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A trajectory attractor of a nonlinear elliptic system in an unbounded domain

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It has been shown above that C _C Ker A and therefore for w E C we have the representation w -- (wl, 0), where w~ e W1 and V(w) = ~r(w,, 0). Let us define a function ~ : W1 --~ R by setting ~(wl) -- V(wl, 0). We denote the set of critical points of the functional ~0 on the set W~ by
Co.
Obviously, C C_ C0 and therefore V(C) C_ ~o(C0). The function ~: W1 ~ R is C ~ and hence, by the Morse-Sard theorem, the Lebesgue measure of ~o(C0) is /~(~(C0))=0. Then /z(V(C))=0 and we have
(V(C))=0.
Thus for any point x 9 B there exists a neighborhood U such that the Lebesgue measure of the set
V(B n U)
is zero. Neighborhoods of the type U cover the set B. In this cover we choose a countable subcover; this is possible, since E is separable. Since a countable union of sets of measure zero has measure zero, it follows that the set
S(B)
has measure zero. The proof of the theorem is complete. IZI
eferences
1. R. Abraham and J. Robbin, Transversal
Mappings and Flows
Benjamin, New York (1967). 2. J. Ills,
Uspekhi Mat. Nauk
[Russ/an
Math.
Surveys], 24, No. 3, 157-210 (1969). 3. S. Smale, Amer. J.
Math.
87, No. 4, 861-866 (1965). 4. J. Kupka, Proc. Amer.
Math.
Soc., 16, No. 5, 954-957 (1965). 5. S. I. Pokhozhaev,
Mat. Sb. [Math. USSR-Sb.] 75
(117), No. 1, 106-111 (1968). 6. Yu. I. Sapronov,
Uspekhi Mat. Nauk [Russian Math. Surveys]
51, No. 1,101-132 (1996). 7. Yu. G. Borisovlch, V. G. Zvyagin, and Yu. I. Sapronov,
Uspekhi Mat. Nauk [Russian Math. Surveys]
32, No. 4, 3-54
1977).
VORONEZH ST TE UNIVERSITY
Translated by I. P. Zvyagin
Mathematical Note~ Vol. 63 No. 1 1998
Trajectory ttractor of a Nonlinear Elliptic System in an Unbounded Domain
S V Zelik
KEY
WORDS
nonlinear elliptic system, Laplace operator, Sobolev space, trajectory attractor, attractive sets.
In an unbounded domain f~ C R '~ , we consider the nonlinear elliptic system
aAu + 7 Du - f(u) =
g x), I)
xEfi, u[0fi=u0. Here u = (u',...,u~), f = (f',..., fk), and g = (g',...,gk) are vector functions, x = (z',...,x"), A is the Laplace operator with respect to the variables x = (xl,..., xn), a 9 E(R ~, R k) is a constant matrix such that a -k a* :> 0, and "r/)u is a first-order differential operator with constant coefficients:
~u = ~-~O~,u, ~ e s k, Rk .
i=l Translated from
Matematicheskie Zametki
Vol. 63, No. I, pp. 135-138, January, 1998. Original article submitted August 6, 1997. 120 0001-.t346/98/6312-0120 20.00 C)1998 Plenum Publishing Corporation
Suppose that the nonlinear function f satisfies the following conditions: (i) f ~ C(R k, R~); (ii)
f(u).u > -C1
+ C21ul r, C2 >
;
(iii) If(u)l -< C(1 + lulr-I), r > 2.
Here and in what follows, the notation
u.v
is used for the inner product in R k . For a cylindrical domedn (n = R+ • w, w ~ R"-~), the behavior of solutions of problem (1) as z ~ cr was examined in [1-4]. A trajectory attractor of system (1) in a cylindrical domain is constructed in [2]. In this paper it is assumed that there exists a distinguished direction /'E R" in R" such that (iv)
Tsf~
C fl for all s > O;
(v) U,>o T_,a = R ,
where
Ts
= TJ: R" ~ R" is defined by
Tsx = x + s~.
