Nonlinear Analysis, Theory. h4ethods & Applications, Vol. 30, No. 8. pp. 5527-5532, 1997 Pror. 2nd World Congress of Nonlinear Analysrs

8 1997 Elsevier Science Ltd

PII: SO362-546X(%)00144-7

F’rinted in Grw Britain. All rights reserved 0362-546X/97 $17.00 + 0.00

ON THE EXISTENCE AND ON THE FRACTAL AND HAUSDORFF

DIMENSIONS OF SOME GLOBAL ATTRACTOR.

ANCA ION and ADELINA GEORGESCU Romanian Academy, Institute of Applied Mathematics,

Bucharest, P.O.Box l-24, 70700, Romania.

Key words and phrases: dynamical systems, asymptotic behavior, attractors

1. INTRODUCTION

In this paper we treat a dynamical system attached to a nonlinear evolution equation coming from the pattern formation problem (more precisely, from the Benard convection theory [5], [8]).We treat both the compact and the noncompact case. In the compact case, we prove the existence of the attractor and give estimates for its Hausdorff and fractal dimensions. In the noncompact case we prove only the existence of the attractor.

2. THE COMPACT CASE. 2.1. EXISTENCE AND UNIQUENESS OF THE SOLUTION AND OF THE

ATTRACTOR

Consider the problem:

$+AAu+2Au+2u+g(u)=O (I)

in R = (-l,I) x (-1,I) C R2

4x70) = uo(x), uo E L2(Q), u(x, t) Ian = 0, &(x,t) Ian = 0,

where g(u) = u3 + Pu2 - (R + l)u, and p and R are arbitrary real constants. Let H = L2(0), with the usual inner product and norm ( , ), 11 and V = Hi(Q) with the inner

product ((u, v)) = Jn Au. Avdx and the induced norm ]] I]. V c H c V’, the embedding of V in H beeing compact. Let Au = AAu + 2Au + 2u, a(u,v)=$nAu.Avdx-2$nVu+Vvdx+2J,u.vdx. The following inequalities are easily proved to hold :

Iah 41 I C II4 . I141 1 4uL, ~1 2 b12, a(u, u) 2 $ 11u112.

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PROPOSITION 2.1. If the solution of the problem (1) exists, then absorbing balls in H exist fool, it.

PROOF. By mulptiplying (1) by u and by integrating over R, we obtain:

J eudx $ a(u, u) t n at I R g(“)udx = O (2)

By using the inequalitiy: and the H ellipticity of a( , ), we find:

f$ I$ t lu12 I cz and using GRONWALL’S LEMMA we have:

Iu(t)12 5 Iu(0)j2 e-2t + Cz(l - e-2t), hence limsup lu(t)j2 5 C2 (where Cz = 4C112).

so, fbr%y p > &, there exists a moment, t,,R such that, any solution v(t) with l,u,-l I: R satisfies the inequality lu.(t)l 5 p for t > t,,R.O

PROPOSITION 2.2. If the solution u of the problem (1) exists, then absorbing balls in V exist for. it.

PROOF. By multiplying (1) by AA 2~ and by integrat,ing over Q, we obtain:

;-$ lAu,12 t s, lAAu12 dz t 2 J, /A1112 dz 5 s, IAAul lg(u)I + 2 s, IAAul . /AllI dz.

The terms in the right,-hand side of the preceeding inequality are estimated by using the Young’s inequality. The most difficult terms to estimate are those of the form JQ u6dx and In v”dx.

For the first one we use the embedding Ht(0) - L6(s2) and the interpolation inequality:

1~12 < (uSI- 11~11~ to obtain: :

,Jfl u6da: 5 kl 1~1~ (Azll’ 5 IcIp4 lAu12, for the second, we use:

an1 we get,: ’ R 1~~

d 1 I 4 Iu(’ II&;(~) 5 k2p2 IAuI' ,

;f lAu12 5 26 lAu12 + 4p”(R t 1)“.

The use of t,he V-coercivity and int,egration between t and t + r of (2) implies :

J J lAu12 dx 2 2C2 + ,?. f cl

With (3) and (4) we are in the hypotheses of UNIFORM GRONWALL’S LEMMA. Its follows : Ibll = IA4 L: PI for k 2 1, R_ + 7,

proves the existence of a.bsorbing balls in V.

