A sharp estimate and change on the dimension of the attractor for Allen-Cahn equations
aa r X i v : . [ m a t h . A P ] F e b A sharp estimate and change on the dimension of the attractor forAllen-Cahn equations ∗ Nikos. I. Karachalios,
Department of Mathematics , University of the Aegean , Karlovassi, 83200 Samos, Greece
Nikos B. Zographopoulos,
Department of Mathematics, , National Technical University of Crete , Chania, 73100 Crete, Greece
Abstract
We consider the semilinear reaction diffusion equation ∂ t φ − ν ∆ φ − V ( x ) φ + f ( φ ) = 0, ν > ⊂ R N . We assume the standard Allen-Cahn-type nonlinearity, while the potential V is eitherthe inverse square potential V ( x ) = δ | x | − or the borderline potential V ( x ) = δ dist( x, ∂ Ω) − , δ ≥ δ = 0). In the subcritical cases δ = 0, N ≥ < µ := δν < µ ∗ , N ≥ µ ∗ is the optimal constant of Hardy and Hardy-type inequalities),we present a new estimate on the dimension of the global attractor. This estimate comes out by an improvedlower bound for sums of eigenvalues of the Laplacian by A. D. Melas (Proc. Amer. Math. Soc. (2003),631-636). The estimate is sharp, revealing the existence of (an explicitly given) threshold value for the ratio ofthe volume to the moment of inertia of Ω on which the dimension of the attractor may considerably change.Consideration is also given on the finite dimensionality of the global attractor in the critical case µ = µ ∗ . Let Ω ⊂ R N , N ≥ ∂ Ω and consider the eigenvalues 0 < λ (Ω) ≤ λ (Ω) . . . ≤ λ m (Ω) ≤ . . . (counting multiplicity) of the Dirichlet Laplacian − ∆ u = λu, in Ω , (1.1) u = 0 , on ∂ Ω . In [16, Theorem 1, pg. 312] it was proved that for any m ≥ m X i =1 λ i (Ω) ≥ N C N N + 2 µ N (Ω) − N m N +2 N , C N = (2 π ) ω − /NN . (1.2)Here ω N denotes the volume of the unit ball in R N and µ N (Ω) denotes the N -dimensional volume of Ω. Thelower bound (1.2) on the sums of the eigenvalues is sharp in view of H. Weyl’s asymptotic formula λ m (Ω) ∼ C N (cid:18) mµ N (Ω) (cid:19) N , as m → ∞ . ∗ Keywords and Phrases: Allen-Cahn equation, singular potential, Hardy inequality, attractors, Hausdorff dimension, fractaldimension. AMS Subject Classification: 35K57, 35B40, 35B41, 37L30.E-mail: [email protected] (N. I. Karachalios) & [email protected] (N. B. Zographopoulos) I (Ω) the “momentof inertia” of Ω, defined as I (Ω) = min α ∈ R N Z Ω | x − α | dx. Then the lower bound (1.2) can be improved as m X i =1 λ i (Ω) ≥ N C N N + 2 µ N (Ω) − N m N +2 N + M N µ N (Ω) I (Ω) m, (1.3) M N = cN + 2 , with c < (2 π ) ω − N N , but c independent of N .