By a
solution
of problem (1) we understand a function u that belongs to the space
o(a) ~oo = [w~,,(n) n L~~ ~'
satisfies (1) in the sense of equality in the space :/Y(n), and has the prescribed trace u0 e ~0(0n) on the boundary Off. Recall that
Wt,p(V),
where V is an open set in R", is the Sobolev space of distributions from 2Y(V) such that their derivatives order < l belong to
Lp(V), II YiIt,p - ][ []w~.,(v),
and the topology in the space O(n) is defined by the family of seminorms R max{
I1 , n R R R .
I1~, n n Bxollo n B~oll0,r } ReR+, ~oe Bxoll~,2, I1~, n n ,
Here B~o is the ball of radius R centered at x0 E R ". It is easy to verify that 0(~) with this topology is a reflexive separable Fr~chet space. We denote the closure of the space C~~ in the topology of e(n) by O0(n) and the space
o n)
equipped with the weak topology, by O~(n). Since the boundary of n is not assumed to be smooth, the trace u0 of a function u E O(n) on 0n is defined as the image of u under the natural projection
~: e n) ~ voCOa) =
e(a)/o0(n)
(u [ 0n -- 7ru). The set of seminorms in the space l~0(On) can be specified as
R O n), [ 0n u0,
[uo, OnnB~nollvo =inf{llw v, nnB.ollo:we
w = v 6 O0 n)},
R E R+, xo E R n.
It is assumed that the right-hand side g =
g(x)
in (1) belongs to the space Z(n) = [w~,l~ ) + L~e(n)] *, where
I/r + 1/q = 1.
Recall that the space .~. consists of all functions g G D~(n) that can be represented in the form
lo
n k
g = g, + g~, where g, e [W~,,~ )] and g~ e [Z~~176
The system of seminorms in E(n) is specified by
R
119 n n B~oll- = inf{ 119~, ~ n B ~ 11-,,~ + II0~, n n B~0110,, :
Z0
toe , g2 e [L~r e R+, xo e R".
=gl g2, gl e
[W~l,2 a)] k
Let ~.w(n) denote the space ~(n) equipped with the weak topology. In addition, we assume that Ilg; Xo,
Rllb
= sup Ilg, fl n T~B~oII- =
C(xo,
R) < oo (2)
s>0
for any ball BzRo C R n 9
121
Theorem 1.
Suppose that the
above
conditions are satisfied. Then for any uo E ~o(O~) problem
(1)
has a solution u E
O(~), and for
any solution u the
estimate
Ilu, B~ n nile < c~(1 + 119,
nR+
n nll.~ + Iluo, R'~+ n Onll~~ (3)
~0
holds. Here ~ > 0 is an arbitrary positive number and the constant C~ depends only on c, R, and the constants introduced in
(i)-(iii)
(and is independent of f~ C R and
x0 E R ). The estimate (3) and conditions (iv) and (v) make it possible to apply the concept of a trajectory attractor of a dynamical system (see [5, 6]) to examine the behavior of solutions of problem (1). Below, we briefly present a scheme of construction of a trajectory attractor using Eq.(1) as an example. First, note that, by condition (iv), the semigroup of translations {T,, s _> 0} ((T~u)(z)
= u(T,z))
acts in the spaces O(f~) and ~(f~), i.e.,
T,e n) c e n) and T,~ n) c ~ n) for s >_ 0.
We shall refer to the set 7/+(g) =
[Tsg, s >_
0]--~ as the
hull
of the right-hand side g in the space .~. (f~). The symbol [. ]x~ denotes closure in Ew(fZ). It is readily verified that, by (2), the hull 7-/+(g) is compact
in z~ a).
Thus (see [7]) there exists a nonempty w-limit set
_>o
~h_ s J
~ of the hull 7-/+(g) with respect to the action of the semigroup {T~, s > 0}. Let us describe the structure of this set. Definition 1. A function ((x) e ~-(R ~) is said to be a
full ,ymbol
of Eq. (1) if linT,( 6
w(g)
for any s 6
R. Here and in what follows li~ denotes the restriction f a function to the domain ~.
The set of
all
full symbols of (1) is denoted by
Z(g).
