Allen–Cahn system

Nonlinear Analysis 52 (2003) 1821 – 1841

www.elsevier.com/locate/na

Global attractor for the Cahn–Hilliard/ Allen–Cahn system Maria Gokielia;∗ , Akio Itob a Interdisciplinary

Centre for Mathematical and Computational Modelling, Warsaw University, Pawinskiego 5a, 02-106 Warsaw, Poland b Department of Architecture, School of Engineering, Kinki University, 1 Takayaumenobe, Higashihiroshima, Hiroshima 739-2116, Japan Received 15 October 2001; accepted 17 May 2002

Abstract We study the coupled Cahn–Hilliard/Allen–Cahn problem with constraints, which describes the isothermal di/usion-driven phase transition phenomena in binary systems. Our aim is to show the existence–uniqueness result and to construct the global attractor for the related dynamical system. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Parabolic systems; Cahn–Hilliard; Allen–Cahn; Long-time behavior; Subdi/erential

1. Introduction In this paper, we consider the coupled system of Cahn–Hilliard and Allen–Cahn equations in the following form, denoted as (CH/AC): wt − 9{−9w +  + f1 (w)} = 0 ut − 9u + + f2 (u) = 0 [; ] ∈ @R2 ˆ (w; u) @w =0 @n ∗

a:e: in Q := × (0; + ∞);

a:e: in Q;

a:e: in Q;

a:e: on  :=  × (0; + ∞);

Corresponding author. Fax: +48-22-554-0801. E-mail addresses: [email protected] (M. Gokieli), [email protected] (A. Ito).

0362-546X/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 2 ) 0 0 3 0 3 - 6

(1.1) (1.2) (1.3) (1.4)

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M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

@ {−9w +  + f1 (w)} = 0 @n @u =0 @n

a:e: on ;

w(0) = w0 ; u(0) = u0

a:e: on ;

(1.5) (1.6)

in :

(1.7)

Here, is a bounded domain in RN (1 6 N 6 3) with a smooth boundary  := @ ; fi (i = 1; 2) are smooth functions from R into itself; ˆ is a proper, non-negative, l.s.c. and convex function on R2 and @R2 ˆ is the subdi/erential of ˆ on R2 ;  and  are positive constants; w0 and u0 are the given initial data. This system has been introduced by Cahn and Novick–Cohen [2] as an extension of the classical Cahn–Hilliard model of phase separation in binary mixtures under quenching. The mixture can represent e.g. alloys, biological populations, two-component liquids; we refer to its components as A and B. In the original approach of Cahn and Hilliard, the phase separation phenomenon is understood as formation in the mixture of domains occupied by only one, or mostly one, component: A or B. Thus, the process (considered originally as isothermal) is described by the evolution of one variable, the relative concentration of one of the components, denoted here by w. Its dynamics is governed by the Cahn–Hilliard equation (1.1), derived in the framework of Ginzburg–Landau non-equilibrium thermodynamics. However, when dealing with alloys, crystallography distinguishes many very ordered structures, but consisting not of separated domains occupied by A or B exclusively, but of some regular distribution of atoms A and B on crystalline grids. Modeling such a phase transformation phenomenon, called ordering, requires introducing some micro-scale considerations into a macro-scale model. The function u that appears here, and Eq. (1.2), known as Allen–Cahn and governing the evolution of u, are a result of such a procedure (cf. [2,3]); u expresses some measure of order between the atoms (or other micro-components) A and B. Moreover,  ¿ 0 and  ¿ 0 are responsible for surface tension, or existence of a transition layer between domains that are being formated. The functions f1 and f2 represent the energies of interactions between the two components; − ˆ is the entropy in our system. It is also its main nonlinearity. As typical examples of the entropy ˆ , which are meaningful from the physical point of view, we give  (r + r2 ) log(r1 + r2 ) + (r1 − r2 ) log(r1 − r2 )    1    +(1 − r1 − r2 ) log(1 − r1 − r2 )       +(1 − r1 + r2 ) log(1 − r1 + r2 ) if (r1 ; r2 ) ∈ K; ˆ (r1 ; r2 ) := (1.8)  ˆ (q1 ; q2 )  lim if [r1 ; r2 ] ∈ @K;   [q1 ;q2 ]∈K    [q1 ;q2 ]→[r1 ;r2 ]      +∞ otherwise;

M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

or

 ˆ (r1 ; r2 ) :=

0

H if (r1 ; r2 ) ∈ K;

+∞

otherwise;

1823

(1.9)

where K := {(r1 ; r2 ) ∈ R2 ; 0 ¡ r1 + r2 ¡ 1; 0 ¡ r1 − r2 ¡ 1}, @K is the boundary of K and KH := K ∪ @K. The mathematical treatment of this model has started with the work of Brochet et al. [1]. They gave the existence–uniqueness result and, for the asymptotic stability as t → ∞, showed the existence of a global attractor and an inertial set (which is sometimes called exponential attractor). However, the mathematical setting did not contain any constraints, that is, they considered the case of ˆ ≡ 0. For the model with constraints, recently, in [3] Gokieli has proved the existence– uniqueness result as well as the large-time behavior for the case in which ˆ is given by (1.8). Especially, she proved that any element of the !-limit set of the initial data [w0 ; u0 ] is a solution of the steady-state problem associated with (CH/AC). The aim in this paper is to Jnd a class of ˆ which guarantees the existence, uniqueness of solutions of (CH/AC) and construct a global attractor for the dynamical system associated with (CH/AC). Our results are, in particular, applicable to both of the cited examples of ˆ . The plan of this paper is as follows: in the rest of this section, we rewrite (CH/AC) in a slightly di/erent and easier to handle form, and we introduce the principal spaces and operators. In Section 2, we state the main deJnitions, assumptions and results. Section 3 is devoted to the proof of the existence–uniqueness theorem. Actually, we show there that (CH/AC) can be written as an evolution equation governed by subdi/erential operator on some suitable Hilbert spaces, so as to use the results of [5] and deduce existence of a weak solution. Establishing this equivalence is the main part of Section 3 (Sections 3.2, 3.3). The obtained solution appears to be a limit of solutions of some regularized problems, which allows us to get more information about the regularity of the solution to (CH/AC). This will be used in Section 4, where we show the existence of a global attractor. We do not follow here the general method described by Hale [4] and Temam [8], because the semigroup generated by our system is continuous only in a very weak topology, which is not satisfactory from the point of view of physics: our objective is to get a compact attractor in the L2 -topology. At Jrst, let us note that the Jrst equation, by virtue of the boundary conditions (1.4), (1.5), is conservative:  d w(t) d x = 0 ∀t ¿ 0; dt i.e.,



