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Asymptotic Dynamics of a New Mechanochemical Model in Biological Patterns.

1 Introduction

In this paper, we study the following mechanochemical model

[partial derivative][phi]/[partial derivative]t = [DELTA][phi]4 + [alpha][phi] - [[phi].sup.3] - [epsilon][phi][psi], (1.1)

[partial derivative][psi]/[partial derivative]t = -[lambda][([DELTA] + 1).sup.2] [psi] - [gamma][psi] + g[[psi].sup.2] - [[psi].sup.3] - [[epsilon]/2 [phi.sup.2] (1.2)

for (x, t) [member of] [OMEGA] x (0, [infinity]), where [OMEGA] [subset] [R.sup.n](n [less than or equal to] 3) is a bounded domain with smooth boundary, and all the parameters are arbitrarily given positive constants.

It was Morales, Rojas, Torres and Rubio [10] who first derived the system (1.1)-(1.2), which is to model ternary mixtures by using the theory of pattern formation and of polyelectrolytes, with mean-field approximations. Recently, Morales, Rojas, Oliveros and Hernandez derived a new mechanochemical model based on Coupled Ginzburg-Landau and Swift-Hohenberg equations in biological patterns of marine animals [9]. The new model was proposed to describe some cellular interactions in three out layers of animal (such as the fish) marine skin. [phi](x, t) represents the concentration difference of at least two pigment, and [psi](x, t) is the difference of dermal cellular densities of at least two types of cells. The [mathematical expression not reproducible] is the flux of pigment concentration difference [phi] in the epidermis, and [mathematical expression not reproducible] is the density of cell flux [psi] of the dermis. The system (1.1)-(1.2) is supplemented by the zero flux boundary condition,

[mathematical expression not reproducible],

where v is the outward unit normal to [partial derivative][OMEGA], that is

[partial derivative][phi]/[partial derivative]v = 0, 2[lambda] [partial derivative][psi]/[partial derivative]v + [lambda] [partial derivative][DELTA][psi]/[partial derivative]f = 0, x [member of] [partial derivative][OMEGA] (b1)

and the natural boundary condition

[partial derivative][psi]/[partial derivative]v = 0, x [member of] [partial derivative][OMEGA]. (b2)

It follows from (b2) that (b1) can be replaced by

[partial derivative][DELTA][psi]/[partial derivative]v (x, t) = 0, x [member of] [partial derivative][OMEGA].

Hence, we consider the Neumann boundary conditions

[partial derivative][phi]/[partial derivative]v (x, t) = [partial derivative][DELTA][psi]/[partial derivative]v (x, t) = 0, t > 0, x [member of] [partial derivative][OMEGA] (1.3)

and the initial condition

[phi](x, 0) = [[phi].sub.0], [psi](x, 0) = [[psi].sub.0], x [member of] [OMEGA]. (1.4)

The dynamic properties of the reaction-diffusion system (1.1)-(1.2), such as the global asymptotical behaviors of solutions and existence of global attractors are important. During the past years, many authors had paid much attention to the higher order equation ( [1,4,6,7,16]) or the reaction-diffusion systems ([2,3,11]). You ([17,18,19]) had proved the existence of global attractor for some Gray-Scott type systems. The main difficulties for treating the problem (1.1)-(1.2) are caused by the nonlinearity of low order terms, and linear higher order terms are not homogeneous. The source type nonlinear low terms and Neumann boundary conditions can not make us use Poincare type inequality directly, thanks to strong absorptive terms -[[phi].sup.3] and -[[phi].sup.3], which guarantees the existence of a global solution and will not blow up.

The paper is arranged as follows. In Section 2, some notations and the main results are stated. We present some estimates in Section 3, and then we prove that problem (1.1)-(1.4) possesses global attractors on [L.sup.2]([OMEGA]) x [H.sup.2]([OMEGA]) in Section 4. Based on this result, we prove the existence of global attractors for problem (1.1)-(1.4) in [H.sup.k] (k [greater than or equal to] 0) space in Section 5.

2 Statement of main results

We first introduce the following abbreviations.

The notation (*, *) for [L.sup.2]-inner product will also be used for the notation of duality pairing between dual spaces, [mathematical expression not reproducible]. We use the same letter C to denote different positive constants, and C(*, *, *) to denote positive constants depending on the quantities appearing in the parenthesis.