Lemma 1. The
relation w(g)
= IInZ(g)
holds. In particular, the set
of ruB
symbols is nonempty.
Definition 2. For any function a e 7~+(g)~ we define K + C O(fl) to be the set of solutions of problem (1) with right-hand side
a(x)
and arbitrary u0 e ]~0(0f~). The set
K
= U
KS
c o (a)
aET~+ g)
we be called the unifed
frajecfory ,pace
of Eq. (1). Lemma 2. The
semigroup {Ts, s >.
0}
acts
in
the space K + ,
i.e.,
TsK + C K +
for s>0. Definition 3. We say that a set A C K + is a
irajectory a~iractor
of (1) if the following conditions hold: (1) A is compact in the space O~(f~); (2) A is strongly invariant with respect to the action of the semigroup {T,, s >_ 0} (i.e., T, fl = ,4) for s_>0;
122
(3) .,4 is an attractive set of the semigroup {Ts, s > 0} in the space K + , i.e., for any neighborhood 0(.,4) of .,4 in the topology of the space 0~(~), there exists an S =
S(O(A))
such that
T~K + C O(A)
(4) for all s >_ S. Remark. The above definition of an attractive set slightly differs from the traditional one (see [7, 8]). Usually, the validity of condition (4) is only required for
bounded
(in some sense) subsets of K + . However, the estimate (3) implies that, in the case under consideration, the attraction condition is satisfied by
all
subsets of g + with the same constant S =
S(O(Jt)) .
Theorem 2.
Suppose that the conditions specit~ed.above are satist~ed. Then
(1) has a
trajectory attractor A that can be
represented as
A= U HaKe
~ez g)
Here
Ke stands
for
the set of all solutions u E
O(R )
of the equation
+ -/(u) = r . e R ,
with a given right-hand side ~ e Z(g).
Corollary.
The definition of the topology of the apace
O~(fl)
and the observation that the sets
O B~0) ~
H,_.,2 B~o) and
O B~0) ~
L._~ B~o)
are compact subsets os the
correspond/rig
spaces
for any e > 0
imply that,/or
an arbitrary
ball Bffo C fl,
we have lim
distnt_..,(a~o){HBRaTsK+,
Ha~0A} = 0,
s---*q-oo
lim
distL._,(Bao){HB~oT~K +,
HB~oA} = O.
$---*+ oo
Here dist.{X, Y} = sup inf I[x - y[[..
zEX yEY
This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00354 and by the CRDF under grant No. 186100. References
1. A. V. Babin,
lzv. Ross. Akad. Nauk Set. Mat. [Russian Acafl. Sci. Izv. Math.],
58, No. 2, 3-18 (1994). 2. M. I. Vishik and S. V. Zelik,
Mat. Sb. [Russian Acad.
ScL Sb.
Math.],
187, No. 12, 21-56 (1996). 3. S. V. Zelik,
Mat. Zametki [Math. Notes],
61, No. 3, 447-450 (1997). 4. A. Galsina, X. Morn, and J. Sola-Morales, J.
Differential Equations,
102, 244-304 (1993). 5. V. V. Chepyzhov and M. I. Vishik, C. R. Acad.
Sci. Paris. S6r. A,
231, 1309--1314 (1005). 6. V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J.
Math. Pures Appl. 9)
1997)
to appear).
7. A. V. Babin and M. I. Vishik,
Attractors of Evolution Equations
[in Russian], Nauka, Moscow (1989). 8. V. V. Chepyzhov and M. I. Vishik, J.
Math. Pures Appi. 9),
73, No. 3, 279-333 (1994). INSTITUTE OF PROBLEMS OF DATA TRANSIVIISSION, RUSSIAN ACADEMY OF SCIENCES Translated by O. V. Sipacheva 123

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