In this paper we note that the improved estimate (1.3) implies a new and sharp estimate on the dimension ofthe global attractor, for the reaction diffusion system φ t − ν ∆ φ − V ( x ) φ + f ( φ ) = 0 , ν > , in Ω , t > , (1.4) φ ( x,
0) = φ ( x ) , for x ∈ Ω , (1.5) φ ( x, t ) = 0 in ∂ Ω , t > , (1.6)where generally Ω ⊂ R N is a bounded domain of N ≥
1. The potential V ( x ) is either the inverse square potential V ( x ) = δ | x | − or the borderline potential V ( x ) = δ dist( x, ∂ Ω) − , δ ≥
0. For the reaction term f : R → R we aremaking the standard assumption that is a polynomial of odd degree with a positive leading coefficient, f ( s ) = γ − X k =1 b k s k , b γ − > . (1.7)With the nonlinearity (1.7) equation (1.4) can be considered as a singular Allen-Cahn-type equation (includingthe classical Allen-Cahn equation when δ = 0). Setting µ := δ/ν , we consider as subcritical the cases δ = 0 for N ≥ < µ < µ ∗ for N ≥ N ≥
3, depending on the type of the potential involved in (1.4). The constant µ ∗ denotes the optimal constant of Hardy and Hardy-type inequalities . The critical value µ ∗ = ( N − / µ Z Ω u | x | dx ≤ Z Ω |∇ u | dx, for all u ∈ C ∞ (Ω) , Ω ⊆ R N , N ≥ , (1.8)which is not attained in H (Ω). Similarly, µ ∗ = 1 / µ Z Ω u d ( x ) dx ≤ Z Ω |∇ u | dx, for all u ∈ C ∞ (Ω) , d ( x ) := dist( x, ∂ Ω) , Ω ⊂ R N , N ≥ , (1.9)which is also not attained in H (Ω).This new estimate, although it simply comes out by incorporating (1.3) in the well known procedure estimatingthe distortion of infinitesimal m -volumes produced by the associated semiflow (cf. [3, 19]), is sharp not onlydue to the optimality of the lower bound (1.3). Regarding its explicit dependence on the parameters of theproblem and the geometric properties of the open set Ω it reveals the existence of a threshold value on the ratio R (Ω) := µ N (Ω) /I (Ω) on which the dimension of the attractor in the subcritical case may considerably change:There exists R thresh ( N, µ, f ) > such that if R (Ω) < R thresh then dim H A ≤ d ( N, µ, f, R (Ω)) . On the otherhand, if R thresh ≤ R (Ω) then dim H A ≤ . The result is given in detail in Section 2. In the case δ = 0, N ≥
1, the result can be viewed as a new conditionon the diffusivity ν for a change of the attractor dimension (Theorem 2.3 & Remark 2.4). Furthermore it isverified that the dimension of the attractor should be actually smaller than the existing estimates indicate, seefor example [1, 10, 19].In Section 3, we comment to the case of the critical potentials µ = µ ∗ . This case can be treated in generalizedSobolev spaces as the results of [2, 4, 5, 6, 7, 13, 18], suggest. Using the Weyl’s type estimates on the eigenvaluesof the critical Schr¨odinger operator − ∆ − V derived in [14], we conclude by presenting the necessary conditionsunder which the dimension of the attractor in the critical case can be (explicitly) estimated. ≤ µ < µ ∗ : Sharp estimates on the Hausdorffdimension of the global attractor A. Inverse square potential V ( x ) = δ | x | − . We consider first the case where Ω ⊂ R N , N ≥
