN ) ≤ u S 1 (RN ) , we get u embedding S 2 (RN ) → S 1 (RN ) is continuous. S 1 (RN ) ≤C u S 2 (RN ) , that is, the Next, we discuss on the operator A = −Pα,β + λu. First, one can check that A is a positive definite sectorial operator on the Hilbert space X = L2 (RN ). Thus, we can define fractional powers of A on X = L2 (RN ) as follows (see [11, Chapter 1]): For θ > 0, we define, as usual, the operator A−θ : X → X as A−θ u = 1 Γ(θ) ∞ tθ−1 e−At udt, u ∈ X. 0 Then A−θ is injective and we define X θ to be the range of A−θ , and Aθ : X θ → X to be the inverse of A−θ . We also set X 0 = X and A0 = IdX . We know that X θ = D(Aθ ) is a separable Hilbert space endowed with the inner scalar product (u, v)X θ = (Aθ u, Aθ v), u X θ = Aθ u X . For any θ, η > 0 and θ η, X θ is continuously embedded into X η . From the above definition, one can see that X 1 = {u ∈ S 1 (RN ) : Au ∈ L2 (RN )} = S 2 (RN ), X 1/2 = S 1 (RN ). We have the following basic estimate (see e.g. [11, Theorem 1.4.3, p. 26]): Aθ e−At Cθ t−θ e−δt for all t > 0, (2.1) Now we deal with the nonlinear term f (u). 1 Lemma 2.2. Suppose (F) − (G) hold. Then, there exists 0 ≤ γ < such that 2 the map fˆ : X 1/2 = S 1 (RN ) → X −γ is Lipschitzian continuous on every bounded subset of S 1 (RN ). Proof. We consider two cases: 2 ˆ Case 1: 0 < ρ ≤ Nα,β −2 . In this case it is easy to check that the map f : 1 N 2 N 1 S (R ) → L (R ) is Lipschitzian continuous on every bounded subset of S (RN ) by using H¨ older inequality and the embedding S 1 (RN ) → L2(ρ+1) (RN ). 2 4 q N ˆ 1 N Case 2: Nα,β −2 < ρ < Nα,β −2 . We will show that f : S (R ) → L (R ), where ∗ 2α,β q := , is Lipschitzian continuous on every bounded subset of S 1 (RN ). ρ+1 A SEMILINEAR STRONGLY DEGENERATE PARABOLIC EQUATIONS 5 By (1.3) we have |f (X, u)| ≤ C(|a||u| + |u|ρ+1 ). Hence for u ∈ S 1 (RN ), by H¨ older’s inequality, we have |f (X, u)|q dX ≤ C RN (|a|q |u|q + |u|q(ρ+1) )dX RN ≤C a q u 2∗ /ρ L α,β (RN ) q 2∗ L α,β (RN ) + u 2∗ α,β 2∗ α,β (RN ) L < +∞ ∗ since S 1 (RN ) is continuously embedded into L2α,β (RN ) and the assumption of a. This shows that fˆ is a map from S 1 (RN ) to Lq (RN ). Let u, v ∈ S 1 (RN ) and u S 1 (RN ) ≤ R, v S 1 (RN ) ≤ R, we have from (1.3) |f (X, u) − f (X, v)|q dX ≤ C |u − v|q (|a|q + |u|qρ + |v|qρ )dX RN RN |a|q |u − v|q dX + C ≤C RN |u|qρ |u − v|q dX + C |v|qρ |u − v|q dX. RN RN Applying H¨ older’s inequality, we have |a|q |u − v|q dX ≤ a RN qρ L 2∗ /ρ α,β (RN ) |u|qρ |u − v|q dX ≤ u qρ |v|qρ |u − v|q dX ≤ v qρ RN RN L L 2∗ α,β (RN ) 2∗ α,β (RN ) u−v q u−v q u−v q L L L 2∗ α,β (RN ) 2∗ α,β (RN ) 2∗ α,β (RN ) , , . ∗ Since S 1 (RN ) is continuously embedded into L2α,β (RN ) and 1 < q < 2∗α,β , there exists a positive number M (R) such that fˆ(u) − fˆ(v) Since 2 Nα,β −2 0 depending on K and independent of n ∈ N. Thus, we can get tmax < tn + T ∗ , for n ∈ N large enough. This contradicts the maximality of tmax . 