Dynamical system for BCF model describing crystal surface growth Текст научной статьи по специальности «Математика»

DYNAMICAL SYSTEM FOR BCF MODEL DESCRIBING CRYSTAL SURFACE GROWTH1

This paper treats the initial-boundary value problem for a nonlinear parabolic equation of forth order which was presented by Johnson — Orme — Hunt — Graff — Sudijono — Sauder — Orr [1] in order to describe the interesting phenomena of crystal surface growth under molecular beam epitaxy (MBE). First we construct unique local solutions in a suitable function space by applying the techniques of abstract parabolic evolution equations. Second we establish a priori estimates to obtain the global existence of solutions. Our goal is then to construct a dynamical system determined from the initial-boundary value problem of the model equation.

Ключевые слова: Dynamical system, BCF model, semilinear parabolic equation, a priori estimate.

Johnson — Orme — Hunt — Graff — Sudijono — Sauder — Orr [1] in order to describe the process growing of crystal surface by a mathematical model. Here, u = u(x,t) denotes a displacement of height of surface from the standard level at a position x £ Q.

By physical experiments one can observe very interesting phenomena of crystal growth on the growing surface under the molecular beam epitaxy (MBE), see [2]. To understand their mechanisms Johnson et al. [1] presented the model given in (1) on the basis of the BCF theory due to Burton — Cabrera — Frank [3] (cf. also [4 — 7]).

The term -aA2u in the equation of (1) denotes a surface diffusion which is caused by the difference of the chemical potential proportional to the curvature of the surface. Therefore the adatoms have tendency to migrate from the positions of a large curvature to those of a small one. The macroscopic representation of the surface diffusion by -aA2u is due to Mullins [8], where a > 0 is a surface diffusion constant.

In the meantime, — jiV ■ denotes the effect of surface roughening. Such

Introduction

We study the initial-boundary value problem for a nonlinear parabolic equation of fourth order

<

on

in

(1)

in a two-dimensional bounded domain Q C R2. Such a problem was presented by

roughening is caused by Schwoebel barriers [9; 10] (cf. also [2]) which prevent adatoms

1rThis work is supported by Grant-in-Aid for Scientific Research (No. 16340046) of the Japan Society for the Promotion of Science.

from hopping from the upper terraces to lower ones. As a consequence, non-equilibrium uphill current is induced. The macroscopic representation of the roughening by —^V • (i+|Vm|2) f°rmulated in the paper Johnson et al. [1], where fj, > 0 is a coefficient of surface roughening. Some numerical simulations for one or two-dimensional model of (1) were performed by the papers [4 — 7].

This paper is devoted to studying (1) by mathematical analysis. Our goal is in fact to construct global solutions and furthermore a dynamical system determined from (1). First we shall show the local existence and uniqueness of solutions for initial functions uo £ H*(Q) by using the theory of abstract parabolic evolution equations (see [11; 12]). More precisely, we will apply the result due to [13] for semilinear abstract parabolic evolution equations. Second we shall obtain a priori estimates concerning the H:-norm for local solutions to show the global existence. Owing to the techniques of abstract evolution equations, one can easily verify continuous dependence of the global solutions with respect to the initial functions. This shows that a continuous semigroup S(t) is determined from the global solutions of (1) in the L2-norm. We shall then be able to construct a dynamical system (see [14; 15]) in universal space L2(Q) the phase space of which is H*(Q).

There may be several possibilities for choosing boundary conditions of u on dQ. In the present paper we will take the homogeneous Neumann type boundary conditions. Since the equation is of forth order, we have to impose the Neumann conditions to Au, too. These boundary conditions imply that, if fQ u0(x)dx = 0, then fQ u(x, t)dx = 0 for every 0 < t < to, i. e., the total mean of displacements is invariant in time. It is however possible to prove similar analytical results even for other types of boundary conditions like the homogeneous Dirichlet boundary conditions, periodic boundary conditions and so on.

Throughout the paper, Q is a bounded domain of C4 class in R2. According to

[16], the Poisson problem — Au = f in Q under the homogeneous Neumann boundary conditions = 0 on dQ enjoys the shift property that if / £ H2(Q), then u £ H4(Q).