3, containing theorigin and δ > N = 2 reduces actually to the case δ = 0, see [18, pg.108]). Let us note that equation(1.4) can be rewritten as ∂ t φ + ν K s φ + f ( φ ) = 0 , ν > , where K s denotes the Schr¨odinger operator K s = − ∆ − µ | x | , µ = δν . In the subcritical case 0 < µ < µ ∗ = ( N − /
4, the Hilbert space H µ (Ω) can be considered (cf. [18]), which isdefined as the completion of C ∞ (Ω) with respect to the norm || u || H µ (Ω) := (cid:20)Z Ω (cid:18) |∇ u | − µ | x | u (cid:19) dx (cid:21) , < µ < µ ∗ . (2.1)Hardy’s inequality (1.8) implies that the norm || u || , = (cid:0)R Ω |∇ u | dx (cid:1) / of the Sobolev space H (Ω) and thenorm (2.1) are equivalent. One has that C µ k| u || H (Ω) ≤ || u || H µ (Ω) ≤ || u || H (Ω) , (2.2)with C µ := 1 − µµ ∗ >
0, if 0 < µ < µ ∗ . The operator K s := − ∆ − µ/ | x | with domain D ( K s ) := (cid:26) u ∈ H µ (Ω) : − ∆ u − µ | x | u ∈ L (Ω) , < µ < µ ∗ (cid:27) , is a nonnegative self-adjoint operator on L (Ω). Due to the standard embedding properties of H (Ω) ≡ H µ (Ω)we have the following Proposition 2.1
Let Ω ⊂ R N , N ≥ be a bounded domain containing the origin, and assume that < µ = δ/ν <µ ∗ . The problem (1.4)-(1.6) with nonlinearity (1.7) and V ( x ) = δ | x | , defines a semiflow S ( t ) : L (Ω) → L (Ω) .The semiflow possesses a global attractor A µ which is bounded in H (Ω) , compact and connected in L (Ω) . We recall that since f is of the form (1.7), there exists κ > f ′ ( s ) ≥ − κ, for all s ∈ R . (2.3)The main result result of the paper is Theorem 2.2
Let Ω ⊂ R N , N ≥ , a bounded domain containing the origin. Let < µ = δ/ν < µ ∗ and considerthe global attractor A µ of the semiflow S ( t ) : L (Ω) → L (Ω) . We define R thresh := κµ ∗ M N ( µ ∗ ν − δ ) . (i) If R thresh ≤ R (Ω) = µ N (Ω) /I (Ω) , then A µ has finite Hausdorff dimension dim H A µ ≤ .(ii) If R (Ω) < R thresh , then A µ has finite Hausdorff dimension dim H A µ ≤ d and finite fractal dimension dim F A µ ≤ d where d = (cid:18) N + 2 N C N (cid:19) N (cid:18) κµ ∗ ( µ ∗ ν − δ ) − M N R (Ω) (cid:19) N µ N (Ω) , N ≥ . (2.4) Proof:
The usual arguments can be applied establishing that for every t >
0, the function φ → S ( t ) φ is Fr´echetdifferentiable. The differential is D ( t, φ ) : ξ ∈ L (Ω) → Φ( t ) ∈ L (Ω), Φ( t ) where Φ( t ) is the solution of the firstvariation equation Φ t + ν H s Φ − f ′ ( S ( t ) φ )Φ = 0 for t > , (2.5)supplemented with the initial and boundary conditionsΦ( t ) = 0 on ∂ Ω and Φ(0) = ξ ∈ L (Ω) . (2.6)We shall examine the development of infinitesimal parallelepipeds spanned by Φ ( t ) , · · · Φ m ( t ) ∈ L (Ω), whereΦ i ( t ) , i = 1 , . . . , m is an infinitesimal vector evolving from Φ i (0) = ξ i ∈ L (Ω). The vectors Φ i , i = 1 , . . . , m are m -solutions of (2.5)-(2.6) starting from the initial conditions Φ i (0) = ξ i ∈ L (Ω) and it is well known that the m -dimensional volume | Φ ( t ) ∧ Φ ( t ) . . . ∧ Φ m ( t ) | of the infinitesimal parallelepiped spanned by Φ i ( t ) is given by | Φ ( t ) ∧ Φ ( t ) . . . ∧ Φ m ( t ) | = | ξ ∧ . . . ∧ ξ m | exp Z t Tr[ L ( S ( s ) φ ) ◦ Q m ( s )] ds, (2.7)where L ( S ( t ) φ ) = − ν H s Φ − f ′ ( S ( t ) φ )Φ , for Φ ∈ L (Ω) . We denote by Q m ( t ), the orthogonal projection in L (Ω) onto span { Φ ( t ) , . . . , Φ m ( t ) } . We fix t for the timebeing and we consider an orthonormal basis e , e , . . . of L (Ω) withspan { e , . . . , e m } = span { Φ ( t ) , . . . , Φ m ( t ) } = span Q m ( t ) L (Ω) . Since Φ i ( t ) ∈ H (Ω), for all i ∈ N and almost all t >