Theorem 1. For any positive parameters [alpha], [lambda], [gamma], g, [epsilon], any [([[phi].sub.0], [[psi].sub.0]).sup.T] [member of] [L.sup.2]([OMEGA]) x [H.sup.2]([OMEGA]), and n [less than or equal to] 3, there exists a global attractor A in the phase space [L.sup.2]([OMEGA]) x [H.sup.2]([OMEGA]) for the solution semiflow [{S(t)}.sub.t [greater than or equal to] 0] on [L.sup.2]([OMEGA]) x [H.sup.2]([OMEGA]) generated by system (1.1)-(1.2) with the Neumann boundary conditions (1.3).

The basic theory of infinite dimensional dynamical systems and global attractors can be seen in [11,15] and references therein. A few definitions are listed for clarity.

Definition 1. Let [{S(t)}.sub.t [greater than or equal to] 0] be a semiflow on a real Banach space X. A bounded subset [B.sub.0] of X is called an absorbing set in X for this semiflow, if for any bounded subset B [subset] X there is some finite time [t.sub.0] [greater than or equal to] 0 depending on B such that S(t)B [subset] [B.sub.0] for all t [greater than or equal to] [t.sub.0].

Definition 2. Let [{S(t)}.sub.t [greater than or equal to] 0] be a semiflow on a real Banach space X whose norm-induced metric is denoted by d(*, *). A subset A of X is called a global attractor for this semiflow, if the following properties are satisfied:

(H1) A is a nonempty, compact, invariant set in the sense that S(t)A=A for any t [greater than or equal to] 0.

(H2) A attracts any bounded set B of X with respect to the Hausdorff distance,

[mathematical expression not reproducible].

Definition 3. A semiflow on a real Banach space X is asymptotically compact if for any bounded sequence [u.sub.n] in X and any sequence [t.sub.n] [subset] (0, [infinity]) with [t.sub.n] [right arrow] [infinity], there exist subsequences [mathematical expression not reproducible] of [u.sub.n] and [mathematical expression not reproducible] of [t.sub.n], such that [mathematical expression not reproducible] exists in X.

Lemma 1. ([15]) Let [{S(t)}.sub.t [greater than or equal to] 0] be a semiflow on a real Banach space X. If the following properties are satisfied:

(1) there exists a bounded absorbing set [B.sub.0] [subset] X for [{S(t)}.sub.t [greater than or equal to] 0],

(2) [{S(t)}.sub.t [greater than or equal to] 0] is asymptotically compact on X, then there exists a global attractor A for [{S(t)}.sub.t [greater than or equal to] 0] in X, which is given by

[mathematical expression not reproducible]. (2.1)

Let us write (1.1)-(1.2) as an evolution problem

[mathematical expression not reproducible],

where u = [([phi], [psi]).sup.T], H := [L.sup.2]([OMEGA]) x [L.sup.2]([OMEGA]),

[mathematical expression not reproducible],

and

[mathematical expression not reproducible].

F(u) and the operator A is considered on the Hilbert space [L.sup.2] with dense domain

D(A) = {u [member of] [H.sup.2]([OMEGA]) x [H.sup.4]([OMEGA]) : [partial derivative][phi]/[partial derivative]v = [partial derivative][DELTA][psi]/[partial derivative]v = [partial derivative][psi]/[partial derivative]v = 0, on [partial derivative][OMEGA]},

D([A.sup.1/2]) = {u [member of] [H.sup.1]([OMEGA]) x [H.sup.2]([OMEGA]) : [partial derivative][phi]/[partial derivative]v = [partial derivative][psi]/[partial derivative]v = 0, on [partial derivative][OMEGA]}.

Let

[mathematical expression not reproducible].

In order to prove the existence of solutions, we shall show A is sectorial, F(u) is locally Lipschitz continuous as the operation between the space D([A.sup.1/2]) and [L.sup.2] x [L.sup.2], denoting by <*, *> the scalar product in [L.sup.2] and

[mathematical expression not reproducible].

where [[delta].sub.0] > 0 is taken sufficient large. For any u = [([phi], [psi]).sup.T], v = [([bar.[phi]], [[bar.[psi]]).sup.T], {u, v} [member of] D(A), noticing [GAMMA] is symmetric we find that

[mathematical expression not reproducible],

which proves the symmetry of A. Next, since [GAMMA] is positive definite, the operator A is bounded below, that is, for each u [member of] D(A),

<Tu, u> = [[integral].sub.[OMEGA]] (-[DELTA][phi][phi] + [lambda][[DELTA].sup.2] [psi][psi]) dx + [[integral].sub.[OMEGA]] [[delta].sub.0][u.sup.T] [GAMMA]udx

= [[integral].sub.[OMEGA]] ([([nabla][phi]).sup.2] + [lambda][([DELTA][psi]).sup.2])dx + [[integral].sub.[OMEGA]] [[delta].sub.0][u.sup.T] [GAMMA]udx [greater than or equal to] [[delta].sub.0] <u, u>.