1. Preliminary

We shall first recall the known results on semilinear evolution equations studied in [13]. Consider the initial value problem

^ + .4 u = F(u), 0 <t<T, (2)

u(0) = u0

in a Banach space X. Here, A is a closed linear operator of X, the spectral set of

which is contained in a sectorial domain £ = (A £ C; | arg A| < u} with some angle

0 < uj < and the resolvent satisfies the estimate

M

|| (A - Ay'W^x) < (3)

with some constant M > 0. Therefore, —A generates an analytic semigroup e-tA on X. U0 is an initial value in D(Aa) with the estimate

l|A“uo||< R, (4)

here a is some exponent such that 0 < a < 1 and R > 0 is a constant. F(•) is a nonlinear mapping from D(An) to X with a < n < 1 and is assumed to satisfy a Lipschitz condition of the form

IIF(u) — F(v)||< ^(||A“u|| + ||Aav||)x

x [||An(u — v)|| + (||Anu|| + ||Anv||)||Aa(u — v)||], u, v £ D(An), (5)

where <^(-) is some increasing continuous function. Then the following theorem is known.

Theorem 1 [13, Theorem 3.1]. Let 0 < a < n < 1 and let (3), (4) and (5) be satisfied. Then (1) possesses a unique local solution in the function space:

u £ C([0, Tr]; D(Aa)) nC 1((0, Tr]; X) nC((0,Tr]; D(A)), t1-au £ B((0, Tr]; D(A)),

where Tr > 0 being determined by R. Moreover, the estimate

t1-“||Au(t)|| + tn-a||Anu(t)|| + ||A“u(t)|| < Cr, 0 <t < Tr,

holds with some constant Cr > 0 determined by R alone.

We shall next list well-known results in the theory of function spaces and of linear operators. Let Q be a bounded C4 domain in R2. For 0 < s < 4, Hs(Q) denotes the Sobolev space of order s, its norm being denoted by || • ||Hs (see [16, Chap. 1] and

[17]). For 0 < s0 < s < s1 < 4, Hs(Q) coincides with the complex interpolation space [hso(Q), HS1 (Q)]e, where s = (1 — 9)s0 + 9s1, and the estimate

! • ||hs < c! • |Ihso II • IIhsi (6)

holds. When 0 < s < 1, HS(Q) C LP(Q), where ^ with continuous embedding

II • ||lp < c! • ||hs• (7)

When s = 1, H 1(Q) C Lq(Q) for any finite 2 < q < to with the estimate

1 _£ £

■ \\u < C|| ' llH1*ll ' llw. (8)

where 1 < p < q < to. When s > 1, HS(Q) C C(H) with continuous embedding

II • ||c < c! • IIhs• (9)

Consider a sesquilinear form given by

a{u,v) = d Vu ■ Vv dx + c uvdx, u, v G H1 (Q)

J Q J Q

on the space H 1(Q), where d > 0 and c > 0 are positive constants. From this form we can define realization A of the Laplace operator — dA + c in L2(Q) under the homogeneous Neumann boundary conditions on dQ (see [18, Chap. VI]). The

realization A > c is a positive definite self-adjoint operator of L2(Q) and its domain is characterized by

V(A) = H2N(Q) = {ue dfn = 0 on (10)

For 0 < 9 < 1, the fractional powers Ad of A are defined and are also positive definite self-adjoint operators of L2(Q). As shown in [19], we can characterize for 0 < 9 < 1, their domains in the form

mA‘>)=lH2"<S1)- when 0 < # < |,

V ' = {ue H2d{Q)- g = 0 on <9Q}, when § < 0 < 1. ' 1

In addition, it is verified that the following estimates

■ || l2 < || • \\h29 < Ce\\Ae ■ ||L2, O<0< 1, 9 ^ | (12)

hold with some constants Cq > 1.