0, we have e i ∈ H (Ω) for all i ∈ N . Then Q m ( t ) e i = e i if i ≤ m and Q m ( t ) e i = 0 otherwise. Thus, for i ≤ m ,( L ( S ( t ) φ ) ◦ Q m ( t ) e i , e i ) L (Ω) = ( L ( S ( t ) φ ) e i , e i ) L (Ω) = − ν (cid:26)Z Ω |∇ e i | dx − Z Ω µe i | x | dx (cid:27) − Z Ω f ′ ( S ( t ) φ ) e i dx. (2.8)Using (2.3), inequality (2.8) becomes( L ( S ( t ) φ ) ◦ Q m ( t ) e i , e i ) L (Ω) ≤ − ν (cid:26)Z Ω |∇ e i | dx − Z Ω µe i | x | dx (cid:27) + κ, (2.9)noticing that R Ω e i dx = 1 by the orthonormality of the e i . ThereforeTr[ L ( S ( s ) φ ) ◦ Q m ( s )] = m X i =1 ( L ( S ( t ) φ ) ◦ Q m ( t ) e i , e i ) L (Ω) ≤ − ν m X i =1 (cid:26)Z Ω |∇ e i | dx − Z Ω µe i | x | dx (cid:27) + κm = − ν m X i =1 ( K s e i , e i ) L (Ω) + κm ≤ − ν m X i =1 Z Ω |∇ e i | dx + κm, ν := νC µ = − ν m X i =1 ( − ∆ e i , e i ) L (Ω) + κm. (2.10)Furthermore, by using the inequality m X i =1 ( − ∆ e i , e i ) L (Ω) ≥ m X i =1 λ i (Ω) , we may insert (1.3) in (2.10), to getTr[ L ( S ( t ) φ ) ◦ Q m ( t )] ≤ − ν N C N N + 2 µ N (Ω) − N m N +2 N − ν M N R (Ω) m + κ m, (2.11)= − ν N C N N + 2 µ N (Ω) − N m N +2 N + κ m, (2.12)where κ = κ − ν M N R (Ω) . (2.13)Assume that κ ≤ R thresh ≤ R (Ω). Then (2.7) implies the exponential decay in time of the m -dimensionalvolume | Φ ( t ) ∧ Φ ( t ) . . . ∧ Φ m ( t ) | , for any m ≥
1. Then Constantin-Foias-Temam theory [19, Proposition 2.1,pg. 364 & Theorem 3.3, pg. 374] implies that the Hausdorff dimension of the attractor is less or equal than m ,for any m ≥
1. This proves claim ( i ) of the theorem.Assume next that κ >
0, i.e. condition R thresh > R (Ω) is satisfied. We define by (2.11) the function g ( x ) = − βν µ N (Ω) − N x N +2 N + κ x, β = N C N N + 2 . This function has the root d = β − N (cid:18) κ ν (cid:19) N µ N (Ω) . Then again the results of [19], imply that dim H A µ ≤ d . In addition, since the function g is a concave functionof the continuous variable x , [10, Corollary 2.2, pg. 815] implies that the fractal dimension of A µ is dim F A µ ≤ d This proves claim ( ii ) of the theorem. (cid:4) In the subcritical case δ = 0 (classical Allen-Cahn equation), we may consider the initial-boundary valueproblem (1.4)-(1.6) in a bounded domain Ω ⊂ R N , N ≥
1. In this case we have
Theorem 2.3
Let Ω ⊂ R N , N ≥ be a bounded domain. Consider the global attractor A of the semiflow S ( t ) : L (Ω) → L (Ω) associated to (1.4)-(1.6) when δ = 0 and with the nonlinearity (1.7). We define R thresh := M N κν . (i) Assume that R thresh ≤ R (Ω) . Then A has finite Hausdorff dimension and dim H A ≤ .ii) Assume that R (Ω) < R thresh . Then A has finite Hausdorff dimension dim H A µ ≤ d and finite fractaldimension dim F A µ ≤ d where d = (cid:18) N + 2 N C N (cid:19) N (cid:16) κν − M N R (Ω) (cid:17) N µ N (Ω) , N ≥ . (2.14) Remark 2.4