 w(x; t) d x =



w0 (x) d x= : m0

∀t ¿ 0:

This means that in our system the global mass quantity is conserved in time.

(1.10)

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M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

So, let us introduce a new function v that we deJne by  1 w0 (x) d x (= : w − m0 ) v=w− | |

(1.11)

and rewrite system (1.1)–(1.7) into (1.12)–(1.18), which we will use afterwards: vt − 9{−9v +  + f1 (v + m0 )} = 0 ut − 9u + + f2 (u) = 0 [; ] ∈ @R2 ˆ (v + m0 ; u) @v =0 @n

a:e: in Q;

(1.14)

a:e: on  :=  × (0; + ∞);

(1.15) a:e: on ;

a:e: on ;

v(0) = v0 := w0 − m0 ;

(1.12) (1.13)

a:e: in Q;

@ {−9v +  + f1 (v + m0 )} = 0 @n @u =0 @n

a:e: in Q := × (0; + ∞);

(1.16) (1.17)

u(0) = u0

in :

Now, by (1.10) and (1.11)  v(x; t) d x = 0 ∀t ¿ 0:

(1.18)

(1.19)



From now on, we consider in our argumentation system (1.12)–(1.18) with the given data v0 , u0 and m0 instead of (1.1)–(1.7) with w0 and u0 . Notation. Throughout this paper we use the following notation. (1) L2 ( ) is the Hilbert space  with the usual inner product (· ; ·) and its subspace (L2 ( ))0 := {z ∈ L2 ( ); z d x=0} of L2 ( ) is also a Hilbert space with the inner product induced from L2 ( ), i.e., (· ; ·)0 := (· ; ·). Let 0 be the usual projection operator from L2 ( ) onto (L2 ( ))0 , i.e.,  1 ( 0 [z])(x) := z(x) − z(y) dy ∀z ∈ L2 ( ): | | (2) H 1 ( ) is the usual Sobolev space and V0 := H 1 ( ) ∩ (L2 ( ))0 is the Hilbert space with the inner product (z; v)V0 := (∇z; ∇v)L2 ( )

∀z; v ∈ V0 :

We denote by V0∗ the dual space of V0 . We also denote by · ; ·0 the duality pairing between V0∗ and V0 and by F0 the duality mapping from V0 onto V0∗ .

M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

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Here we note that V0∗ is the Hilbert space with the inner product (v; z)V0∗ := v; F0−1 z0 (=z; F0−1 v0 )

∀z; v ∈ V0∗ :

(3) Let X and Y be (real) Hilbert spaces with inner products (· ; ·)X and (· ; ·)Y , respectively. Then, the Cartesian product space X × Y is also a Hilbert space equipping the inner product (U1 ; U2 )X ×Y := (x1 ; x2 )X + (y1 ; y2 )Y

∀Ui := [xi ; yi ] ∈ X × Y (i = 1; 2):

We deJne the product Hilbert spaces W, V, H and F by W := (H 2 ( ) ∩ (L2 ( ))0 ) × H 2 ( ); H = (L2 ( ))0 × L2 ( );

V = V0 × H 1 ( );

F = V0∗ × L2 ( ):

Here, we note that W ,→,→ V ,→,→ H ,→ F: (4) −)N is the maximal monotone operator on (L2 ( ))0 with the domain  @z D(−)N ) = z ∈ (L2 ( ))0 ∩ H 2 ( ); = 0 in H 1=2 () : @n We note that if v ∈ D(−)N ) then −)N v = F0 v

in (L2 ( ))0 :

2. Main theorems Throughout this paper, we consider system (CH/AC) under the following assumptions. (A1) ˆ is a proper, non-negative, l.s.c. and convex function from R2 into R ∪ {+ ∞}. We denote by D( ˆ ) and @R2 ˆ the e/ective domain of ˆ and the subdi/erential of ˆ in R2 , respectively. Moreover, we suppose that D( ˆ ), the closure of D( ˆ ) in R2 , is compact and Int D( ˆ ), the interior of D( ˆ ) in R2 , is non-empty. (A2) fi (i = 1; 2) are Lipschitz continuous on [,i; 1 ; ,i; 2 ], where ,i; j (i; j = 1; 2) are deJned by ,i; 1 := inf {xi ; [x1 ; x2 ] ∈ D( ˆ )};

,i; 2 := sup{xi ; [x1 ; x2 ] ∈ D( ˆ )}

for each i = 1; 2. We Jx a primitive fˆ i of fi so that fˆ i ¿ 0 on [,i; 1 ; ,i; 2 ]. (A3)  ¿ 0,  ¿ 0 and m0 ∈ (,1; 1 ; ,1; 2 ) are constants and there exists m1 ∈ R such that (m0 ; m1 ) ∈ Int D( ˆ ). Remark 2.1. (i) It is easy to check by (A1) that there exist positive constants ci (i = 1; 2) such that ˆ is bounded from below by a quadratic function ˆ (r1 ; r2 ) ¿ c1 (|r1 |2 + |r2 |2 ) − c2