We remark that, even when 9 = |, it is true that T>(Ai) C H2(H) continuously. We shall finally consider realization of —dA in L2(Q) under the homogeneous Neumann boundary conditions. The operator —dA is a nonnegative self-adjoint operator of L2(Q) with the same domain D(—dA) = HN(Q) as A. Clearly, the constants functions are an eigenfunction of the eigenvalue 0 of — dA. Consider the orthogonal complement of the space of constant functions, namely,

L2m(Q) = (u £ L2(Q); m(u) = 0},

where m(u) be the integral mean

m{u) = 7—- [ u(x)dx, u<eL2(Q). (13)

|Q|

Then —dA is a self-adjoint operator of Lmm(Q) with domain HNN(Q) n Lmm(Q). On account of the Poincare — Wirtinger inequality

||u — m(u)|L2 < C||Vu|L2, u £ H 1(Q)

(cf. [20, p. 194]), we verify that

(—dAu,u) = d||Vu|||2 > ^HullL, u £ HN(Q) n L2m(Q),

with some 5 > 0. This means that — dA is positive definite in L^(Q) with the estimate ||— dAu||L2 > 5||u||L2, u £ H2n(Q) n L2m(Q). (14)

2. Local solutions

We shall construct local solution to our problem (1) by handling it as an abstract equation of the form (2). The underlying space X is set as X = L2(Q).

The linear operator A is defined by A = A2, where A is the realization of

— \J~aA + 1 in L2(Q) under the homogeneous Neumann boundary conditions, i. e.,

d = y/a, c = 1. Clearly, A > 1 is also a positive definite self-adjoint operator of X. Consequently, A is a sectorial operator of X. In addition, we can verify the following properties.

Proposition 1. For 0 < 9 < 1,9 = 3/8, 7/8, we have (V(A0) = H4d(Q), when 0 < 9 < §,

< V{Ae) = = {uG H4e(Q); g* = 0 on <9Q}, when I <9 < |, (15)

\v{A0) = H%{n) = {«G H48(Qy, | = £Au = 0 on 1<0<1.

Moreover,

Dsl\\Ae • ||L2 < || • \\H4s < De\\Ae • ||L2, 0 < 9 < 1, 9^ §, f, (16)

with some constants Dq > 1.

Proof. Since A = yl2, it holds that Ae = A20 for 0 < 9 < 1. Hence, if 0 < 9 < (15)

and (16) are verified directly from (11) and (12), respectively.

Consider the case when 9 =1. By definition, u £ D(A2) if and only if u £ D(A) and Au G V(A)] so, Au G H2(Q) with |^ = 0 on dfl. Then, the shift property implies that u £ H4(Q). Thus, (15) and (16) are verified for the particular case when 9 = 1.

It remains to consider the intermediate case when | < 9 < 1, because (15) and (16) are already verified for 9 = 1. But they can be shown to hold by the same

interpolation techniques as in [19]. So, the detailed proof will be omitted. □

3 3

We remark that, even when 9 = 3/8, 7/8, it is true that V{As) C H^{0) and

7 7

V{As) C Ft2(H), respectively, with continuous embedding.

Fix two exponents a and rj in such a way that a = | and rj = |. In view of

—A = —{—y/aA + l)2 = —aA2 + 2\faA — 1,

the nonlinear operator F is defined by

{ \/ ?/ \ 7 7

F(u) = —jjX/ ■ I ----- ) — 2\J~aAu + u, u G V{A%) c H2(Q).

\1 + |Vu|2/

Then, we verify the following Lipschitz condition on F.

Proposition 2. The operator F satisfies

\\F(u)-F(v)\\<C[\\A1Hu-v)\\ +

7 7 1 7

+ (11^4.81/11 + — w)||], u,v<eV(Az). (17)

Proof. Clearly it suffices to consider the term /iV • ( 1+|^(|2 ] only.

We note that

V

f Vu \ f Vv

Vi + |v«|2y ~~ ' vi + |Vw|2

V-

- V

V(tt — v)

1 + |Vm|2_

' (\Vu\2 - \Vv\2)Vv

' (1 + |V«|2)(1 + |Vw|2)

I + II.