The existence of the solut,ion of problem (1) is proved by the Galerkin-Faedo method, the onlh difficult point being that of proving that g(um) -+ g(u,) weakly in L2(Q), where u, is the sequence converging weakly in L’(O, T; V) and weakly-star in Lm(O, T : H) t,o the solution u. This difficulty is overcome by using a Lemma of Lions [7].

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The uniqueness and continuity of the solution as a function of initial data are proved by means of classical methods.

Having proved the existence and uniqueness of the solution we can define the semigroup of the continuous operators S(t) : H - H, S(t)u, = u(t). Moreover, the operators S(2) are uniforml) compact for t large (see [9]).

Indeed, by PROPOSITIONS 2.1 and 2.2, for any R + 0 there exists a. moment, t1 = tp,~ + 1’ such that for any uu E B(0, R) C H we have S(t)~u E B(O,pl) c V, t > tl.

it follows: ,g S(t)B(O, R) c B(O,p,) c V.

Since the embedding of V in H is compact, it follows that U S(1)B(O, R) is relatively compact t>t1

in H. With the PROPOSITION 2.1. we place ourselves in the hypotheses of the theorem of existence of

attractors of [9], so the existence of the global compact convex connected attractor is proved.

2.2. ESTIMATES OF THE DIMENSIONS OF THE ATTRACTOR

We consider the linearization of (1) arround the solution corresponding to the initial data U(O) = 210.

- zz -AU - g’(u)u at (5)

U(z,O) = [ in R. The semigroup S(t) is differentiable [6].

Let L(t,uo) E L(H, H) be the Fr&het differential; then L(~,uo)[ = U(t), where U(t) is the solution of (5), with u solution of (1) corresponding to u. (0) = uu.

We proved tha.t for some to > 0 we have : SW I~(to,~~oNc(~, 4 cc. UOEH

For the definitions of Liapunov numbers and exponents and for the THEOREM providing the estimates of the dimensions of the attractor, we send to [9].

In order to apply the theorem we have t,o estimate:

q~ =liyzp 2~5 SF; ($ J,” Tr F’IS(r)uo) 0 QN(~)~T) ,

le, I<’ ,=I, ,N

where

with {~3}3=1,~ is an orthonormal (in L2(fl) ) basis for the space L(Ul,...UN), Ul,...,IjN being solutions of (5) with initial conditions &, . . . . &I. we have :

where: g’(u) 2 --ICI = $ + R+ 1.

In order to estimate the first term in the right-hand side we apply the Lieb-Thirring inequality and for this, some lower bounds for the eigenvalues of A will be necessary. It is not easy to find explicit

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eigenvalues for A on H:(0) and we overcome this difficulty by applying a form of Lieb-Thirring

inequality which uses the eigenvalues of A= +AA on W = H&,(Q). Then, since Ha(Q) c H&_,(Q) the inequality obtained is valid for our functions.

Let us define: cy(u,, v) =: f s, Au Avdx.

Then:

and the Lieb-Thirring inequality implies:

N with p(r) = C #(x),from where, with the Holder inequality:

j=l

valid for any orthonormal family (~1, . ..pN} in W&.(fl), where X = & If qj E Hz(a) C W&,(Q), j E 1, . . . . N, we have:

C!! 4~,, ~3) 2 &z .In p(x)dx j=l

which, together with (8), leads to:

Inequalities (6) and (9) imply:

hence:

YN < LlN - &$J3 =: f(N).

Using the quoted theorem and one of its consequences we obtain:

PROPOSITION 2.3. If& x & thend~(A) = d&d) = 0

PROPOSITION 2.4. If kl + 0, and

2) h(d) 5 NI (I- &) .

