The universal attractor for a class of nonlinear wave equations

Guo,

etc.:

Universal

u = &3&

Attractor

for

31

‘.

- 06F)y2 + c1y + cq

(32)

1 Z14 = -F 2 215

= 9@ut

(33)

+ 58-4c+6, - 5cs;@ - +F 217

=

(34) (35)

58-1002ya,

z18 = 96-s0,y - 9e-geta,

- 5e-7d2, + 5e-%,a2y

(36)

and cl, c2 are constants. All discussion above are valid only for & # 0. When & = 0, we can suppose that vz = 0 at the same time, otherwise exchange of < and 71will lead to (A) and (B) types of reductions again. Furthermore, we suppose that &, # 0 (or qy # 0), otherwise would be a function of t only, we’ll give all the possible nonequivalent solutions of this case lately.

References [I] [2] [3] [4] [5]

P.J.Olver, Application of Lie Group to Differential Equation, Berlin Springer(1986). P.A.Clarkson and M.D.Kruskal, J. Math.Phys.,301(1989):2201. S-y Lou, Phys. Lett.A, 151(1990):133. J-f Zhang, H-d Yu, Commun. Theor. Phys., 22(1994):245. S-y Lou etc. Commun. Theor. Phys., S-y Lou, J. Phys. A, 23(199O):L649. 15(1991):465. S-y Lou etc. J. Phys. A, 24(1991):1455. [6] H-y Ruan, S-y Lou, Acta Physica Sinica, 41(8)(1992):1213. [7] V.E.Zakharow, S.V.Manakov, S.P.Novikov, L.P.Pitayevsky, “Theory of Solitons: The Method of the Inverse Scattering Problem”, Nauka, Moscow (1980) (in Russian).

The Universal Equations 1

Attractor

for a Class of Nonlinear

Boling GUO & Weiguo ZHANG t (Institute of Applied Physics and Computational Mathematics, 100088, China) t (Changsha Railway University, Changsha 410075, China)

Wave

P.O. Box 8009, Beijing

This paper deals with the universal attractor of initial-boundary value problem for nonlinear wave equations 21tt = 2~,,t + g(u,), - f(u). Under the assumptions o(s) E C2, a’(s) > CO> 0, we obtain the existence of universal attractor of this problem. Its Hausdorff and fractal dimensions are proved to be finite. Key words: Nonlinear wave equation, Universal attractor, Hausdorff dimension, Fractal dimension. Abstract:

lThe

paper

was received

on Feb.

29, 1996

32

Communications

in Nonlinear

Science

& Numerical

Simulation

Vol.1,

No.2

(Apr.

1996)

The nonlinear wave equation utt = uzst + a(~,), is an important nonlinear wave equation, which appears in the vibration of bar with viscous effect. The existence and uniqueness of solution of initial-boundary value problem for this equation were studied in [l-3]. In this paper, we consider the following more general initial-boundary value problem

utt = wzzt+ @(Kz)z- f(u) + g(z), u(z, 0) = uo(z), Ut(z, 0) = ‘L11 (xc),

(1.1)

u(0, t) = 0, u(1, t) = 0.

(1.3)

U.2)

We obtain better result on the global existence under the weaker conditions. Furthermore, we prove that there is the existence of the universal attractor for the problem (l.l)-(1.3), and the universal attractor has finite Hausdorff and fractal dimensions. Theorem

(1) c(.s)~C’, (2) f(s)EC’,

1. Suppose that the following

conditions are satisfied: g’(s) is lower bounded, i.e.,a’(s)>ae (cro need not be positive); f(0) = 0, let F(s) = Jl f(z)&, f(.z) satisfies

(3) g(z)EH;, uo(z)EH,lrlH3, ul(z)EH;. Then the problem (l.l)-(1.3) exists a unique weak solution u(z, t)~L~(0, u,cL”(O, T; H;)nL2(0, T; Hz), utt, uzttEL2(0, T; H-l). Furthermore, we assume that (1) a(s)EC2, g’(s)>(To > 0, (2) ,s$~mi~f~ZO,

T; HAnH3),

(1.4)

and there exists LJ > 0, such that lim inf sf(s) -s2Ws) I+~

>() _ ,

(1.5)

For7 the sake of convenience, we denote by R the domain (0, l), by 1.10the L2 norm, by ].Ico the LM norm, E. = (H~(0)fW2(R))xL2(R), El = (H~(s2)nH3(0))xH~(O). Lemma 1. Suppose that (1.4)-(1.6) hold, ( uc,ui)~&, problem (l.l)-(1.3), we have

b& + 14; + IwlMKe

--UPt+ A42(1 -

where the constant M1 depends on ]](uo, ul)]lE, and the constant p satisfies

Lemma 2. Under the assumption solution ~(2, t) of problem (l.l)-(1.3)

epwpt ) + hf3, t>O,

of

(1.7)

and )glo, MZ and Ma depend only on ]g]c;

0 < p < min{3XI, here Xi is the first eigenvalue of A = -a,,

then for the solution u(z,t)

T},

with zero boundary

(1.8) condition.

of Lemma 1, the following estimates hold for the

--pt + 11(~7w)ll&I~4e

Ms(l - e -““) + Mfj, vt>o,

(1.9)

Guo: etc.: Universal Attractor

where Al, and 1Ms depend on ]](~c,~r)]]~~,

33

for

AIs depends only on ]g]o.