3. Existence of a global attractor in S 1 (RN ) By Theorem 2.1, we can define a continuous semigroup S(t) : S 1 (RN ) → S 1 (RN ) as follows S(t)u0 := u(t), where u(t) is the unique global mild solution of (1.1) subject to u0 as initial datum. 3.1. Existence of bounded absorbing sets. For the sake of brevity, in the following lemmas, we give some formal calculations, the rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in [20]. We first prove the existence of a bounded absorbing set for S(t) in S 1 (RN ). Lemma 3.1. Suppose (F) − (G) hold. Then the semigroup S(t) generated by (1.1) has a bounded absorbing set in S 1 (RN ), that is, there exists a positive constant ρ, such that for every bounded subset B in S 1 (RN ), there is a number T = T (B) > 0, such that for all t ≥ T , u0 ∈ B, we have u(t) S 1 (RN ) ≤ ρ. Proof. Taking the inner product of (1.1) with u in L2 (RN ) we get 1 d u 2 dt 2 L2 (RN ) (|∇x u|2 + |∇y u|2 + |x|2α |y|2β |∇z u|2 )dX + RN +λ u (3.1) 2 L2 (RN ) + f (X, u)udX = (g, u)L2 (R) . RN A SEMILINEAR STRONGLY DEGENERATE PARABOLIC EQUATIONS 7 Using (1.5), we have |u|2 dX − f (X, u)udX ≥ −µ RN RN C1 (X)dX. (3.2) RN By the Cauchy inequality, the right-hand side of (3.1) is estimated as follows |(g, u)L2 (R) | ≤ g L2 (RN ) u L2 (RN ) ≤ λ−µ u 2 2 L2 (RN ) + 1 g 2(λ − µ) 2 L2 (RN ) . (3.3) It follows from (3.1)-(3.3) that d u dt 2 L2 (RN ) ≤ 2 C1 (|∇x u|2 + |∇y u|2 + |x|2α |y|2β |∇z u|2 )dX + (λ − µ) u +2 L1 (RN ) RN + 1 g λ−µ 2 L2 (RN ) 2 L2 (RN ) . (3.4) Hence, in particular, we have d u(t) dt 2 L2 (RN ) + (λ − µ) u(t) 2 ≤ 2 C1 L1 (RN ) + 1 g λ−µ 2 L2 (RN ) . Using the Gronwall inequality, we obtain u(t) 2 L2 (RN ) ≤ e−(λ−µ)t u0 2 L2 (RN ) 2 C1 L1 (RN ) 1 + g λ−µ (λ − µ)2 + 2 L2 (RN ) . Hence we deduce the existence of a bounded absorbing set in L2 (RN ): There are a constant R and a time t0 ( u0 L2 (RN ) ) such that for the solution u(t) = S(t)u0 , u(t) L2 (RN ) ≤R for all Integrating (3.4) on (t, t + 1), t ≥ t0 ( u0 t ≥ t0 ( u0 L2 (RN ) ), L2 (RN ) ). (3.5) and using (1.6), we find that t+1 |∇x u(s)|2 + |∇y u(s)|2 + |x|2α |y|2β |∇z u(s)|2 dX ds t RN t+1 +(λ − µ) u(s) 2 L2 (RN ) ds ≤C u(t) 2 L2 (RN ) +1+ g t ≤ C R2 + 1 + g 2 L2 (RN ) . 2 L2 (RN ) (3.6) 8 C.T. ANH On the other hand, multiplying (1.1) by −Pα,β u and integrating over RN , then integrating by parts and using (1.4), we obtain d dt |∇x u|2 + |∇y u|2 + |x|2α |y|2β |∇z u|2 dX + RN |Pα,β u|2 dX RN |∇x u|2 + |∇y u|2 + |x|2α |y|2β |∇z u|2 dX +λ RN f (X, u)Pα,β udX − = RN gPα,β udX RN fu (X, u) |∇x u|2 + |∇y u|2 + |x|2α |y|2β |∇z u|2 dX − =− RN |∇x u|2 + |∇y u|2 + |x|2α |y|2β |∇z u|2 dX + ≤ RN + 1 4 gPα,β udX