∀[r1 ; r2 ] ∈ R2 :

(2.1)

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M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

(ii) It follows from (A1) that all ,i; j (i; j =1; 2) are Jnite. Without loss of generality, we may therefore assume (A2) instead of (A2): (A2) fi are globally Lipschitz continuous on whole R with compact supports, fˆi ¿ 0 and bounded on R. We denote by L(fi ) the Lipschitz constants of fi (i = 1; 2). (iii) Let us recall here (1.19) and ,1; 1 6 v + m0 6 ,1; 2 a.e. in . In our model, the condition m0 ∈ [,1; 1 ; ,1; 2 ] is a compatibility condition under which (CH/AC) has at least one solution. Moreover, for the case when m0 = ,1; 1 (resp. m0 = ,1; 2 ) the component v(x; t) = 0 for any (x; t) ∈ × [0; + ∞), i.e., our problem is no longer a system. So, from now on, we only consider the case when m0 ∈ (,1; 1 ; ,1; 2 ). Before giving the deJnition of a solution to the Cauchy problem (CH/AC), let us introduce in H the functions / and 0, being, in terms of physics, the minus entropy and the convex part of the free energy of the system. They are associated with ˆ in the following way:   ˆ (y + m0 ; z) d x if U := [y; z] ∈ H     /(U ) := (2.2) with ˆ (y + m0 ; z) ∈ L1 ( );      +∞ otherwise;

0(U ) :=

 2 |y|V0 + 2 |∇z|2L2 ( )   2   

+ 12 |z|2L2 ( ) + /(U )

+∞

if U := [y; z] ∈ V with ˆ (y + m0 ; z) ∈ L1 ( );

(2.3)

otherwise:

Note here that due to (A1), / and 0 are proper, non-negative, l.s.c. and convex H functions on H. We denote by D(0), D(0) and @H 0 the e/ective domain of 0, the closure of D(0) in H and the subdi/erential of 0 on H, respectively. We give now the deJnition of solutions to the Cauchy problem (CH/AC). Denition 2.1. Let [v0 ; u0 ] ∈ H and m0 ∈ (,1; 1 ; ,1; 2 ) satisfy the assumption in (A3). Then, a pair [v; u] of functions v : R+ → (L2 ( ))0 and u : R+ → L2 ( ) is called a solution to (CH/AC) if the following properties hold: (w1) For each T with 0 ¡ T ¡ + ∞, 1 [v; u] ∈ C 0 ([0; T ]; F) ∩ Hloc ((0; T ]; F) ∩ L2 (0; T ; V) ∩ L2loc ((0; T ]; W):

(w2) There exist functions  : (0; + ∞) → (L2 ( ))0 and : (0; + ∞) → L2 ( ) such that [; ] ∈ L2loc ((0; + ∞); H); [(t); (t)] ∈ @H /(v(t); u(t))

(2.4) a:e: t ¿ 0;

(2.5)

M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841



F0−1



  d v(t) ; y − ()N v(t); y) + ((t) + dt 0

∀y ∈ (L2 ( ))0

0 [f1 (v(t)

and a:e: t ¿ 0:

1827

+ m0 )]; y)0 = 0 (2.6)

d (u(t); z) + (∇u(t); ∇z) + ( (t) + f2 (u(t)); z) = 0 dt ∀z ∈ H 1 ( )

and a:e: t ¿ 0:

(2.7)

(w3) v(0) = v0 and u(0) = u0 . Remark 2.2. Let us note that we use @H in our deJnition, on the contrary to (1.14), where we have [; ] ∈ @R2 ˆ . These two subdi/erentials are not equal, and our formulation weakens the notion of solution to (1.12)–(1.18). We are now in a position to formulate our main results. H

Theorem 2.1. Let [v0 ; u0 ] ∈ D(0) . Then, problem (CH/AC) has a unique solution. Thanks to Theorem 2.1, we can deJne a family of operators {S(t)} := {S(t); t ∈ R+ }, H representing a dynamical system associated with (CH/AC) on D(0) : for each t ¿ 0 H and [v0 ; u0 ] ∈ D(0) S(t)[v0 ; u0 ] := [v(t); u(t)], where [v; u] is the solution to (CH/AC). Since our system is autonomous, it is easy to check that {S(t)} satisJes the following semigroup properties: H (S1) S(0) = I on D(0) , H (S2) S(t + s) = S(t)S(s) on D(0) for any s; t ¿ 0. Theorem 2.2. There exists a subset A of H such that the following properties are ful=lled: (i) A is non-empty, compact in H and connected in F. (ii) A is invariant with respect to the semigroup {S(t)}, i.e., S(t)A = A; ∀t ¿ 0. H (iii) For each bounded subset B ⊂ D(0) of F, lim dist H (S(t)B; A) = 0;

t→+ ∞

(2.8)

where for any A; B ⊂ H  distH (A; B) := sup inf |a − b|H : a∈A

b∈B

Throughout this paper, we call the above set A the global attractor for the dynamical H system {S(t)} on D(0) . Let us underline that, as already stated in the introduction, these existence, uniqueness and asymptotic behavior results are valid for very di/erent

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M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

forms of the nonlinearity, or, in physical terms, of the entropy, embracing in particular the logarithmic and non-smooth forms. 3. Proof of Theorem 2.1 3.1. Uniqueness Uniqueness of solutions to (CH/AC) follows directly from the following proposition. H

Proposition 3.1. Let [v0i ; u0i ] ∈ D(0) and [vi ; ui ] be the solutions to (CH/AC) corresponding to the initial data [v0i ; u0i ], (i = 1; 2), respectively. Then, there exist positive constants :1 and :2 , independent of the initial data [v0i ; u0i ] (i = 1; 2), such that |v1 (t) − v2 (t)|2V0∗ + |u1 (t) − u2 (t)|2L2 ( ) 