A is self-adjoint and bounded below, which means that A is itself sectorial. We prove the local Lipschitz continuity of nonlinear function F(u)

[mathematical expression not reproducible].

Notice [H.sup.2] [??] [L.sup.[infinity]] and [H.sup.1] [??] [L.sup.6] for n [less than or equal to] 3.

We find by differentiation that, for [for all]u, v [member of] U [subset] D ([A.sup.1/2])

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

The general theory of ([5]) guarantees the existence of a local solution for system (1.1)-(1.2). The priori estimate Lemma 2-Lemma 5 implying the local solution can be a global one for t. Moreover, the family of operators S[(t).sub.t[greater than or equal to] 0] forms a strongly continuous semigroup on the space D([A.sup.1/2]), which has the property

u [member of] C([0, [T.sub.max]); [H.sup.1] x [H.sup.2]) [intersection] [C.sup.1]((0, [T.sub.max]); [H.sup.1] x [H.sup.2]) [intersection] [L.sup.2]([0, [T.sub.max]); [L.sup.2] x [L.sup.2]).

3 Absorbing sets

Lemma 2. For any given R > 0 there exists a constant [M.sub.1](R) > 0 such that if the initial data [u.sub.0] = [([[phi].sub.0], [[psi].sub.0]).sup.T] [member of] H and [[parallel][u.sub.0][parallel].sup.2.sub.H] [less than or equal to] R, then S(t)[u.sub.0] = [([phi], [psi]).sup.T] [member of] H for all t [greater than or equal to] 0, and

[[parallel]S(t)[u.sub.0][parallel].sup.2.sub.H] [less than or equal to] [M.sub.1](R), for t [greater than or equal to] 0.

Proof. Taking the inner products <(1.1),2[phi]>, and <(1.2),2[psi]>, then summing up the resulting equalities and by the Neumann boundary conditions, we get

[mathematical expression not reproducible],

that is

d/dt ([[parallel][phi][parallel].sup.2] + [[parallel][psi][parallel].sup.2]) + [kappa] ([[parallel][phi][parallel].sup.2] + [[parallel][psi][parallel].sup.2]) + 2 [[parallel][nabla][phi][parallel].sup.2] + 2[lambda] [integral] [[([DELTA] + 1)[psi]].sup.2]dx

[less than or equal to] 2[LAMBDA]([phi], [psi]), (3.1)

where

[mathematical expression not reproducible].

The Young inequalities yield

[mathematical expression not reproducible],

where [[epsilon].sub.1], [[epsilon].sub.2], [[epsilon].sub.3] are arbitrarily positive constants. Hence, we have

[mathematical expression not reproducible].

We take [[epsilon].sub.1] = 1/2, [[epsilon].sub.2] = 1/2g, [[epsilon].sub.3] = 1/(9[[epsilon].sup.2]/8+[kappa]/2) and 0 < [kappa] [less than or equal to] 2, then

[LAMBDA]([phi], [psi]) [less than or equal to] ([kappa]/4[[epsilon].sub.1] + g[[epsilon].sup.-3.sub.2] + (9[[epsilon].sup.2]/8 + [kappa]/2) 1/2[[epsilon].sub.3]) [absolute value of [OMEGA]] [equivalent to] 1/2 [C.sup.*] ([kappa], g, [epsilon], [absolute value of [OMEGA]]). (3.2)

From (3.1) and (3.2), we obtain

d/dt ([[parallel][phi][parallel].sup.2] + [[parallel][psi][parallel].sup.2]) + [kappa] ([[parallel][phi][parallel].sup.2] + [[parallel][psi][parallel].sup.2]) [less than or equal to] [C.sup.*] ([kappa], g, [epsilon], [absolute value of [OMEGA]]), for t [member of] [0, [T.sub.max]),

then, applying the Gronwall inequality, we deduce that

([[parallel][phi][parallel].sup.2] + [[parallel][psi][parallel].sup.2]) [less than or equal to] [e.sup.-[kappa]t] ([[parallel][[phi].sub.0][parallel].sup.2] + [[parallel][[psi].sub.0][parallel].sup.2]) + [C.sup.*]/[kappa], for t [member of] [0, [T.sub.max]).