RN |Pα,β u|2 dX RN |g|2 dX. RN (3.7) Hence d ds (|∇x u|2 + |∇y u|2 + |x|2α |y|2β |∇z u|2 )dX RN |∇x u|2 + |∇y u|2 + |x|2α |y|2β |∇z u|2 dX + ≤ RN 1 g 4 (3.8) 2 L2 (RN ) . Combining (3.6), (3.8), and applying the uniform Gronwall inequality, we have (|∇x u(t)|2 + |∇y u(t)|2 + |x|2α |y|2β |∇z u(t)|2 )dX ≤ C R2 + 1 + g RN 2 L2 (RN ) (3.9) for all t ≥ t0 . Using (3.5), we finish the proof. We now derive uniform estimates of the derivative of solutions in time. Lemma 3.2. Suppose (F)−(G) hold. Then for every bounded subset B in S 1 (RN ), there exists a constant T = T (B) > 0 such that ut (s) where ut (s) = 2 L2 (RN ) d dt (S(t)u0 )|t=s ≤ ρ1 for all u0 ∈ B, and s ≥ T, and ρ1 is a positive constant independent of B. Proof. By differentiating (1.1) in time and denoting v = ut , we get ∂v ∂f − Pα,β v + λv + (X, u)v = 0. ∂t ∂u Taking the inner product of the above equality with v in L2 (RN ), we obtain 1 d v 2 dt 2 L2 (RN ) |∇x v|2 + |∇y v|2 + |x|2α |y|2β |∇z v|2 dX + RN +λ v 2 L2 (RN ) + RN ∂f (X, u)|v|2 dX = 0. ∂u By (1.4), it follows that d v dt 2 L2 (RN ) ≤2 v 2 L2 (RN ) . (3.10) A SEMILINEAR STRONGLY DEGENERATE PARABOLIC EQUATIONS 9 On the other hand, integrating (3.7) from t to t + 1 and using (3.9), we obtain t+1 ut (s) 2 L2 (RN ) ds ≤ C(ρ, g 2 L2 (RN ) ) (3.11) t as t large enough. Combining (3.10) with (3.11), and using the uniform Gronwall inequality, we have ut (s) 2L2 (RN ) ≤ C(ρ, g 2L2 (RN ) ). The proof is complete. We now show the existence of a bounded absorbing set in S 2 (RN ). Lemma 3.3. Let assumptions (F) − (G) hold. Then there exists a positive number R such that for any solution u of problem (1.1) we have Pα,β u 2 L2 (RN ) + u 2 L2 (RN ) ≤ R2 . (3.12) Proof. Taking the L2 -inner product of (1.1) with −Pα,β u + u, we have Pα,β u ≤− 2 L2 (RN ) +λ u 2 L2 (RN ) + f (X, u)udX RN ut − Pα,β u + u dX + (λ + 1) RN uPα,β udX − RN f (X, u)Pα,β udX RN g − Pα,β u + u dX. + RN Using (1.5) and integrating by parts, we get Pα,β u ≤− 2 L2 (RN ) + (λ − µ) u 2 L2 (RN ) (|∇x u|2 + |∇y u|2 + |x|2α |y|2β |∇z u|2 )dX ut − Pα,β u + u dX − (λ + 1) RN − RN + C1 RN fu (X, u)(|∇x u|2 + |∇y u|2 + |x|2α |y|2β |∇z u)|2 )dX + g − Pα,β u + u dX RN L1 (RN ) . Using (1.4) and the Cauchy inequality, we have Pα,β u 2 L2 (RN ) + u 2 L2 (RN ) ≤C ut 2 L2 (RN ) + u 2 S 1 (RN ) + g 2 L2 (RN ) + C1 L1 (RN ) . Hence, by Lemmas 3.1 and 3.2, we get (3.12). 