+ :1

s

t

{|v1 (;) − v2 (;)|2V0 + |∇(u1 (;) − u2 (;))|2L2 ( ) } d;

6 e:2 (t−s) {|v1 (s) − v2 (s)|2V0∗ + |u1 (s) − u2 (s)|2L2 ( ) } ∀s; t 0 6 s 6 t ¡ + ∞:

(3.1)

Proof. We put v := v1 − v2 and u := u1 − u2 . Then, [v; u] satisJes



  d F0−1 v(t) ; y − ()N v(t); y)0 dt 0 + ( 0 [1 (t) − 2 (t) + f1 (v1 (t) + m0 ) − f1 (v2 (t) + m0 )]; y)0 = 0 ∀y ∈ (L2 ( ))0

and for a:e: t ¿ 0

(3.2)

and d (u(t); z) + (∇u(t); ∇z) + ( 1 (t) − 2 (t) + f2 (u1 (t)) − f2 (u2 (t)); z) = 0 dt ∀z ∈ H 1 ( )0

and a:e: t ¿ 0:

(3.3)

Now, we substitute y = v(t) ∈ V0 and z = u(t) ∈ H 1 ( ) for a.e. t ¿ 0 in (3.2) and (3.3), respectively (cf. (w1)). Then, we add these results to get for a.e. t ¿ 0 1 d {|v(t)|2V0∗ + |u(t)|2L2 ( ) } + |v(t)|2V0 + |∇u(t)|2L2 ( ) 2 dt  + {(1 (t) − 2 (t))v(t) + ( 1 (t) − 2 (t))u(t))} dt

= − (f1 (v1 (t) + m0 ) − f1 (v2 (t) + m0 ); v(t)) − (f2 (u1 (t)) − f2 (u2 (t)); u(t)):

M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

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Here, it follows from the monotonicity of @H / on H and by (2.4) that  {(1 (t) − 2 (t))v(t) + ( 1 (t) − 2 (t))u(t))} dt ¿ 0;

while (A2) implies that |(f1 (v1 (t) + m0 ) − f1 (v2 (t) + m0 ); v(t))| 6 L(f1 )|v(t)|2(L2 ( ))0 ; |(f2 (u1 (t)) − f2 (u2 (t)); u(t))| 6 L(f2 )|u(t)|2L2 ( ) : Furthermore, the following interpolation inequality holds, see [6, Lemma 5.1]: ∀< ¿ 0; ∃C(<) ¿ 0

s:t: |y|2(L2 ( ))0 6 <|y|2V0 + C(<)|y|2V0∗ ∀y ∈ V0 :

Consequently, we arrive at the bound d {|v(t)|2V0∗ + |u(t)|2L2 ( ) } + 2{ −
a:e: t ¿ 0:

We Jx < ¿ 0 so that  −
∀[y; z] ∈ H;

(3.4)

G(U ) := [ 0 [f1 (y + m0 )]; f2 (z) − min{max{z; ,2; 1 }; ,2; 2 }]; ∀U := [y; z] ∈ H:

(3.5)

The following properties of 0, which is deJned in (2.3), A and G, are straightforward conclusions from their deJnitions: (0) We see from (2.1) that 0(U ) ¿ c1 |U |2H − c2 | |

∀U ∈ H:

(3.6)

Moreover, there exists a positive constant c3 such that 0(U ) ¿ c3 |U |2V − c2 | |

∀U ∈ V:

(3.7)

Hence, we see that for any r ∈ R the level set {U ∈ H; 0(U ) 6 r} is compact in H (and not only closed). (A) A is a linear, continuous, symmetric and positive operator on H. So, we can consider the operator A1=2 , the square root of A, deJned by the identity (AU1 ; U2 )H = (A1=2 U1 ; A1=2 U2 )H

∀Ui ∈ H; i = 1; 2:

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M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

Moreover, it is easy to see that |A1=2 U |H = |U |F

∀U ∈ H:

(3.8)

(G) G is a Lipschitz continuous operator on H , its range R(G) is bounded in H. We deJne a functional Gˆ on H by   ˆ ) := fˆ 1 (y + m0 ) d x + G(U fˆ 2 (z) d x





−



g(z) ˆ dx

∀U := [y; z] ∈ H;

where gˆ : R → R is the primitive namely,  ,2;2 2    − ,2; 2 r +   2    2 r g(r) ˆ :=  2     2    r + ,2; 1 − ,2; 1 2

of the function g(r) := min{max{r; ,2; 1 }; ,2; 2 }, ∀r ¿ ,2; 2 ; ∀,2; 1 6 r 6 ,2; 2 ; ∀r 6 ,2; 1 :

Then Gˆ is non-negative, bounded and Gˆateau di/erentiable on H (cf. (A2) ). 1; 2 (R+ ; Moreover, the Gˆateau derivative ∇Gˆ coincides with G and, for each U ∈ Wloc H), ˆ ) ∈ W 1; 2 (R+ ); G(U loc

 d ˆ d G(U (t)) = G(U (t)); U (t) dt dt H

for a:e: t ¿ 0:

(3.9)

Remark 3.1. We note that G(U ) = [ 0 [f1 (y + m0 )]; f2 (z) − z]

H

∀U := [y; z] ∈ D(0) :

3.3. Existence of solution In this part, we show that (CH/AC) can be reformulated as an evolution equation governed by the subdi/erential @H 0 on H, namely, d AU (t) + @H 0(U (t)) + G(U (t))  0 dt U (0) = U0 := [v0 ; u0 ]:

in H a:e: t ¿ 0;

(3.10) (3.11)

We will be then able to apply the results of [5, Section 8] to get the existence of its solution. To state this equivalence, we pass through minimization of a convex functional deJned by J , and through approximating problems, where ˆ is regularized.