Let [M.sub.1] be the constant [M.sub.1] = [C.sup.*]/[kappa] + 1. The proof is completed.

Lemma 3. For any given R > 0 there exists a constant [M.sub.2](R) > 0 such that if the initial data [u.sub.0] = [([[phi].sub.0], [[psi].sub.0]).sup.T] [member of] E and [[parallel][u.sub.0][parallel].sup.2.sub.E] [less than or equal to] R, then S(t)[u.sub.0] = [([phi], [psi]).sup.T] [member of] E for all t [greater than or equal to] 0, and

[[parallel]S(t)[u.sub.0][parallel].sup.2.sub.E] [less than or equal to] [M.sub.2](R), for t [greater than or equal to] 0.

Proof. Take the inner products <(1.1), -2[DELTA][phi]>, and ((1.2), -2[DELTA][psi]). Then sum up the resulting equalities

[mathematical expression not reproducible],

that is

d/dt ([[parallel][nabla][phi][parallel].sup.2] + [[parallel][nabla][psi][parallel].sup.2]) + [[kappa].sup.2] ([[parallel][nabla][phi][parallel].sup.2] + [[parallel][nabla][psi][parallel].sup.2]) [equivalent to] [[LAMBDA].sub.2] ([phi],[psi]), (3.3)

where

[mathematical expression not reproducible].

Then, applying the Nirenberg inequality (n [less than or equal to] 3), we deduce that

[mathematical expression not reproducible].

Using the Young inequality and Lemma 2, we have

[mathematical expression not reproducible]

and

([[kappa].sub.2] + [[epsilon].sup.2]/6 + 2/3 [g.sup.2] - 2[gamma] - 2[lambda]) [[parallel][nabla][psi][parallel].sup.2] [less than or equal to] [lambda]/4 [[parallel][nabla][DELTA][psi][parallel].sup.2] + C ([[kappa].sub.2], [epsilon], g, [gamma], [lambda]).

Summing up the resulting equalities, we have

[[LAMBDA].sub.2]([phi], [psi]) [less than or equal to] [[integral].sub.[OMEGA]] [([square root of 6][phi][nabla][phi] + [epsilon][nabla][psi]/[square root of 6]).sup.2] dx - [[integral].sup.[OMEGA]] [([square root of 6][psi][nabla][psi] [square root of 6]/3 g[nabla][psi]).sup.2] dx

- [[parallel][DELTA][phi][parallel].sup.2] - [lambda] [[parallel][nabla][DELTA][psi][parallel].sup.2] + [C.sup.*.sub.2] ([alpha], [gamma], g, [lambda], [[kappa].sub.2], [epsilon], [absolute value of [OMEGA]]. (3.4)

From (3.3) and (3.4), we find that

d/dt ([[parallel][nabla][phi][parallel].sup.2] + [[parallel][nabla][psi][parallel].sup.2]) + [[kappa].sub.2] ([[parallel][nabla][phi][parallel].sup.2] + [[parallel][nabla][psi][parallel].sup.2]) [less than or equal to] [C.sup.*.sub.2] ([alpha], [gamma], g, [lambda], [[kappa].sub.2], [epsilon], [absolute value of [OMEGA]]. (3.4)

then, applying the Gronwall inequality, we deduce that

[mathematical expression not reproducible].

Let [M.sub.2] be the constant [M.sub.2] = [C.sup.*.sub.2]/[[kappa].sub.2] + 1. The proof is completed.

Lemma 4. For any given R > 0 there exists a constant [M.sub.3](R) > 0 such that if the initial data [u.sub.0] = [([[phi].sub.0], [[psi].sub.0]).sup.T] [member of] [E.sub.1] and [mathematical expression not reproducible], then S(t)[u.sub.0] = ([phi], [psi]).sup.T] [member of] [E.sub.1] for all t [greater than or equal to] 0, and

[[parallel]S(t)[u.sub.0][parallel].sup.2.sub.E] [less than or equal to] [M.sub.3](R), for t [greater than or equal to] 0.