Lemma 3. Assume that (1.4)-(1.6) hold, if ( us, ‘1~r)~Er, then for the solution ~(2, t) of problem (l.l)-(1.3), we have

1~1~+ ll(~,wj1/~05~7e+ + Ms(~- e+) + M9, where y = min{l

(1.10)

Vt20,

- 2wo,p}, 0 < WC,< f.

Lemma 4. Under the conditions of Lemma 3, there exists S > 0: such that the solution ZL(Z,t) of problem (l.l)-(1.3) satisfies

ll(~,w)JI~llMl0e

4t + Mrr(l

(1.11)

- eebt) + M12, t>O,

where Nero depends on ]](uo,‘~L~)]]~~, Mrr depends on ]](~,ur)]]~~,

MI2 depends only on

1.9lO~ Proposition 1. Assume that (1.4)-(1.6) hold, then there exists constant p > 0, such that for every (~Lo,‘~~I)E& with

there exists T(R),

such that

That is, B2

= {(‘1L,%)EEl,

l/b4w)ll~I
(1.12)

is a bounded absorbing set of S(t) in El. Theorem

2. The set A = w(&)

= f-j U S(t)B 2 is a universal attractor $20 t>s

for problem

(l.lj-(1.3). That is, the set A satisfies (i) A is bounded and weakly compact in El S(t)A = A,

Vt>O;

(ii) For any bounded set B in El lim d”(S(tjB, t-+03 (iii) A is the maximal topology in El.

set satisfying

(i) and (ii),

A) = 0; and A is connected in the sense of weak

3. Assume that (1.4)-(1.6) hold and rr(s)~C~, fcC2. Then the universal A of problem (l.l)-(1.3) has finite Hausdorff and factual dimensions.

Theorem

attractor

References [l] T.M.Greenberg, R.C.Maccamy, V.J.Mizel, J.math.Mech., [2] G.Andrews, J.Diff.Equ., 35(1980), 200-231.

17(1968), 707-728.

34

Communications

in Nonlinear

Science

& Numerical

Simulation

Vol.1,

No.2

(Apr.

1996)

[3] Liu Yacheng and Liu Dacheng, Chinese Ann. of Math., A, 4(1988), 459-470. [4] T.M.Ghidaglia, J.Diff.Equ., 74(1988), 369-390.

Global Attractor for Axially Symmetric Kuramoto-Sivashinsky Equation in Annular

Domains

1

Boling GUO & Hongjun GAO (Center of Nonlinear Studies, Institute of Applied Physics and Computational Mathematics, P. 0. Box 8009, Beijing 100088, China) Abstract: In this paper, the finite dimensional global attractor for axially symmetric Kuoamoto-Sivashinsky equation in annular domains is obtained. Key Words: global attractor, Kuramoto-Sivashinsky equation, annular domains.

Introduction In [l]-[4], the global attractor and the existence of inertial manifolds in H1 Sobolev space for Kuramoto-Sivashinsky equation in one space dimension was proved. In [5], the global asymptotic behavior of the Kuramoto-Sivashinsky equation in thin 2D domains was obtained. The dissipativity of the general two (and higher)- dimensional problem has been open for some time, the essential difficulty being lack of a proof of the existence of an absorbing set. Our goal in this paper is to obtain the global attractor and its dimensional estimate for axially symmetric Kuramoto-Sivashinsky equation, that is ~+~+~~+(l+$)~+f~+~,~,z=o,

(1)

where 0 < TO5 r < ri, t > 0. We consider (1) with the following boundary condition:

u(q), t) = u(r1, t) = 0, $(ro;t)

= $;t)

= 0, t 2 0.

The initial value is u(z, 0) = uo(z)

r E I = [To, q].

(3)

Define Hilbert space H = L’(1) with the norm denoted by 1~1: = s ]~]~dr, where s dr = ST; dr, and V = H2(1) fl Hi(I) with the norm ]]u]]~ = s Iu,,(r)12dr. Our main results is: We consider the dynamical system for (l)-(3). This dynamical system possesses an finite dimensional attractor A which is maximal, connected, and compact in H,1 (I). ‘The

paper

was received

on Mar.

1, 1996