M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

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For each [y∗ ; z ∗ ] ∈ H, let us introduce a functional J deJned on H by J ([y; z]) := 0([y; z])–(y∗ ; y)0 − (z ∗ ; z)

∀[y; z] ∈ H:

It is easy to check that J is proper, l.s.c., convex and coercive on H as well as strictly convex on the e/ective domain D(0). So, we see that J has one and only one minimizer [y0 ; z0 ] in D(0). Of course, we then have [y∗ ; z ∗ ] ∈ @H 0(y0 ; z0 ): The next proposition, which is a generalization of [5, Proposition 6.1], characterizes the above relation in terms of a variational inequality and a regular elliptic problem. Proposition 3.2. The following statements (a) – (c) are equivalent to each other: (a) [y0 ; z0 ] is a minimizer of J , i.e., J ([y0 ; z0 ]) = min J ([y; z]): [y; z]∈H

(b) [y0 ; z0 ] ∈ V and (∇y0 ; ∇(y0 − y)) + (∇z0 ; ∇(z0 − z)) + (z0 ; z0 − z)  ˆ (y0 (x) + m0 ; z0 (x)) d x +

∗

∗

6 (y ; y0 − y)0 + (z ; z0 − z) +



ˆ (y(x) + m0 ; z(x)) d x

∀[y; z] ∈ V: ˆ (y + m0 ; z) ∈ L1 ( ): (c) [y0 ; z0 ] ∈ W and there exists [; ] ∈ H such that [; ] ∈ @H /([y0 ; z0 ]);

(3.12)

− 9y0 +  = y∗

(3.13)

a:e: in ;

− 9z0 + z0 + = z ∗

a:e: in ;

(3.14)

@y0 =0 @n

a:e: on ;

(3.15)

@z0 =0 @n

a:e: on :

(3.16)

It is a standard argument to prove (a) ↔ (b) and (c) → (b), hence, (c) → (a). (Note that the presence of “6” in (b) should not be surprising, it is related with the deJnition of the subdi/erential, or, in other words, results from the presence of the constraint). From uniqueness of the minimizer [y0 ; z0 ] of J , it is enough to show that system (3.12)–(3.16) has a solution in W. However, in order to prove this existence result, we have to consider some suitable approximate problems.

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M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

In order to do this, we approximate the function ˆ in the following way: for each < ∈ (0; 1] 2 ˆ • ˆ < is the Yosida  approximation of in R , ˜ ˆ • < (r1 ; r2 ) := R2 C< (z1 ; z2 ) < (r1 − z1 ; r2 − z2 ) d z1 d z2 ∀(r1 ; r2 ) ∈ R2 ,

where C< is a molliJer. Note that ˜ < is a C ∞ and convex function. Using this function ˜ < for each < ∈ (0; 1], we consider the following system (3.17)–(3.20) as an approximate problem for (3.12)–(3.16):   @ ˜< − )N y0< + 0 (y0< + m0 ; z0< ) = y∗ a:e: in ; (3.17) @r1 

@ ˜< − )N z0< + z0< + (y0< + m0 ; z0< ) = z ∗ a:e: in ; (3.18) @r2 @y0< =0 @n

a:e: on ;

(3.19)

@z0< =0 @n

a:e: on :

(3.20)

Then, we see that [y0< ; z0< ] is one and only one minimizer of the functional J< on H deJned by J< ([y; z]) :=

 2  1 |y| + |∇z|2L2 ( ) + |z|2L2 ( ) 2 V0 2 2  ˜ < (y + m0 ; z) d x − (y∗ ; y)0 − (z ∗ ; z) +

∀[y; z] ∈ H:

Next, we give some uniform estimates for {[y0< ; z0< ]}. Lemma 3.1. There are positive constants <0 ∈ (0; 1] and M0 such that for all < ∈ (0; <0 ]   

   @ ˜< @ ˜<   |[y0< ; z0< ]|W +  0 (y0< + m0 ; z0< ); (y0< + m0 ; z0< )  6 M0 :   @r1 @r2 H

Proof. By using (A3) and the fact that [y0< ; z0< ] is the minimizer of J< , we can see that there exist a number <0 ∈ (0; 1] and a positive constant M such that J< ([y0< ; z0< ]) 6 M

∀< ∈ (0; <0 ]:

Also, as ˜ < is bounded from below (cf. (2.1) and the deJnition of ˜ < ), this implies min{; ; 1} |[y0< ; z0< ]|2V − (y∗ ; y0< )0 − (z ∗ ; z0< ) 6 M + c2 | |; 2 where c2 is the same number as in (2.1).

M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

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Using the Schwarz and Young inequality for the last two terms, we arrive at min{; ; 1} |[y0< ; z0< ]|2V 6 M + c2 | | + C|[y∗ ; z ∗ ]|H ¡ + ∞ 4 for some suitable positive constant C. Now, in order to obtain boundedness in H 2 ( ), let us multiply (3.17) and (3.18) by [ − 9y0< ; −9z0< ]. Here, we use the following fact which is a direct consequence of the convexity of ˜ < (namely, of the positive deJniteness of its second derivatives matrix):   

 @ ˜< @ ˜< (y0< + m0 ; z0< ) ; −9y0< + (y0< + m0 ; z0< ); −9z0< ¿ 0: 0 @r1 @r2 0

It follows that   1 ∗2 1 ∗2 |9y0< |2(L2 ( ))0 + |9z0< |2L2 ( ) 6 |y |(L2 ( ))0 + |z |L2 ( ) ; 2 2 2 2 where again we used the Schwarz and Young inequalities. Hence, we got an estimate in W, and it also yields that      @ ˜< @ ˜< (y0< + m0 ; z0< ) ; (y0< + m0 ; z0< ) 0 @r1 @r2 0¡<6<0

is bounded in H. Proof of Proposition 3.2. Due to Lemma 3.1, we can choose a sequence {
in H and weakly in W;


weakly in H

as n → ∞, where /˜ < is associated with ˜ < by the same way as in (2.2) instead of . From the Mosco convergence of /˜
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M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841



(?)