Proof. Taking the inner products <(1.1),2[phi]>, and ((1.2),2[[DELTA].sup.2][psi]), by the Neumann boundary conditions, we get

[mathematical expression not reproducible].

Then summing up the resulting equalities, we see that

[mathematical expression not reproducible]. (3.5)

By the Nirenberg inequalities, we get

[mathematical expression not reproducible].

Using the Young inequalities, Lemma 2 and above inequalities, we obtain

[mathematical expression not reproducible] (3.6)

and

[mathematical expression not reproducible].

Similarly, we have

[mathematical expression not reproducible]. (3.7)

From (3.5) and (3.6)-(3.7), we obtain

[[LAMBDA].sub.3]([phi], [psi]) [less than or equal to] - [lambda]/4 [[parallel][[DELTA].sup.2][psi][parallel].sup.2] - 1/2 [[parallel][nabla][phi][parallel].sup.2] + [C.sup.*.sub.3] ([gamma], [epsilon], [lambda], [alpha], [[kappa].sub.3], [absolute value of [OMEGA]])

[less than or equal to] [C.sup.*.sub.3] ([gamma], [epsilon], [lambda], [alpha], [[kappa].sub.3], [absolute value of [OMEGA]]),

that is

d/dt ([[parallel][phi][parallel].sup.2] + [[parallel][DELTA][psi][parallel].sup.2] + [[kappa].sub.3] ([[parallel][phi][parallel].sup.2] + [[parallel][DELTA][psi][parallel].sup.2]) [less than or equal to] [C.sup.*.sub.3]

Applying the Gronwall inequality, we deduce that

[mathematical expression not reproducible].

Taking [M.sub.3] = [C.sup.*.sub.3]/[[kappa].sub.3] + 1, the proof is completed.

4 Asymptotic compactness

Lemma 5. For any given R > 0 there exists a constant [M.sub.4](R) > 0 such that if the initial data [u.sub.0] = [([[phi].sub.0], [[psi].sub.0]).sup.T] [member of] [E.sub.2] and [mathematical expression not reproducible], then S(t)[u.sub.0] = [([phi], [psi]).sup.T], [member of] [E.sub.2] for all t [greater than or equal to] 0, and

[mathematical expression not reproducible].

Proof. Take the inner products <(1.1),-2[DELTA][phi]>), and <[DELTA](1.2),-2[[DELTA].sup.2][psi]>). By the Neumann boundary conditions, we get

[mathematical expression not reproducible].

Then summing up the resulting equalities, we see that

[mathematical expression not reproducible]. (4.1)

Using the Nirenberg inequalities (n [less than or equal to] 3), we obtain

[mathematical expression not reproducible].

From Lemma 3, and noticing [H.sup.1]([OMEGA]) [??] [L.sup.4]([OMEGA]) for n [less than or equal to] 3, we deduce that

[mathematical expression not reproducible]. (4.2)

Similarly, we obtain

[mathematical expression not reproducible]. (4.3)

From (4.1) and (4.2)-(4.3), we obtain

[[LAMBDA].sub.4]([phi], [psi]) [less than or equal to] - 3/4 [[parallel][DELTA][phi][parallel].sup.2] - 3/4 [[parallel][nabla][[DELTA].sup.2][phi][parallel].sup.2] + [C.sup.*.sub.4] ([gamma], [epsilon], [lambda], [alpha], [[kappa].sub.3], [absolute value of [OMEGA]])

that is

d/dt ([[parallel][nabla][phi][parallel].sup.2] + [[parallel][nabla][DELTA][psi][parallel].sup.2]) + [[kappa].sub.4]

([[parallel][nabla][phi][parallel].sup.2] + [[parallel][nabla][DELTA][psi][parallel].sup.2]) [less than or equal to] [C.sup.*.sub.4]

Applying the Gronwall inequality, we deduce that

[mathematical expression not reproducible]

Taking [M.sub.4] = [C.sup.*.sub.4]/[[kappa].sub.4] + 1, the proof is completed.

We now finish the proof of Theorem 1.