3.2. Asymptotic compactness of the semigroup S(t). In order to prove the existence of a global attractor, we mainly have to prove the asymptotic compactness of S(t). Firstly, in Lemma 3.4 below, we prove a tail-estimate of solutions. Then, using this estimate, we prove the asymptotic compactness of S(t) in L2 (RN ). Finally, using the singular Gronwall inequality and properties of analytic semigroup generated by sectorial operators, we obtain the asymptotic compactness of the semigroup S(t) in S 1 (RN ). Lemma 3.4. Let ϑ¯ ∈ C 1 ([0, ∞)) be a function such that ϑ¯ ∈ [0, 1] and ¯ = 0 for s ∈ [0, 1] and ϑ(s) ¯ = 1 for s ∈ [2, ∞). ϑ(s) 10 C.T. ANH Setting ϑ = ϑ¯2 and the functions ϑ¯k : RN −→ R and ϑk : RN −→ R be defined by 2 ϑ¯k (X) = ϑ¯ 2 |X| ) k2 and ϑk (X) = ϑ |X| k2 for any k ∈ N. Then whenever u : [0, ∞) −→ S 1 (RN ) is a solution of (1.1), for every η > 0 there exist two constants K0 > 0 and T0 > 0 we have 2 ϑk (X)|u(t, X)| dX < η for all t ≥ T0 , k ≥ K0 . (3.13) RN Proof. We follow the arguments in the proof of Theorem 6.5 in [19]. Since ∇ϑk (X) = |X|2 2 |X|2 2 ϑ X and ∇ϑ¯k (X) = 2 ϑ¯ X, we have 2 2 k k k k2 sup |∇ϑk (X)| ≤ X∈RN Cϑ k and sup |∇ϑ¯k (X)| ≤ X∈RN Cϑ¯ . k Let Vk : S 1 (RN ) −→ R+ be a function defined by Vk (u) = 1 2 ϑk (X)|u(X)|2 dX, u ∈ S 1 (RN ). RN It is easy to check that Vk is Fr´echet differentiable and ϑk (X)u(X)v(X)dX = (ϑk u, v)L2 (RN ) , u, v ∈ S 1 (RN ). (Vk (u)) (v) = RN Therefore, if u : [0, ∞) −→ S 1 (RN ) is a positive semi-trajectory of the semigroup S(t), we have (Vk ◦ u) (t) = ϑk (X)u(t)ut (t)dX = (ut (t), ϑk u(t))L2 (RN ) . RN Let q be a positive number such that q + µ < λ. We have (Vk ◦ u) (t) + 2q(Vk ◦ u)(t) = (ut , ϑk u(t))L2 (RN ) + 2q(Vk ◦ u)(t) ϑk (X)|u|2 dX − =(Pα,β u, ϑk (X)u)L2 (RN ) − λ RN − ϑk (X)|u|2 dX gϑk (X)udX + q RN f (X, u)ϑk (X)udX RN RN ϑk (X)|u|2 dX ≤(Pα,β u, ϑk u) + (q + µ − λ) RN − g(X)ϑk (X)udX + RN C1 (X)ϑk (X)dX, RN where we have used (1.5). Since ϑk (X) |u|2 + |Pα,β u|2 dX ≤ (Pα,β u, ϑk u) ≤ RN |u|2 + |Pα,β u|2 dX, |X|≥k A SEMILINEAR STRONGLY DEGENERATE PARABOLIC EQUATIONS 11 we have ϑk (X)|u|2 dX (Pα,β u, ϑk u) + (q + µ − λ) RN − g(X)ϑk (X)udX + RN C1 (X)ϑk (X)dX RN |u|2 + |Pα,β u|2 dX − (λ − q − µ − ε) ϑ¯k u ≤ |X|≥k + 1 ¯ ϑk g 4ε 2 RN 2 L2 (RN ) |C1 (X)||ϑk (X)|dX + RN |u|2 + |Pα,β u|2 dX + ck , ≤ |X|≥k where the assumption 0 < ε < λ − q − µ has been used, and 1 ¯ 2 ϑk g RN + |C1 (X)||ϑk (X)|dX. ck = 4ε RN It follows from here that |u|2 + |Pα,β u|2 dX + ck , (Vk ◦ u) (t) + 2q(Vk ◦ u)(t) ≤ ∀t ≥ 0. |X|≥k Hence, we have t (Vk ◦ u)(t) ≤ e−2qt |u|2 + |Pα,β u|2 dXdξ + ck + e−2qt (Vk ◦ u)(0). e2qξ |X|≥k 0 For any η > 0 given, since C1 ∈ L1 (RN ), g ∈ L2 (RN ), there exists K1 > 0 such that for all k ≥ K1 , we have η ck < . 3 On the other hand, by Lemma 3.3, there exist K2 > 0 and T1 > 0 such that for all t ≥ T1 , k ≥ K2 , t e−2qt e2qξ 0 |u|2 + |Pα,β u|2 dXdξ < |X|≥k η . 3 Obviously, there exists T3 > 0 such that for all t ≥ T3 , η e−2qt (Vk ◦ u)(0) < . 