Furthermore,

A(u — v) V(u — v) ■ V|Vu|2

I = -------------77---------77---------7777^--- = 1 1 + i2-

1 + |Vu|2 (1 + |Vu|2)2

We observe that V|Vu|2 = 2 ( Re

d2u du d2u du

+

, Re

dx\ dx1 dx1 dx2 dx2

Therefore, on account of (16),

\\h\\ < ||A(u - v)\\L2 < C\\A*(u - v

and

d2u du d2u du

+

dx1 dx2 dx1 dx2 dx2

(18)

21| < C||V(u — v)|L21|(1 + |Vu|2)-2V|Vu|2||L» <

d2u

dxidxj

< C\\u - v\\Hi\\u\\ 7 < C\\A*(u - v)II\\AsU\

h 2

(due to (9)).

In the meantime, we see that

(|Vu|2 — |Vv|2)Av V(|Vu|2 — |Vv|2) -Vv

-/-/ —---“----:-—i .------:--—TXT — ~-----:--—T^TTT----:-—TXrH'

(1 + |Vu|2)(1 + |Vv|2) (1 + |Vu|2)(1 + |Vv|2)

(|Vu|2 — |Vv|2)[(1 + |Vv|2)V|Vu|2 + (1 + |Vu|2)V|Vv|2] ■ Vv

+

(1 + |Vu|2)2(1 + |Vv|2)2

II1 + II2 + IIs.

On account of (16) it is easy to estimate II1 as

№11 <C'HV(^ - v)\\L2\\Av\\L™ <C\\u - v\\Hi\\v\\H7<C\\Ai(u - w)|| 11^8^1 Similarly, using (18), we obtain that

IIII2II < C

|u — v|h2 + ||u — v|h 1 E (

d2u

dxidxj

+

d2v

dxidxj

<

I I 7 7

< C[\\A2(u - u)|| + ||A*(u - w)||(||A^|| + ||A^|

The similar arguments are available for II3, too. As a result,

1 7 7

\\Ih\\<C\\AHu-v)\\(\\Asu\\ + \\Asv\

We have thus proved the proposition. □

As is obvious, (17) means that F fulfils (5) with a = 1/4 and n = 7/8. Theorem

1 then provides the following local existence of solution.

Theorem 2. For any Uo G V(A^) = H1(Q), there exists a unique solution to (1) in the function space:

fu £C([0,To]; H 1(Q)) nC1 ((0, To]; L2(Q)) nC((0,T„]; H4N 2 (Q)), \tlueB((0,To};H4m.

Here, T0 > 0 is determined by the norm ||u0||H 1 alone. Moreover,

3 5

ti\\u(t)\\H4+ts\\u(t)\\H7 + \\u(t)\\Hi < Co, 0<t<To, (19)

C0 > 0 being determined by ||u0||H 1 alone.

3. Global solutions

We shall establish a priori estimates for the local solutions.

Let u0 £ H 1(Q) and let u be any local solution of (1) on interval [0,Tu] in the solution space:

u £ C ([0, Tu ]; H 1(Q)) nC1 ((0,Tu]; L2(Q)) nC ((0,Tu]; HN 2 (Q)). (20)

Proposition 3. There exists a constant C > 0 independent of u0 such that the estimate

|u(t)|H1 < C(|u0|H1 + 1), 0 < t < Tu, (21)

holds for any local solution u in the space (20).

Proof. Integrate the equation of (1) in Q. Then, by the boundary conditions of (1), we observe that

d [ d [ du 1

— udx=—a -—Audx — 11 —-——-- dx = 0.

dtjn Jdndn ^ Jdndn l + \Vu\2

Hence,

m(u(t)) = m(u0), 0 < t < Tu, (22)

where m(-) is given by (13).

Multiply next the equation of (1) by u and integrate the product in Q. After integration by parts using again the boundary conditions, we have

1 d f , l2 , f , . l2 , f |Vu|2

\u\ dx = —a \Au\ dx + fi dx.

2 dt J q Jq Jq 1 + |Vu|

Therefore,

[ \u\2dx + a [ \Au\2dx < n\Q\.