Proof. [Proof of Theorem 1] First, by Lemma 4, the solution semiflow S[(t).sub.t [greater than or equal to] 0] of reaction-diffusion system (1.1)-(1.2) has a bounded absorbing set [B.sub.0] in [E.sub.1]. Second, according to Lemma 5 and due to that Sobolev imbedding [E.sub.2] [??] [E.sub.1] is compact, this solution semiflow [S(t).sub.t[greater than or equal to] 0] is asymptotically compact in [E.sub.1], then by Lemma 1, there exists a global attractor A for S[(t).sub.t[greater than or equal to] 0 in [E.sub.1], which is given by (2.1).

5 The [H.sup.k] global attractor

In order to consider the global attractor for the system (1.1)-(1.2) in [H.sup.k] space, we introduce the definition as follows:

H = [L.sup.2]([OMEGA]), [H.sub.1/2] = {u [member of] [H.sup.2]([OMEGA]), [partial derivative]u/[partial derivative]n [|.sub.[partial derivative][OMEGA]] = 0},

[H.sub.1] = {u [member of] [H.sup.4]([OMEGA]), [partial derivative]u/[partial derivative]n [|.sub.[partial derivative]u [OMEGA]] = 0}.

In this paper, we used to assume that the linear operator

L = -[lambda][[DELTA].sup.2] : [H.sub.1] [right arrow] H

is a sectorial operator, which generates an analytic semigroup [e.sup.tL], and L induces the fractional power operators and fractional order spaces as follows

[L.sup.[alpha]] = [(-L).sup.[alpha] : [H.sub.[alpha]] [right arrow] H, (i = 1, 2), [alpha] [member of] R,

where [H.sub.[alpha]] = D ([L.sup.[alpha]]) is the domain of [L.sup.[alpha]]. By the semigroup theory of linear operators, [H.sub.[beta]] [subset] [H.sub.[alpha]] is a compact inclusion for any [beta] > [alpha]. For details of the space [H.sub.[alpha]] see [8].

Then, we have the following lemma on the existence of global attractor which is equivalent to Lemma 1 and the proof is similar to [12,13,14].

Lemma 6. Assume that ([phi](t), [psi](t)) = S(t)([[phi].sub.0], [[psi].sub.0]) (([[phi].sub.0], [[psi].sub.0])) [member of] H x H),t [greater than or equal to] 0) is a solution of (1.1) and S(t) the semigroup generated by (1.1). Assume further that [H.sub.[alpha]] is the fractional order space generated by L and

(1) For some [alpha] [greater than or equal to] 0, there is a bounded set B [subset] [H.sub.[alpha]+3/4] x [H.sub.[alpha]+3/4], which means that for any ([[phi].sub.0], [[psi].sub.0]) [member of] [H.sub.[alpha]+1/4] x [H.sub.[alpha]+3/4], there exists [t.sub.0] [greater than or equal to] 0 such that

([phi](t), [psi](t)) [member of] B, [for all]t > [t.sub.0];

(2) There is a [beta] > [alpha], such that for any bounded, set U [subset] [H.sub.[beta]+1/4] x [H.sub.[beta]+3/4], there are T > 0 and C > 0,

[mathematical expression not reproducible].

Then (1.1) has a global attractor A [subset] [H.sub.[alpha]+1/4] x [H.sub.[alpha]+3/4] which attracts any bounded set of [H.sub.[alpha]+1/4] x [H.sub.[alpha]+3/4] in the [H.sub.[alpha]+1/4] x [H.sub.[alpha]+3/4] norm.

For sectorial operators, we also have the following lemma which is important for this paper and can be founded in [12, 13, 14].

Lemma 7. Assume that L is a sectorial operator which generates an analytic semigroup T(t) = [e.sup.tL]. If all eigenvalues [lambda] of L satisfy Re[lambda] < -[[lambda].sub.0] for some real number [[lambda].sub.0] > 0, then for [L.sup.[alpha]] (L = -L) we have

(1) T(t) : H [right arrow] [H.sub.[alpha]] is bounded for all [alpha] [member of] R and t > 0;

(2) T(t) [L.sup.[alpha]]x = LT(t)x, [for all]x [member of] [H.sub.[alpha]];

(3) For each t > 0, [L.sup.[alpha]]T(t) : H [right arrow] H is bounded, and

[parallel][L.sup.[alpha]]T(t)[parallel] [less than or equal to] [C.sub.[alpha]][t.sup.-[alpha]][e.sup.-[delta]t],

where some [delta] > 0 and [C.sub.[alpha]] > 0 is a constant depending only on [alpha];

(4) The [H.sub.[alpha]-norm can be defined by [mathematical expression not reproducible].