For each U0 ∈ D(0)

H

the system {(3:9); (3:10)} has at least one

  solution U in the sense of DeJnition 2:1 such that    t 1=2 d (A1=2 U ) ∈ L2 (0; T ; H); t0(U ) ∈ L∞ ([0; T ]):  dt   In particular; if U ∈ D(0) then; 0  A1=2 U ∈ W 1; 2 (0; T ; H);

         

0(U ) ∈ L∞ ([0; T ]):

Explicitly, this implies d U ∈ L2loc ((0; T ]; F); dt

U ∈ L∞ (0; T ; V);

U (t) ∈ D( ˆ )

for a:e: t ∈ [0; T ]:

which, by (2.5) and the Lipschitz regularity of G, yields U ∈ L2loc ((0; T ]; W): Thus, U = (v; u) is the solution to (CH/AC) in the sense of DeJnition 2.1. In addition, we know that U is the limit of UF , solutions of approximate, regularizing problems (CH=AC)F := {(3:21); (3:22)} for F → 0 (see [5, Section 8]): d AF UF (t) + @H 0F (UF (t)) + G(UF (t)) = 0 dt

in H for a:e: t ¿ 0;

UF (0) = U0 := [v0 ; u0 ];

(3.21) (3.22)

where 0F is the Yosida approximation of 0 on H:  1 H 2 H 0F (U ) := inf |U − U |H + 0(U ) (¿ 0) UH ∈H 2F

∀U ∈ H;

G is the same as in (3.4) and AF is deJned, for each F ∈ (0; 1], by AF U := [(−)N )−1 y + Fy; (1 + F)z]

∀U := [y; z] ∈ H:

By the basic theory of nonlinear semigroups, it follows from (A2) and from the Lipschitz continuity of A−1 on H that for each F ∈ (0; 1] the approximate problem F (CH=AC)F has one and only one solution UF ∈ W 1; 2 (0; T ; H) with 0F (UF ) ∈ L∞ (0; T ) for any T with 0 ¡ T ¡ + ∞. Let us now give some uniform estimates on the solutions of the approximate problems. 3.4. Regularity of solutions to (CH/AC) Note at Jrst that property (0), stated in Section 3.2 above, yields an analogous lower bound on 0F : (0F ) There exists F0 ∈ (0; 1] such that c1 0F (U ) ¿ |U |2H − c2 | | ∀U ∈ H; ∀F ∈ (0; F0 ] (3.23) 2

M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

1835

and 0F (U ) ¿

c3 |U |2V − c2 | | 2

∀U ∈ V; ∀F ∈ (0; F0 ];

(3.24)

where ci (1 6 i 6 3) are the same numbers as in (2.1) and (3.7) in property (0). We can give now the uniform estimates. Lemma 3.2 (cf. Kenmochi [5, Lemma 8.1]). Let T be a =nite positive real number. Then, the following properties are satis=ed: (i) There exist positive constants R1 and R2 , such that |UF |2C([0; T ];F) + F|UF |2C([0; T ];H) + |UF |2L2 (0; T ;H) + |0F (UF )|L1 (0; T ) 6 R1 (|U0 |2F + F|U0 |2H ) + R2 (T + 1)

∀F ∈ (0; F0 ]:

(ii) There exists a positive constant R3 such that      dUF 2  dUF 2     + sup |0F (UF (t))| + F  dt  2  dt  2 t¿0 L (0; + ∞;F) L (0; + ∞;H) 6 R3 (|0(U0 )| + 1)

∀F ∈ (0; F0 ]:

(iii) There exist positive constants R4 and R5 such that      1=2 dUF 2  1=2 dUF 2   t t + F + sup {t|0F (UF (t))|}   dt L2 (0; T ;F) dt L2 (0; T ;H) 06t6T 6 R4 (|U0 |2F + F|U0 |2H ) + R5 (T + 1)

∀F ∈ (0; F0 ]:

Proof. (i) Let H0 ∈ D(0) be Jxed. We multiply (3.21) by UF (t) − H0 and use (3.23), the boundedness of R(G) and the deJnition of the subdi/erential @H 0F to get d {|UF (t) − H0 |2F + F|UF (t) − H0 |2H } + 0F (UF (t)) 6 :1 ; dt a:e: t ∈ [0; T ] and ∀F ∈ (0; F0 ]

(3.25)

for some positive constant :1 . Integrating (3.25) on the time interval [0; t] ⊂ [0; T ], we obtain  t 2 2 |UF (t) − H0 |F + F|UF (t) − H0 |H + 0F (UF (s)) ds 0

6 :1 T + |U0 − H0 |2F + F|U0 − H0 |2H ; ∀t ∈ [0; T ] and ∀F ∈ (0; F0 ]: By (3.23) again, (3.26) implies directly estimate (i).

(3.26)

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M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

(ii) We multiply (3.21) by (d=dt)UF (t) := UF (t) and use (3.9) to conclude |UF (t)|2F + F|UF (t)|2H +

d ˆ F (t))} = 0 {0F (UF (t)) + G(U dt

a:e: t ¿ 0:

(3.27)

We integrate (3.27) on [0; t] and obtain  t  t  2 |UF (s)|F ds + F |UF (s)|2H ds + 0F (UF (t)) 0

0

ˆ F (t)) 6 0(U0 ) + G(U ˆ 0) + G(U

∀t ¿ 0:

(3.28)

The last inequality, together with the non-negativity and boundedness of Gˆ and with (3.23), gives estimate (ii). (iii) We multiply (3.27) by t and integrate this resultant on [0; t](⊂ [0; T ]). Then it follows that  t  t  2 s|UF (s)|F ds + F s|UF (s)|2H ds + t0F (UF (t)) 0

0

ˆ F )|L1 (0; T ) 6 |0(UF )|L1 (0; T ) + |G(U

∀t ∈ [0; T ]:

(3.29)

Combining estimate (i) with (3.29) and using the boundedness of Gˆ again, gives us estimate (iii). As a consequence of Lemma 3.2, we obtain the following regularity properties of solutions U := [v; u] to (CH/AC). H

Proposition 3.3. Let U0 ∈ D(0) . Then there exist positive constants R6 and R7 such that 2   dU  + sup {t|U (t)|2V } |U |2C 0 ([0; T ];F) + |U |2L2 (0; T ;V) + t 1=2 dt L2 (0; T ;F) 06t6T 6 R6 |U0 |2F + R7 (T + 1): If U0 ∈ D(0) then additionally    dU 2   + sup {t|U (t)|2V } 6 R3 (|0(U0 )| + 1);  dt  2 t¿0 L (0; + ∞;F) where R3 is the same constant as in Lemma 3.2(ii).

4. Proof of Theorem 2.2—construction of the global attractor In this section, we construct a global attractor A for the dynamical system {S(t)} H on D(0) .

M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

1837

At Jrst, note that, in addition to properties (S1) and (S2) stated in Section 2, the semigroup S satisJes (S3) For each t ¿ 0, S(t) is continuous with respect to the strong topology in F. Indeed, (S3) is a direct consequence of Proposition 3.1. Next, we show the existence of a compact absorbing set in H, this being a key step in our construction of the global attractor. Lemma 4.1. Let H0 ∈ D(0) be the same element as in the proof of Lemma 3.2. Then there exist positive constants R8 and R9 such that H

0(U ) + (G(U ); U − H0 )H ¿ R8 |U |2H − R9

∀U ∈ D(0) :

Proof. We use (3.6) and the boundedness of R(G) to conclude 0(U ) + (G(U ); U − H0 )H ¿ c1 |U |2H − c2 | | − |G(U )|H |U |H − |G(U )|H |H0 |H

2  M1 c1 + M1 |H0 |H + c2 | | ; ¿ |U |2H − 2 2c1 where M1 := supU ∈H |G(U )|H . We put R8 := c1 =2 and R9 := M12 =(2c1 ) + M1 |H0 |H + c2 | |, the latter giving us the assertion. Lemma 4.2. There exists a bounded subset B0 in F such that  H for each bounded subset B(⊂ D(0) ) in F  (∗∗)   there exists a finite time t(B) ¿ 0 such that S(t)B ⊂ B0 ;

∀t ¿ t(B):

Proof. We Jx the same element H0 ∈ D(0) as in Lemma 4.1 and then multiply (3.21) by U (t) − H0 . It follows from the deJnition of @H 0 that  d 1 |U (t) − H0 |2F + 0(U (t)) − 0(H0 ) dt 2 +(G(U (t)); U (t) − H0 )H 6 0

a:e: t ¿ 0:

Using Lemma 4.1 and the continuity of the embedding of H in F, it follows that d |U (t) − H0 |2F + R10 |U (t) − H0 |2F 6 R11 dt

a:e: t ¿ 0

for some positive constants R10 and R11 . By the Gronwall lemma we derive the inequality |U (t) − H0 |2F 6 e−R10 t |U0 − H0 |2F +

R11 R10

∀t ¿ 0:

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M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

At last, we put  B0 :=



W ∈ F; |W |F 6

2R11 1 + 6|H0 |2F + R10

 H

∩ D(0) :

The set B0 is the desired absorbing set. Indeed, for each bounded subset B ⊂ D(0) such that supW ∈B |W |F 6 RB (for some RB ) we choose the Jnite time

H

t(B) := max{0; (2=R10 ) log(2RB )} and we see that S(t)B ⊂ B0 for any time t ¿ t(B). In the next lemma, we construct an absorbing set B1 which will be compact in H. H

Lemma 4.3. There exists a compact subset B1 (⊂ D(0) ) in H satisfying the same condition as in Lemma 4.2, (∗∗). Proof. Let B0 be the same absorbing set as in Lemma 4.2. We put H

B1 := S(1)B0 :

(4.1)

Then, it is easy to check that B1 is also absorbing for {S(t)}. The time step, Jxed to 1, could be of course any other non-zero Jnite positive constant. To complete the proof, we show the compactness of B1 in H. To do this, we use the estimate in Proposition 3.3. Since B0 is bounded in F, it follows that  sup {t 1=2 |v(t)|V0 } + sup {t 1=2 |u(t)|H 1 ( ) } 6 R12 := 2(R6 M22 + 3R7 ) 06t62

06t62

for any U0 := [v0 ; u0 ] ∈ B0 , where [v(t); u(t)] := S(t)U0 and M2 := supW ∈B0 |W |F , i.e., |v(1)|V0 + |u(1)|H 1 ( ) 6 R12

∀U0 ∈ B0 :

(4.2)

This implies that S(1)B0 is bounded in V, hence, B1 is compact in H. Remark 4.1. It follows from Proposition 3.3 and Lemma 4.3 that for each subset  H B(⊂ D(0) ) bounded in F, the set t¿ S(t)B is relatively compact in H for any  ¿ 0. We are now in a position to construct a global attractor for {S(t)}. Let us deJne the set A by A :=



H

S(t)B1 :

(4.3)

s¿0 t¿s

In other words, A is the !-limit set of B1 in H. We will show that A satisJes conditions (i) – (iii) in Theorem 2.2. First, we note that the following

M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

fact:

  U ∈ A if and only if  

1839

there exist sequences {tn } and {Un } ⊂ B1 such that

(4.4)

tn ↑ + ∞; S(tn )Un → U in H (n → ∞):  (i) Since it follows from Remark 4.1 that the set t¿s S(t)B1 is relatively compact in H for each s ¿ 1, we see that A is non-empty and compact in H. We show the connectivity of A in F by contradiction. Assume there exist two non-empty compact subsets A1 and A2 in F such that A1 ∪ A2 = A