3 Choosing T0 = max{T1 , T2 , T3 } and K0 = max{K1 , K2 }, we get the estimate (3.13). Lemma 3.5. The semigroup S(t) is asymptotically compact in the strong L2 (RN ) topology. Proof. The proof is similar to that of Theorem 6.6 in [19], however, we recall it here for convenience of the reader. Let (un ) be bounded in S 1 (RN ), (tn ) be a number sequence such that lim tn = n→∞ +∞. We have to show that there exist a strictly increase sequence (nk ) and u ∈ S 1 (RN ) such that lim S(tnk )unk = u in L2 (RN ). n→∞ Let un S 1 (RN ) ≤ R, ∀n ≥ 1. Since orbits of bounded sets are bounded, we have S(t)un S 1 (RN ) ≤R ∀t ≥ 0, ∀n ≥ 1. 12 C.T. ANH Put α is the Kuratowski measure of non-compactness on L2 (RN ) α(B) = inf{l > 0 : B admits a finite cover by sets of diameter ≤ l}, where B is a bounded subset in L2 (RN ). Using properties of this measure, we have for every k, n0 , α{un (tn )|n ∈ N} ≤α{(1 − ϑk )un (tn ) | n ∈ N} + α{ϑk un (tn ) | n ∈ N} = α{(1 − ϑk )un (tn ) | n ∈ N} + α{ϑk un (tn ) | n ≥ n0 }. Noting that tn → ∞ when n → ∞, then using (3.13), we have for every ε > 0, ∃k, n0 : ϑk un (tn ) L2 (RN ) < ε, ∀n ≥ n0 . Thus, α{un (tn )|n ∈ N} ≤ α{(1 − ϑk )un (tn )|n ∈ N} + . Since 1 − ϑk ∈ C01 (RN ), it is easy to check that the map u → (1 − ϑk )u is compact from S 1 (RN ) to L2 (RN ) by using the compactness of the embedding S 1 (Ω) → L2 (Ω) when Ω is bounded (see [21]). Hence α{(1 − ϑk )un (tn )|n ∈ N} = 0. We therefore obtain that α{un (tn ) | n ∈ N } = 0, i.e., {un (tn )} is precompact on L2 (RN ). This completes the proof. Theorem 3.1. Suppose (F) and (G) hold. Then the semigroup S(t) generated by problem (1.1) has a compact global attractor A in the space S 1 (RN ). Proof. We use some ideas in Lemma 2.2 and Theorem 2.3 in [18]. By Lemma 3.1, in order to prove the existence of the global attractor in S 1 (RN ), we only need to prove that S(t) is asymptotically compact in S 1 (RN ). Let {un }n≤1 be a bounded sequence in S 1 (RN ) and tn → +∞. Fix T > 0, since {un } is bounded and orbits of bounded sets are bounded, {S(tn − T )un } is bounded in S 1 (RN ). Since S(t) is asymptotically compact in L2 (RN ), there is a subsequence {S(tnk − T )unk } and v ∈ S 1 (RN ) such that vk = S(tnk − T )unk v weakly in S 1 (RN ) and strongly to S(t)v in L2 (RN ) as k → ∞. We will prove that S(tnk )unk = S(T )vk converges strongly to S(T )v in S 1 (RN ), and thus S(t) is asymptotically compact. Denote vk (t) = S(t)vk , v(t) = S(t)v, we have t e−A(t−s) (fˆ(vk (s)) + g)ds, vk (t) = e−At vk − 0 t e−A(t−s) (fˆ(v(s)) + g)ds. v(t) = e−At v − 0 Hence applying the estimate (2.1), we have t vk (t)−v(t) S 1 (RN ) ≤ C4 vk −v (t−s)−1/2−γ vk (s)−v(s) L2 (RN ) +C5 S 1 (RN ) ds. 0 By the singular Gronwall inequality [11, Chapter 7], there is a constant C such that, for t ∈ (0, T ], vk (t) − v(t) S 1 (RN ) ≤ Ct−1/2 vk − v L2 (RN ) . In particular, vk (T ) − v(T ) S 1 (RN ) ≤ Ct−1/2 vk − v L2 (RN ) . A SEMILINEAR STRONGLY DEGENERATE PARABOLIC EQUATIONS 