QQ

Since

llu(t)|L2 = llu(t) — m(u(t))|L2 + Hm(u(t))HL2 ,

it follows from (22) that

~~r f \u — m(u)\2dx + a [\Au\2dx < /i\Q\.

2 dt Q Q

Meanwhile, by (14), there exists a constant p > 0 such that

a

p\\u - m(u)W2L2 < -\\Au\\2l2, ueH2(Q).

Therefore,

~~T f \u ~ m(u)\2dx + p f \u — m(u)\2dx + — f \Au\2dx < /i\Q\. (23)

2 dt Q Q 2 Q

Regarding this as a differential inequality of ||u(t) — m(u(t))|||2, we deduce that

\\u(t) - m(u(t))\\2L2 < e~2pt\\u0 - m(u0)\\2L2 + 0 < t < Tu. (24)

P

Hence, on account of (22),

\\u(t)\\2L2<\n\m(u0)2 + e-2pt\\u0\\2L2 + ^, 0 <t<Tu. (25)

P

Multiply now the equation of (1) by Au and integrate the product in Q. Then, by some calculations,

1 d f l2 f o f Vu ■ VAu

Vtt dx = —a VAu\ dx — /i ------- dx <

2 dt JQ JQ JQ 1 + |Vu|

<—a | VAu\2dx + n \VAu\dx.

QQ

As n fQ \VAu\ dx < a JQ \VAu\2dx + we conclude that

is lrft’

Meanwhile, since we have

a

p\\Vu\\2L2<-\\Au\\2L2 + CM\h> ueH2(Q),

with a suitable constant Cp > 0, it is seen that

f |V«|2cir + P [ \Vu\2dx < ^||Z\tt||^2 + Cp\\u\\2L2 +

2 dt Q Q 2 a

Summing up this and (23), we obtain that

1 d

' (|u — m(u)|2 + |Vu|2)dx + p (|u — m(u)|2 + |Vu|2)dx <

QQ

< Cp\\u(t)\\2L2 + (1 + f)/^|n|.

Therefore, solving this, we deduce that

||V^(t)|||2 < e-2pt\\u0\\2Hi + [* e~2p{t-s) [2Cp\\u(s)\\2L2 + 2(1 + ^)ti\n\]ds.

0

Due to (25) it finally follows that

\\Vu(t)\\h — + ~p^j e

CP|Q| / \2 i L . CP , V \ ^|Q|

+ ^^m(w0)2+ 1 + ^ + - —, 0 <t<Tu. (26)

p \ p a J p

This together with (25) yields the desired estimate. □

The estimates (24) and (26) show the following result.

Corollary 1. If an initial function u0 £ H 1(Q) satisfies m(u0) = 0, then any local solution of (1) also satisfies m(u(t)) = 0 for every 0 < t < Tu. Furthermore there exist an exponent p > 0 and a constant Cp > 0 which are independent of u0 such that

||u(t)||H 1 < Cp[e-pt||u0||H 1 + 1], 0 < t < Tu. (27)

As an immediate consequence of a priori estimates, we can prove the global existence of solution.

Theorem 3. Let u0 £ H 1(Q). Then, (1) possesses a unique global solution in the function space:

u £ C([0, to); H 1(Q)) nC1((0, to); L2(Q)) nC((0, to); H^(Q)). (28)

Proof. By Theorem 2, there exists a unique local solution u on an interval [0,T0]. Moreover, by Proposition 3, ||u(T0)||H1 is estimated by ||u0||H 1 alone. This then shows that the solution u can be extended as a local solution on an interval [0, T0 + t], where t > 0 is determined by ||u(T0)||H 1, and hence depends only on ||u0||H1. Repeating this procedure, we obtain the result. □

By Proposition 3 we clearly verify that the global solution also satisfies the estimate

||u(t)||H 1 < C(11u01|h + 1), 0 < t< to, (29)

where C > 0 is the same constant as in (21).

Moreover we can extend the estimate (19) to the global solutions.