The main result of this paper is given by the following theorem, which provides the existence of global attractors of Eq.(1.1) in any kth space [H.sup.k].

Theorem 2. Assume that [OMEGA] denotes an open bounded domain in [R.sup.3], then for any k [greater than or equal to] 0, the initial-boundary value problem (1.1)-(1.2) has a global attractor A in [H.sup.k] x [H.sup.k+2], and A attracts any bounded subset of [H.sup.k] x [H.sup.k+2] in the [H.sup.k] x [H.sup.k+2]-norm.

By Lemma 6, in order to prove Theorem 2, we first prove the following lemma.

Lemma 8. For any [sigma] [greater than or equal to] 0, the solution ([phi], [psi]) of (1.1)-(1.2) is uniformly bounded in [H.sub.[sigma]+1/4] x [H.sub.sigma]+3/4], i.e. for any bounded set U [subset] [H.sub.[sigma]+1/4] x [H.sub.[sigma]+3/4], there exists C > 0 such that

[mathematical expression not reproducible].

Proof. For any ([[phi].sub.0], [[psi].sub.0]) [member of] [L.sup.2]([OMEGA]) x [H.sup.2]([OMEGA]), the solutions ([phi], [psi]) of (1.1)-(1.2) can be expressed as

[mathematical expression not reproducible],

where [L.sub.1] = [DELTA], [L.sub.2] = -[lambda][[DELTA].sup.2]. By Theorem 1, there exists attractor in the phase space [L.sup.2] x [H.sup.2]. Using the same way, it is not difficult to prove that there exists attractor in [H.sup.1] x [H.sup.3], which means ([phi], [psi]) [member of] [H.sub.1/4] x [H.sub.3/4].

Step1. We shall prove that for any bounded set

U [subset] [H.sub.[sigma]+1/4] x [H.sub.[sigma]+3/4] (0 [less than or equal to] [sigma] < 1/4), there exists C > 0 such that

[mathematical expression not reproducible]. (5.1)

We claim that [F.sub.2] : [H.sub.1/4] x [H.sub.1/2] [right arrow] H is bound. Based on Lemma 5 and embedding theorem [H.sup.1] [??] [L.sup.6] for n [less than or equal to] 3, we have

[mathematical expression not reproducible],

where C depends on [lambda], [gamma], g, [epsilon], [absolute value of [OMEGA]] but independent of [[phi].sub.0] and [[psi].sub.0]. Hence, we obtain

[mathematical expression not reproducible], (5.2)

where [beta] = [sigma] + 3/4 and 0 [less than or equal to] [beta] < 1.

Similarly, we claim that [F.sub.1] : [H.sub.1/4] x [H.sub.1/4] [right arrow] H is bound. Based on Lemma 5 and embedding theorem [H.sup.1] [??] [L.sup.6], [H.sup.1] [??] [L.sup.4] for n [less than or equal to] 3, we have

[mathematical expression not reproducible],

where C depends on [alpha], [epsilon], [absolute value of [OMEGA]] but independent of [[phi].sub.0] and [[psi].sub.0]. Hence,

[mathematical expression not reproducible], (5.3)

where 0 [less than or equal to] 2[sigma] + 1/2 < 1. From (5.2)-(5.3), then (5.1) is proved.

For [sigma] = 1/8, we have [phi] [subset] [H.sub.3/8], this meaning [phi] [subset] [H.sup.3/2] ([OMEGA]) (for fractional order Sobolev space). By the embedding theorems of fractional order spaces, we deduce that

[phi](t.x) [subset] [H.sup.3/2] [??] [C.sup.0]([OMEGA]) [intersection] [H.sup.1]([OMEGA]). (5.4)

Step2. We shall prove that for any bounded set U [subset] [H.sub.[sigma]+1/4] x [H.sub.[sigma]+3/4] (1/4 [less than or equal to] [sigma] < 1/2), there exists C > 0 such that

[mathematical expression not reproducible]. (5.5)

We claim that [F.sub.2] : [H.sub.1/4] x [H.sub.3/4] [right arrow] [H.sub.1/4] is bound. Based on Lemma 5 and embedding theorem [H.sup.3] [??] [L.sup.[infinity]] for n [less than or equal to] 3, we have

[mathematical expression not reproducible],

where we used (5.4).