We remark that the assumption R thresh ≤ R (Ω) in Theorem 2.3 which can be viewed as M N R (Ω) κ ≤ ν, (2.15) gives a new condition on the diffusivity ν in order to have a global attractor of small Hausdorff dimension. Werecall that if ν is sufficiently large (cf. [19, Remark 1.2, pg. 88]), || φ ( t ) || L (Ω) decreases exponentially to as t → ∞ , without necessarily implying that A = { } . The attractor A may contain one or many heteroclinic curves.On the other hand, when R (Ω) < R thresh meaning that M N R (Ω) κ > ν, (2.16) we get an improved upper bound (2.4) compared with existing upper bounds on the dimension of the global attractorfor the Allen-Cahn equation [10, 19]. The estimate (2.14) shows that the Hausdorff dimension is indeed smallerdue to the appearance of the term − M N R (Ω) . For the case δ > the conditions of Theorem 2.2, can be similarlyimplemented. B. Borderline potential V ( x ) = δd − ( x ) . In the case of the borderline potential, we start with the case ofΩ ⊂ R N , N ≥
2. Now equation (1.4) can be rewritten as ∂ t φ + ν H s φ + f ( φ ) = 0 , ν > , where K s denotes the Schr¨odinger operator H s = − ∆ − µd ( x ) , µ = δν . In the subcritical case 0 < µ < µ ∗ = 1 /
4, and motivated by [6], we may consider the Hilbert space W µ (Ω) definedas the completion of C ∞ (Ω) with respect to the norm || u || W µ (Ω) := (cid:20)Z Ω (cid:18) |∇ u | − µ u d ( x ) (cid:19) dx (cid:21) , < µ < µ ∗ . (2.17)Similarly to the inverse potential case, the Hardy-type inequality (1.9) implies that the usual Sobolev norm andthe norm (2.17) are equivalent. One has that C µ k| u || H (Ω) ≤ || u || W µ (Ω) ≤ || u || H (Ω) , this time with C µ := 1 − µ >
0, if 0 < µ < µ ∗ . The operator H s := − ∆ − µd ( x ) with domain D ( H s ) := (cid:26) u ∈ W µ (Ω) : − ∆ u − µ ud ( x ) ∈ L (Ω) , < µ < µ ∗ (cid:27) , is a nonnegative self-adjoint operator on L (Ω). Working exactly as for the proof of Theorem 2.2 we have Theorem 2.5
Let Ω ⊂ R N , N ≥ , a bounded domain. Let < µ = δ/ν < µ ∗ and consider the global attractor A µ of the semiflow S ( t ) : L (Ω) → L (Ω) associated to (1.4)-(1.6) with the nonlinearity (1.7) and V ( x ) = δd − ( x ) .We define R thresh := κM N ( ν − δ ) . (i) Assume that R thresh ≤ R (Ω) . Then A µ has finite Hausdorff dimension dim H A µ ≤ .(ii) Assume that R (Ω) < R thresh . Then A µ has finite Hausdorff dimension dim H A µ ≤ d and finite fractaldimension dim F A µ ≤ d where d = (cid:18) N + 2 N C N (cid:19) N (cid:18) κ ( ν − δ ) − M N R (Ω) (cid:19) N µ N (Ω) , N ≥ . µ = µ ∗ In the critical case µ = µ ∗ the initial-boundary value problem (1.4)-(1.6) still defines a semiflow S ( t ) : L (Ω) → L (Ω). For the definition of the semiflow and the existence and finite dimensionality of the global attractorgeneralized Sobolev spaces come into play, as well as, Weyl’s type estimates on the eigenvalues of the criticalSchr¨odinger operators. These results can be used under some further geometric restrictions on Ω. A. Critical inverse square potential V ( x ) = ( N − | x | . We assume that Ω ⊂ R N , N ≥