Proposition 4. There exist increasing functions p (■) such that, for any global solution with initial function u0 £ HD(Q), it holds that

3

\\u(t)\\H4 < (1 + t~^)p(\\u0\\Hi), 0 <t<oo, (30)

\\u(t)\\Hr < (l + t~^)p(\\u0\\Hi), 0 <t<oo. (31)

Proof. We notice that (29) is valid. In view of this fact, let 0 < s < to and let us apply

(19) with initial time s and initial function u(s). Then, there exists t = t(||u0||H 1) > 0, t being decreasing function of the norm ||u0||H 1, for which we have

3

\\u(t)\\H4 <(t- s)~ip(\\u(s)\\Hi), S <t< S + T.

When 0 < t < t, taking s = 0, we have

3

\\u(t)\\H4 < t-4p(\\u0\\Hi), 0 < t < T.

When r < t < 00, taking s = t — we obtain that

_ 3 3

\\u(t)\\H4 < (I) 4 p(\\u(t - l)\\Hi) < T~ip(\\u0\\Hi), T<t<oo.

Hence, (30) is proved.

It is the same for (31). □

We will conclude this section by verifying the Lipschitz continuity of solutions with respect to initial functions. Let B be a closed ball of initial functions

B = {u0 £ H 1(Q); ||u,||H1 < R}

with arbitrarily fixed radius R > 0. By Theorem 3, there exists a unique global

solution to (1) for each u0 £ B.

Proposition 5. Let u (resp. v) be the solution to (1) with initial function u0 £ B

(resp. v0 £ B). Then, for each T > 0 fixed, there exists some constant CR,T > 0

depending on R and T alone such that

7 1

t*\\u(t) - v(t)\\Hr + ti\\u(t) - v(t)\\Hi +

+ ||u(t) — v(t)|L2 < CR,T|u0 — V0IU2, 0 < t < T. (32)

Proof. For 0 < 9 < 1, we have the formula

Ad [u(t) — v(t)] = Ad e-tA (u0 — v0) +

W Ade-(t-s)A[F(u(s)) — F(v(s))]ds, 0 <t < T.

0

From (17) and (31), it follows that for 0 < t < T t$ |A^ [u(t) — v(t)] || < Ae |u0 — v0 || +

+ CRAete [ (t-S)-0{||^[w(S)-i;(S)]||+S-l||^[w(S)-i;(S)]||}^, (33) 0

where Ag = sup te | A0e tA ||. Then, we put the function

0<t<T

p (t) = t? || [u{t) — v{t)] || + t± || A^ [u{t) — v{t)] ||, 0 < t < T.

Using (33) with 0 = § and we easily obtain that for 0 < t < T

p(t) < C\\u0 — v0\\ + C f [t^(t — s)~^ +t^(t — s)~^][s~^ + s~^]p(s)ds. (34)

0

We can verify from (34) the desired estimate. Firstly, let 0 < t < e. It follows

that

r t 1 11 11 7

pit) < C\\u0 — Wo || + C [t^ (t — s)~^ + t^{t — s)“3][S“l -f sup p (s) <

J0 0<S<£

< C\\u0 — Woll + Ces sup p(s).

0<s<£

Therefore, taking e > 0 sufficiently small, we conclude that

sup p (t) < C||u0 — v0||.

0<t<£

Secondly, let e < t < T. On account of this estimate, (34) can be written in the form

pit) < c(l + [t*(t-s)~* + t^(t- s)~^][s~1 + s~l]ds^j ||mo-^o|| +

+ Ce~i [t^{t - s)-* +t*(t - s)~^]p (s)ds.

Hence, by Gronwall’s inequality, we conclude that

P (t) < C||u0 — v0||, e < t < T.

Using (34) with 9 = 0, we finally conclude that

f* I i

|| u{t) — v(t) || < 11 tt0 — foil + C [s 2 -f s »]p (s)ds < C\\u0 — w0||, 0 < t < T.