A1 ∩ A2 = ∅:

(4.5)

Let Ui ∈ Ai , i = 1; 2, and choose the sequences {ti; n } with ti; n ↑ + ∞ and {Ui; n } ⊂ B1 so that in H;

S(ti; n )Ui; n ∈ B1 ; S(ti; n )Ui; n → Ui

sup {|S(ti; n )Ui; n |V } ¡ + ∞

n¿1

(4.6)

for i = 1; 2 due to Lemma 4.3 and (4.4). Without loss of generality, we may assume t1; n ¡ t2; n ¡ t1; (n+1) ; t2; n − t1; n ¿ n

∀n = 1; 2; : : : :

Now, we put U2; n := S(t2; n − t1; n )U2; n and since B1 is absorbing for {S(t)} we see that U2; n ∈ B1 for any n ¿ n0 , where n0 is a suRciently large number. Moreover, it follows from (4.6) and (S2) that S(t1; n )U2; n = S(t2; n )U2; n → U2

in H (n → ∞):

(4.7)

Next, we note that B1 is arc-wise connected in F because of the same property for B0 and the continuity of the operator S(1) with respect to the strong topology of F, resulting from (S3). Thus, there exists a function Ln : [0; 1] → H, continuous with respect to the strong topology F so that Ln (0) = U1; n ;

Ln (1) = U2; n Ln (t) ∈ B1

∀t ∈ [0; 1]:

(4.8)

This implies that for suRciently large n S(t1; n )Ln (t) ∈ B1

∀t ∈ [0; 1]:

(4.9)

Now take two neighborhoods Ui of Ai , i = 1; 2, in F so that U1 ∩ U2 = ∅. Since, by (4.6), (4.7), S(t1; n )U1; n ∈ U1 and S(t1; n )U2; n ∈ U2 for suRciently large n, there is a sequence {tn } ⊂ (0; 1) such that S(t1; n )Ln (tn ) ∈ U1 ∪ U2

(4.10)

for suRciently large n. On the other hand, it follows from (4.9) and from compactness of B1 in H that {S(t1; n )Ln (tn )} has at least one accumulation point U˜ in B1 with respect to the topology H, so F. Furthermore, again by (4.9) and by the deJnition

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M. Gokieli, A. Ito / Nonlinear Analysis 52 (2003) 1821 – 1841

of A (cf. (4.4)), this point belongs to A, which contradicts (4.5). Hence, A must be connected in F. (ii) Let U be any element in A. Then, there exist sequences {tn } and {Un } ⊂ B1 satisfying (4.4). For each t ¿ 0, it follows from (S3) that S(t + tn )Un = S(t)S(tn )Un → S(t)U

in F:

(4.11)

Furthermore, since {S(t + tn )Un } ⊂ B1 for any n ∈ N we may assume that S(t + tn )Un → U˜

in H (n → ∞)

(4.12)

for some U˜ ∈ B1 . From (4.11) and (4.12), we see that U˜ = S(t)U and it follows from (4.4) that S(t)U ∈ A. Hence, S(t)A ⊂ A. On the other hand, for each t ¿ 0, by putting Un := S(tn − t)Un for any n with tn ¿ t, we see from Lemma 4.3 that Un ∈ B1 for any n ¿ n1 and {Un } is relatively compact in H if n1 is suRciently large. Therefore, we may assume that Un → U˜

in H (n → ∞)

for some U˜ ∈ A. By the continuity property of S(t) we get S(t)Un = S(tn )Un → S(t)U˜

in F:

(4.13)

Combining (4.13) with (4.4), we get S(tn )Un → U = S(t)U˜

in H (n → ∞);

namely, U = S(t)U˜ ∈ S(t)A. Thus, A ⊂ S(t)A. (iii) Taking into account Lemma 4.3, it is enough to show (2.8) in the case of B = B1 . We suppose that this assertion is not true. Then, there would exist a positive constant <0 , sequences {tn } with tn ↑ + ∞ and {Un } ⊂ B1 such that |S(tn )Un − U |H ¿ <0 ;

∀U ∈ A ∀n = 1; 2; : : : :

(4.14)

Since {S(tn )Un } is relatively compact in H for a suRciently large n2 , there must exist a subsequence {nk } and an element UH ∈ A such that S(tnk )Unk → UH in H as k → ∞. This contradicts (4.14), and ends the proof of (iii). We have thus proved all the required properties of the attractor: compactness, connectivity, invariance, and attractivity. References [1] D. Brochet, D. Hilhorst, A. Novick-Cohen, Inertial sets for Cahn–Hilliard/Cahn–Allen systems, Appl. Math. Lett. 7 (1994) 83–87. [2] J.W. Cahn, A. Novick-Cohen, Evolution equations for phase separation and ordering in binary alloys, J. Statist. Phys. 79 (1994) 877–909. [3] M. Gokieli, Asymptotic behaviour for di/usive problems, Ph.D. Thesis of UniversitTe de Franche-ComtTe, Besan con, 2002. [4] J.K. Hale, Asymptotic Behavior of Dissipative Systems, in: Mathematical Surveys and Monographs, Vol. 25, American Mathematical Society, Providence, RI, 1988.

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[5] N. Kenmochi, Systems of nonlinear PDEs arising from dynamical phase transitions, in: A. Visintin (Ed.), Phase Transitions and Hysteresis, Lecture Notes in Mathematics, Vol. 1584, Springer, Berlin, 1994, pp. 39–86. [6] J.L. Lions, Quelques mTethodes de rTesolution des problTemes aux limites nonlinTeaires, Dunod, Gauthier-Villars, Paris, 1969. [7] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. Math. 3 (1969) 510–585. [8] R. Temam, InJnite Dimensional Systems in Mechanics and Physics, in: Applied Mathematical Sciences, Vol. 68, Springer, Berlin, 1998.