(7)

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3. THE NONCOMPACT CASE. EXISTENCE AND UNIQUENESS OF THE SOLUTION. EXISTENCE OF THE ATTRACTOR

In this case? the problem reads

$+AAzl+zAv+2a+g(u)=O on R2,

4x30) = uo(2), 110 E P(7P). The existence and uniqueness of a weak solution u(.. t) E H = L2(7Z2) is proven in the framework

of the monotonr operator theory [S]. We define the weighted spaces: He,., = L:,, = {u, : R” -+ 72: j?,.jt”._, = J’?~~p~,~,dr 4 CQ},

v,,, = gf, = {u, : R2 + ‘R. j/u& = .I’ u2pc,& + ,f [VU/~ pc,-,dz + .I’ JAu12 ~t,r& -c co}, with the weight function: P~,~(.x) = Cl,~c2f,,2j1. and the semigroup S’(t) : H - H,~,. 71( , t) = S’(t)n+

PROPOSITION 2.1. Th,ere e.czsts absorb@ ball.? zn, H,,,.

PROOF. The proof follows the lines of the proof of PROPOSITION 1.1 but it also uses t,he properties of the function Q~,-~.

PROPOSITION 2.2. There e.zd absorbing balb in V,,,.

PROOF. The proof is of the same type of that of PROPOSITION 1.2. with some difficulties coming from the estimates:

J lu.18 PC.& F K1 121.14 2 Ilull 2 , <,a f’3 ./ bl4 Pt.-& I K2 I”lf.3 II+

If we take y + 3 then Iu./~.: 4 ~1, and Iw,/~,~ 1 + p2 (PROPOSITION 2.1 applied for th? weight,:, pe,:, p,,:). With an integral form of the UNIFORM GRONWALL’S LEMMA the result is obtained.

In order to overcome the lack of compactness of the embedding of V,,, in H,,7 we use a reasoning similar to that used in [l].

Let

@(z.t) = (1 +f212/2)~~n’+’ (1 -exp(-a)).

A simple but lengthy algebra leads to:

slip J (21.(X, t)j2.J,(Z. tp 5 C’ 1<t_<cc

u.(x. t) being a solution corresponding to (?@I < R (with C’ depending only on R)

PROPOSITION 2.3. For amy bounded set f?c H. them exists some tB such that U S(t)K? 1.9 t>tp

datively compact in. HF.,.

PROOF. Let un = S(&)b,, b, E I?. By PROPOSITIONS 2.1 and 2.2, (‘u,},,~~ is bounded in H,,, and V,,,.

There exists a subsequence such that ~6,~ --t x in HC,pi weakly. There exists some A4 + 0 with:

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I ,,

In order to prove the second inequality. we define:

We then have:

$ &]>A4 14412 ~W)~~~ 5 6 M large enough and t large enough (since ,Ji’_“oo ~(z, t) = t).

Since the embedding V,,,( 1x1 5 A4)

with {un., (I~I+I!z}~~~~~

- Hc,J 1x1 5 M) is compact we find a subsequence {u,,},,.~

convergent.Then the wea.k convergence of u.,, to y implies

X1. IzI<M + X JIl<M.

With (10). (11) we come t,o: u.~,~ --, x in HeI,.

PROPOSITIONS 3.1 and 3.3. imply that, the THEOREM OF EXISTENCE OF THE ATTRACTOR [Y] can be applied.

REFERENCES

[l] F. Abergel, Existence and Finite Dimensionality of the Global Attractor for Evolution Equations on Unbounded

Domains, Journal of Differential Equations 83, 85-108 (1990);

[2] R.S.Adams, Sobolev Spaces, Academic Press, New York, 1975.

[3] H. Brezis, Operateurs maximaux monotones et semigroups de contraction dans les Espaces de Hilbert, Nor01

Holland Mathematical Studies, 1993:

[4] R. Dautray, J. L. Lions, Analyse mathematique et calcul numerique, Masson, 1988;

[5] H. Haken. Advanced Synergetirs, Instability hierarchies of self-organizing systems and devices, Springer-Verlag.

Berlin,l983;

[6] D. He&y; Geometric theory of semilinear parabolic equations, Springer-Verlag, Berlin. 1981.

[7] J. L. Lions ; Quelques methods de resolution des problems aux ;limites non-line&es, Dunod. Paris.1969;

[8] A. C. Newell, T. Passot, J. Lega, Order Parameter Equations for Patterns, Annual Review of Fluid Mechanics.

25 (1993);

[9] R. Temam, Infinite dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciencr

68, Springer-Verlag, New York 1988.