0

Thus we have proved (32). □

4. Dynamical system

We already know by Corollary 1 that, if u0 £ H 1(Q) satisfies m(u0) = 0, then

the global solution u(t; u0) of (1) also satisfies the same condition for every 0 < t < to

and in addition satisfies a dissipative estimate

l|u(t; u0)||H 1 < Cp[e-pt||u0|H 1 + 1], 0 < t < to (35)

with the same p and Cp as in (27). In view of this fact, we set a phase space

Hm(Q) = {u £ H 1(Q); m(u0) = 0}.

For u0 £ Hm(Q), set S(t)u0 = u(t; u0), 0 < t < to. Then, S(t) defines a nonlinear

semigroup acting on Hm(Q).

For each 0 < R < to, let BR be a ball of Hm(Q) such that

Br = {u £ Hm(Q); ||u||h 1 < R}.

We then put

Kr = |J S(t)BR. (36)

0<t<xi

In view of (35), we observe that Br C Kr C B (r?+\)- Clearly, Kr is an invariant

set of S(t), i. e., S(t)KR C Kr for every 0 < t < to. Moreover, by Proposition 5, S{t) is continuous in B ic (-fl2+1) with respect to the L2-norm, i. e., (t,Uo) 1—► S(t)uo

is continuous from [0, 00) x B^jc (-fl2+1) int° L2(Q) with respect to the L2-norm. Of

course, the correspondence is continuous from [0, to) x Kr into Kr, too, with respect to the L2-norm. Hence we have verified the following theorem.

Theorem 4. For each 0 < R < to, (S(t), Kr, L2(Q)) is a dynamical system.

Put C = y/2Cp, where Cp is the constant appearing in (35). Then we observe that, for any Kr, there exists a time tR such that

S(t)KR C Bg for all t £ [tR, to).

In this sense Bg is an absorbing set. Furthermore, in this sense, every dynamical

system (S(t),KR, L2(Q)) is reduced to the dynamical system (S(t),Kg, L2(Q)) as

t ^ to, where Kg is the space given by (36) with R = C We finally put

K = S(1)Kg C Kg.

Then, K is an invariant set of S(t). In addition, (19) yields that

|u||h4 — ||S(1)u0|n4 < C||u0||h1 , u — S(1)u0 £ K, u0 £ K(

which shows that K is a bounded subset of H4(Q). We have thus arrive at the following theorem.

Theorem 5. There is a dynamical system (S(t), K, L2(Q)) the phase space of which is a bounded subset of H4(Q). In addition, for any phase space Kr C Hm(Q), there exists a time tR > 0 such that S(t)KR C K for all t £ [tR, to).

We can verify that S(t) defines also a dynamical system in the Sobolev space H0 (Q) for 0 < 9 < 4.

Corollary 2. For each 0 < 9 < 4, (S(t), K, H0(Q)) defines a dynamical system. Proof. Using (6) with s0 = 0, s = 9, s1 = 4, we observe that

9 4-8

4 II ______|| 4

||S'(i)tt0 - S(t)v0\\He < C\\S(t)u0 - S^vqW^WS^uq - S(t)v0\\J

Therefore, by the boundedness of K in H4(Q) and by Proposition 5, we obtain that

i_0

\\S(t)u0 - S(t)v0\\нв < Ст\\и0 ~ v0\\L24, 0 < t < T; щ, v0 G /С,

where 0 < T < to is any fixed time.

In the meantime, for v0 G K, we have v0 = S(1)v-i with some v_i G Kс. So,

S(t)v0 = S(t + 1)v_1. In view of (28), this means that S(t)v0 is a continuous function

for 0 < t < to with values in H4(Q), a fortiori in H0(П).

For 0 < s, t < T and u0, v0 G K,

||S(t)u0 - S(s)v0\\ns < ||S(t)u0 - S(t)v0||я« + ||S(t)v0 - S(s^Hh^ <

< C7111 Mo — ^0 ||^2 4 + C|| S(t + I)!1-! — + Y)V- \ ||я4 > Щ =

Therefore the mapping (t,u0) ^ S(t)u0 is continuous from [0,T] x K into K in the

H0-norm, T > 0 being arbitrarily fixed time. □

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