13 Since vk → v in L2 (RN ), vk (T ) → v(T ) in S 1 (RN ) as k → +∞. This implies that S(t) is asymptotically compact. Therefore, the semigroup S(t) generated by (1.1) has a compact connected global attractor A in S 1 (RN ). Remark 3.1. Thanks to the Lyapunov function Φ, one can show that the semigroup S(t) is a gradient system. Therefore, by Theorem 10.13 in [20], we have the structure of the global attractor obtained, namely A = W u (E), the unstable manifold of the set E of fixed points of the semigroup S(t). In particular, we have lim dist (S(t)u0 , E) = 0 for every u0 ∈ S 1 (RN ). t→+∞ Acknowledgements. This work was completed when the author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the Institute for its hospitality. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2012.04. 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Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, 1997. [24] B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D 179 (1999), 41-52. Department of Mathematics, Hanoi National University of Education 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam E-mail address: anhctmath@hnue.edu.vn [...]... Electron J Differ Equ 32 (2008), 1-11 [2] C.T Anh and T.D Ke, Existence and continuity of global attractor for a semilinear degenerate parabolic equation, Electron J Differ Equ 61 (2009), 1-13 [3] C.T Anh and L.T Tuyet, Strong solutions to a strongly degenerate semilinear parabolic equation, Viet J Math 41 (2013), 217-232 [4] C.T Anh and L.T Tuyet, On a semilinear strongly degenerate parabolic equation. .. ANH [17] M Mihilescu, G Moroanu and D Stancu-Dumitru, Equations involving a variable exponent Grushin-type operator, Nonlinearity 24 (2011), 2663-2680 [18] M Prizzi, A remark on reaction-diffusion equations on unbounded domains, Disc Cont Dyn Syst 9 (2003), 281-286 [19] M Prizzi and K Rybakowski, Attractors for reaction-diffusion equations on arbitrary unbounded domains, Topol Methods Nonlinear Anal... Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes Math., Vol 840, Springer, Berlin, 1981 [12] A. E Kogoj and E Lanconelli, On semilinear ∆λ -Laplace equation, Nonlinear Anal 75 (2012), 4637-4649 [13] A Kogoj and S Sonner, Attractors for a class of semi-linear degenerate parabolic equations, J Evol Equ 13 (2013), 675-691 [14] E Lanconelli and A. E Kogoj, X-elliptic operators and... Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001 [21] P.T Thuy and N. M Tri, Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations, Nonlinear Diff Equ Appl (NoDEA) 19 (2012), 279-298 [22] P.T Thuy and N. M Tri, Long-time behavior of solutions to semilinear parabolic equations involving strongly degenerate. .. visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM) He would like to thank the Institute for its hospitality This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2012.04 References [1] C.T Anh, P.Q Hung, T.D Ke and T.T Phong, Global attractor for a semilinear parabolic equation involving Grushin operator,... and X-control distances, Ric Mat 49 (2000), Suppl., 223-243 [15] V .A Markasheva and A. F Tedeev, Local and global estimates of the solutions of the Cauchy problem for quasilinear parabolic equations with a nonlinear operator of Baouendi-Grushin type, Math Notes 85 (2009), 385-396 [16] V .A Markasheva and A. F Tedeev, The Cauchy problem for a quasilinear parabolic equation with gradient absorption, Mat Sb... degenerate elliptic differential operators, Nonlinear Diff Equ Appl (NoDEA) 20 (2013), 1213-1224 [23] R Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, 1997 [24] B Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D 179 (1999), 41-52 Department of Mathematics, Hanoi National University of Education 136 Xuan Thuy, Cau... B Franchi and E Lanconelli, Une m´ etrique associ´ ee une classe d’op´ erateurs elliptiques d´ eg´ en´ er´ es, Conference on linear partial and pseudodifferential operators (Torino, 1982) Rend Sem Mat Univ Politec Torino 1983, Special Issue, 105-114 (1984) [8] B Franchi and E Lanconelli, An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality, Comm Partial... (RN ) ≤ Ct−1/2 vk − v L2 (RN ) In particular, vk (T ) − v(T ) S 1 (RN ) ≤ Ct−1/2 vk − v L2 (RN ) A SEMILINEAR STRONGLY DEGENERATE PARABOLIC EQUATIONS 13 Since vk → v in L2 (RN ), vk (T ) → v(T ) in S 1 (RN ) as k → +∞ This implies that S(t) is asymptotically compact Therefore, the semigroup S(t) generated by (1.1) has a compact connected global attractor A in S 1 (RN ) Remark 3.1 Thanks to the Lyapunov... Differential Equations 9 (1984), 12371264 [9] V.V Grushin, A certain class of elliptic pseudo differential operators that are degenerated on a submanifold, Mat Sb., 1971, vol.84 ( 126), 163-195; English transl in : Math USSR Sbornik 13( 1971), 155-183 [10] Y Han and Q Zhao, A class of Caffarelli-Kohn-Nirenberg type inequalities for generalized Baouendi-Grushin vector fields, Acta Math Sci., Ser A, Chin ... attractor for a semilinear degenerate parabolic equation, Electron J Differ Equ 61 (2009), 1-13 [3] C.T Anh and L.T Tuyet, Strong solutions to a strongly degenerate semilinear parabolic equation, ... Phong, Global attractor for a semilinear parabolic equation involving Grushin operator, Electron J Differ Equ 32 (2008), 1-11 [2] C.T Anh and T.D Ke, Existence and continuity of global attractor. .. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes Math., Vol 840, Springer, Berlin, 1981 [12] A. E Kogoj and E Lanconelli, On semilinear ∆λ -Laplace equation, Nonlinear Anal 75 (2012),