3, is a bounded domaincontaining the origin. In the case of the inverse square potential, the critical initial-boundary value problem isfor the equation ∂ t φ + ν K φ + f ( φ ) = 0 , ν > , (3.1)supplemented with the initial and boundary conditions (1.5)-(1.6). Here K denotes the critical Schr¨odingeroperator − ∆ − ( N − | x | . The operator K : D ( K ) → L (Ω), with its domain defined as D ( K ) := (cid:26) u ∈ H (Ω) : − ∆ u − ( N − | x | u ∈ L (Ω) (cid:27) , is a nonnegative self-adjoint operator on L (Ω). Here H (Ω) is the Hilbert space defined in [18, Section 4.1], asthe completion of C ∞ (Ω) in the norm || u || H (Ω) := (cid:20)Z Ω (cid:18) |∇ u | − ( N − | x | u (cid:19) dx (cid:21) . (3.2)The following crucial improvement of (1.8) has been proved in [18, Theorem 2.2, pg. 108]: There exists somepositive constant C ( r, Ω) such that Z Ω |∇ u | dx − Z Ω ( N − | x | u dx ≥ C ( r, Ω) || u || W ,r (Ω) , for all u ∈ C ∞ (Ω) , ≤ r < , (3.3)implying that the embedding H (Ω) ֒ → W ,r (Ω) , ≤ r <
2. From this, we actually infer the compact embeddings H (Ω) ֒ → ֒ → L (Ω), H (Ω) ֒ → ֒ → H s (Ω) , ≤ s <
1, and hence the existence of a complete orthonormal basis { φ j } j ≥ of L (Ω) consisting of eigenfunctions of K with the eigenvalue sequence0 < λ ≤ λ ≤ · · · ≤ λ j ≤ · · · → ∞ , as j → ∞ . (3.4)Furthermore, it was shown in [14, Theorem 2.1] that if2 NN + 2 < q < , (3.5)the eigenvalues (3.4) satisfy the Weyl’s estimate λ j ≥ C ( q, Ω)e − µ N (Ω) − Nq − N +2 qNq j Nq − N +2 qNq , j → ∞ . (3.6)With the estimate (3.6) in hand we have Theorem 3.1
Let Ω ⊂ R N , N ≥ a bounded domain containing the origin. The initial-boundary value problem(3.2)-(1.5)-(1.6) with nonlinearity (1.7), defines a semiflow ˜ S ( t ) : L (Ω) → L (Ω) possessing a global attractor ˜ A . There exists a constant C ( q, Ω , N ) > such that dim H ˜ A ≤ ˜ d with ˜ d = C ( q, Ω , N ) − NqNq − N +2 q (cid:16) κν (cid:17) NqNq − N +2 q µ N (Ω) . (3.7) Proof:
We just note that the process followed in Theorem 2.2, leads to the corresponding estimate for (3.1)Tr[ L ( S ( t ) φ ) ◦ Q m ( t )] ≤ − νC ( q, Ω) µ N (Ω) p ∗− p ∗ m p ∗− p ∗ + κm, p ∗ = qNN − q . (3.8)We remark that the requirement p ∗ − p ∗ > d and the estimate on the dimension follows from [19], [10, Corollary 2.2, pg. 815]. (cid:4) Note that the estimate (3.7) lacks the exponent N/ r = 2 and consequently, Weyl’s estimate (3.6) is not valid in the critical value q = 2.This is not the case for the subcritical problem 0 < µ < µ ∗ . B. Critical borderline potential V ( x ) = d ( x ) . We assume in this case that Ω ⊂ R N , N ≥
2, is a bounded smooth and convex domain. In the case of the critical borderline potential, the initial-boundary value problem isfor the equation ∂ t φ + ν H φ + f ( φ ) = 0 , ν > , (3.9)supplemented with the initial and boundary conditions (1.5)-(1.6). Now H denotes the critical the criticalSchr¨odinger operator − ∆ − d ( x ) . The operator H : D ( H ) → L (Ω), with domain D ( H ) := (cid:26) u ∈ W (Ω) : − ∆ u − u d ( x ) ∈ L (Ω) (cid:27) , is a nonnegative self-adjoint operator on L (Ω). The Hilbert space W (Ω) is the completion of C ∞ (Ω) in the norm || u || W (Ω) := (cid:20)Z Ω (cid:18) |∇ u | − u d ( x ) (cid:19) dx (cid:21) . Improving the results of [6], it was shown in [2], that there exists some positive constant C such that Z Ω |∇ u | dx − Z Ω u d ( x ) dx ≥ C (cid:18)Z Ω |∇ u | q dx (cid:19) q , for all u ∈ C ∞ (Ω) , ≤ q < . Thus as in A. we have the compact embeddings H (Ω) ֒ → ֒ → L (Ω), H (Ω) ֒ → ֒ → H s (Ω) , ≤ s <