[mathematical expression not reproducible], (5.6)

where 0 [less than or equal to] [sigma] + 1/2 < 1.

Similarly, we claim that [F.sub.1] : [H.sub.1/4] x [H.sub.1/4] [right arrow] [H.sub.1/4] is bound.

[mathematical expression not reproducible],

where C depends on [alpha], [epsilon], [absolute value of [OMEGA]] but independent of [[phi].sub.0] and [[psi].sub.0]. Therefore

[mathematical expression not reproducible], (5.7)

where 0 [less than or equal to] 2[sigma] < 1. From (5.6)-(5.7), then (5.5) is proved.

Lemma 9. For any [sigma] > 0, (1.1)-(1.2) has a bounded absorbing set in [H.sub.[sigma]+1/4] x [H.sub.[sigma]+3/4]. That is, for any bounded set U [subset] [H.sub.[sigma]+1/4] x [H.sub.[sigma]+3/4] there are T > 0 and a constant C > 0 independent of [[phi].sub.0] and [[psi].sub.0], such that

[mathematical expression not reproducible]. (5.8)

Proof. Step1. We shall show that for any 0 [less than or equal to] [sigma] < 1/4, (1.1)-(1.2) has a bounded absorbing set in [H.sub.[sigma]+1/4] x [H.sub.[sigma]+3/4]. The solution ([phi], [psi]) can be expressed as

[mathematical expression not reproducible].

On the other hand, note that

[mathematical expression not reproducible],

where [[lambda].sub.1] > 0 is the first eigenvalue of the equation

-[DELTA]u = [lambda]u, u[|.sub.[partial derivative][OMEGA]] = 0.

By assertion (1) of lemma 7, for any given T > 0 and 0 [less than or equal to] [sigma] < 1/4, we have

[mathematical expression not reproducible].

Using assertion (3) of lemma 7, we have

[mathematical expression not reproducible],

where [beta] = [sigma] + 3/4, 0 [less than or equal to] [beta] < 1, C > 0 is a constant independent of [[psi].sub.0]. Similarly, we have

[mathematical expression not reproducible],

where 0 [less than or equal to] 2[sigma] + 1/2 < 1, C > 0 is a constant independent of [[phi].sub.0].

Step2. We shall show that for any 1/4 [less than or equal to] [sigma] < 1/2, (1.1)-(1.2) has a bounded absorbing set in [H.sub.[sigma]] x [H.sub.[sigma]+3/4].

[mathematical expression not reproducible],

where C > 0 is a constant independent of [[psi].sub.0].

[mathematical expression not reproducible].

by iteration, we can obtain (5.8).

Proof. [Proof of Theorem 2.] By Lemma 8 and Lemma 9, we immediately conclude the proof of Theorem 2 is completed.

Conclusions

Based on the regularity estimates for the semigroups and the classical existence theorem of global attractors, we prove that the system possesses a global attractor in the space [H.sub.k+1/4] x [H.sub.kk+3/4]. Comparing this paper with [12,13,14]. The system (1.1)-(1.2) is a two-component model. We define the product Hilbert spaces, using the Lumer-Phillips theorem and the generation theorem for analytic semigroups. The main difficulties for treating the problem (1.1)-(1.2) are caused by the nonlinearity of low order terms, and linear higher order terms are not homogeneous. The existence of the attractor in [H.sup.k] x [H.sup.k+2], guarantee a solution of the model equations for any value of the control parameters. This explains the following: 1) the solutions are robust and not sensitive to changes in the value of its control parameters and 2) the diversity of patterns that explain different biological systems (pigmentation vertebrate) and inhere animals (membranes porous medium and ternary mixtures with surfactants).

https://doi.org/10.3846/13926292.2017.1292324

Acknowledgement

The authors would like to thanks the referees' valuable suggestions for the revision and improvement of the manuscript. This work is supported by the Jilin Scientific and Technological Development Program (No. 20170101143JC).

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Aibo Liu (a) and Changchun Liu (b)

(a) School of Mathematics, Liaoning Normal University 116029 Dalian, China

(b) School of Mathematics, Jilin University 130012 Changchun, China E-mail(corresp.): liucc@jlu.edu.cn

Received September 14, 2016; revised January 21, 2017; published online March 1, 2017
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