1, and theexistence of a complete orthonormal basis { ˜ φ j } j ≥ of L (Ω) made of eigenfunctions of H with the eigenvaluesequence 0 < ˜ λ ≤ ˜ λ ≤ · · · ≤ ˜ λ j ≤ · · · → ∞ , as j → ∞ . (3.10)The convexity condition can not be relaxed since it is known that if Ω is not convex we may have λ > −∞ , [6].By making use of the inequality in [11, Theorem 1.1, pg. 492], Z Ω |∇ u | p dx − (cid:18) p − p (cid:19) p Z Ω | u | p d p dx ≥ ˜ C (Ω) (cid:18)Z Ω | u | r dx (cid:19) pr , for all u ∈ C ∞ (Ω) , (3.11)which holds for 1 < p < N and p ≤ r < NpN − p and its sharp estimates on the optimal constant c ( p, r, N ) D n − p − Npr int ≥ ˜ C (Ω) ≥ c ( p, r, N ) D n − p − Npr int , D int := 2 sup x ∈ Ω d ( x ) , (3.12)it was shown in [14, Theorem 2.3] that if (3.5) holds, the eigenvalues (3.10) satisfy the Weyl’s type estimate˜ λ j ≥ ˆ C (Ω)e − µ N (Ω) − Nq − N +2 qNq j Nq − N +2 qNq , j → ∞ . (3.13)The constant ˆ C (Ω) satisfies the upper and lower estimates c ( q, N ) D N ( q − q int ≥ ˆ C (Ω) ≥ c ( q, N ) D N ( q − q int . (3.14)However, when N ≥ q = 2, given in [12, Theorem3.4, pg. 46] , Z Ω |∇ u | dx − Z Ω u d dx ≥ C ( N, D int ) (cid:18)Z Ω | u | NN − dx (cid:19) N − N , for all u ∈ C ∞ (Ω) , (3.15)we get the Weyl’s type estimate [14, Theorem 2.4]˜ λ j ≥ C ( N, D int )e − µ N (Ω) − N j N , j → ∞ . (3.16)With the estimates (3.13) and (3.16) we have Theorem 3.2 (i) Let Ω ⊂ R N , N ≥ a bounded, smooth and convex domain. The initial-boundary value problem(3.9)-(1.5)-(1.6) with nonlinearity (1.7), defines a semiflow ˆ S ( t ) : L (Ω) → L (Ω) possessing a global attractor ˆ A . There exists a constant C ( q, N, D int ) > such that dim H ˜ A ≤ ˆ d with ˆ d = C ( q, N, D int ) − NqNq − N +2 q (cid:16) κν (cid:17) NqNq − N +2 q µ N (Ω) . (3.17) (ii) Let Ω ⊂ R N , N ≥ a bounded, smooth and convex domain. Then there exists C ( N, D int ) > such that dim H ˜ A ≤ ˆ d and dim F ˜ A ≤ ˆ d with ˆ d = C ( N, D int ) − N (cid:16) κν (cid:17) N µ N (Ω) . (3.18)We conclude by mentioning that in the critical cases µ = µ ∗ , a semiflow may not be defined in H (Ω) asthe non-existence results of [4, 7, 18] indicate. However, the semiflows can be defined in the correspondinggeneralized Sobolev phase spaces H (Ω) , W (Ω) (possibly under appropriate smallness conditions on the growth ofthe nonlinearity) and they satisfy the appropriate energy equations [8, 9]. We also refer to our recent work [15]. References [1] A. V. Babin and M. I. Vishik, Attractors for Partial Differential Evolution Equations in an UnboundedDomain, Proc. Roy. Soc. Edinb., (1990), 221–243.[2] G. Barbatis, S. Filippas and A. Tertikas
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