Do Global Attractors Depend on Boundary Conditions?

We consider global attractors of innnite dimensional dynamical systems given by dissipative partial diierential equations u t = u xx + f(x; u; u x) on the unit interval 0 < x < 1 under separated, linear, dissipative boundary conditions. Global attractors are called orbit equivalent, if there exists a homeomorphism between them which maps orbits to orbits. The global attractor class is the set of all equivalence classes of global attractors arising for dissipative nonlinearities f. We show that the global attractor class does not depend on the choice of boundary conditions. In particular, Dirichlet and Neumann boundary conditions yield the same global attractor class. The results are based on joint work with Carlos Rocha. Parabolic partial diierential equations modelling reaction, diiusion, and drift are an important class of nonlinear innnite dimensional dynamical systems. Aside from applied motivation, much of the mathematical interest has centered on the dynamics of their nite dimensional global attractors. See for example Hal88], Lad91], BV89], Tem88], and the references there. The innuence of boundary conditions has mainly been investigated in connection with stability of equilibria and shape of the underlying spatial domain, see for example MM83], Mat84]. Equations in one-dimensional domains have been studied in much detail, see mostly under Dirichlet or under Neumann boundary conditions, separately. In the present paper, we follow an approach developed more recently by FR91], Fie94], FiRo94]. There the emphasis

is on Neumann boundary conditions.Here, we indicate the necessary adaptations for general separated, linear, dissipative boundary conditions.Although the global attractors for a given nonlinearity will in general depend on our choice of boundary conditions, HR87], we will show that the set of their orbit equivalence classes does not.
To be speci c, we consider scalar equations u t = u xx + f(x; u; u x ) (1.1) on the unit interval 0 x 1. Fixing 0 0 ; 1 1, we impose boundary conditions (1 )u + @ u = 0 (1.2) at x = 0; 1.Here = 0; 1; @ u = u x indicates the outward "normal\ derivative with + at x = 1; at x = 0, and subscripts t; x indicate partial derivatives of solutions u = u(t; x).For the nonlinearities f 2 C 2 we impose dissipation conditions f(x; u; 0) u < 0; (1.3) for juj C 1 and, with continuous functions a; b as well as an exponent < 2 jf(x; u; p)j a(u) + b(u)jpj (1.4) at all arguments (x; u; p) of f.The estimators C 1 ; a; b; are allowed to depend on f.This setting is xed throughout this paper.
The dissipation conditions (1.3), (1.4) guarantee the local semi ow of x-pro les of solutions u(t; ) 2 X ; t 0, to be globally de ned and dissipative: any solution eventually remains in a xed large ball B X .See Ama85], theorem 5.3 for a reference.In fact, we can choose B such that juj < C 1 and jpj < C 2 on B. The state space X is the Sobolev space H 2 intersected with boundary conditions (1.2), = ( 0 ; 1 ).
By dissipativeness, equations (1.1), (1.2) possess a global attractor A f X .This is the maximal compact invariant subset of X or, here equivalently, the set of bounded solutions u(t; ); t 2 IR.Yes, including negative t.This global attractor is our principal object of study here.We call a global attractor A f orbit equivalent to A g , A f = A g ; (1.5) if there exists a homeomorphism H : A f !A g which maps orbits fu(t; ) j t 2 IRg on A f onto orbits in A g .Obviously, = is an equivalence relation and de nes orbit equivalence classes of global attractors.
Let E f denote the set of equilibrium solutions u t = 0 of (1.1), (1.2).Clearly, E f A f .We assume all equilibria to be hyperbolic: all eigenvalues of corresponding Sturm-Liouville eigenvalue problem are nonzero, for linearizations at equilibria.This is a generic nondegeneracy condition on f, for any given .
For given boundary conditions , we de ne the attractor class A( ) as the set of orbit equivalence classes of global attractors A f .Here f 2 C 2 are assumed to be dissipative, as in (1.3), (1.4), with only hyperbolic equilibria.

Documenta Mathematica 1 (1996) 215{228
Theorem 1.1 In the above setting, the global attractor class A( ) does not depend on the boundary conditions (1.2) given by = ( 0 ; 1 ) 2 Q := 0; 1] 2 .In other words, let ; 2 Q.Then A( ) = A( ): (1.6) Speci cally, for any dissipative f 2 C 2 with hyperbolic equilibria E f there exists a dissipative g 2 C 2 , also with hyperbolic equilibria E g , such that the respective global attractors A f ; A g are orbit equivalent A f = A g ; (1.7) in the sense of de nition (1.5).
In section 2 we prove theorem 1.1.We conclude with a discussion of our result, in section 3.
We use a rescaling argument by Rafael Ortega.Let with some smooth amplitude function A > 0 satisfying A( ) = (1 )A( ) + A( ) = 1 (1.9) at = 0; 1.Then the transformation (1.8) de nes a linear isomorphism between the state spaces u 2 X and v 2 X associated to boundary conditions and .Also, v satis es an equation (1.1) with an appropriately rescaled dissipative nonlinearity g instead of f.Therefore A f = A g , by (1.8), (1.9), and A( ) = A( ) in the non-Dirichlet cases.(Strictly speaking, though, the transformation does not preserve the precise form (1.3) of our dissipation condition.)Our slightly more involved proof, given in section 2, will include even the Dirichlet case.In particular, the Neumann and the Dirichlet attractor classes will be shown to coincide.Note that all spaces X , including the Dirichlet cases, are closed linear subspaces of X = H 2 depending continuously on the parameters ; in particular all these spaces are isomorphic from an abstract view point.
We brie y outline the Morse-Smale structure behind our proof of theorem 1.1, in the remainder of the present section.Following FiRo96], we rst normalize f, for simplicity, such that f(x; u; p) = u (1.10) for jxj C 1 or jpj C 2 .Such a normalization can be achieved without changing A f or the ow on it, by dissipation conditions (1.3), (1.4).For u 2 X consider functionals V (u) := Z 1 0 F(x; u; u x )dx: (1.11) Following Mat88] we observe that (1.12) along solutions u(t; x) of (1.1), (1.2), if F satis es pF ppu fF ppp + F ppx = f p F pp ; (1.13) for all (x; u; p) and obeys the boundary condition F p u t = 0 (1.14) at x = 0; 1: By the standard method of characteristics, Joh82], it is easy to nd a solution w = w(x; u; p) of the rst order equation pw u fw p + w x = f p ; (1.15) see also (2.1).In fact, normalization condition (1.10), guarantees global solvability of the characteristic equation of (1.15) which is studied in more detail in section 2 below.Solving then F pp = exp(w) (1.16) we have solved (1.13).The boundary conditions (1.14) hold trivially in the Dirichlet case.We require F p = 0; as an initial condition for (1.16) with respect to p, along the lines in (x; u; p)-space given by the boundary conditions (1.2), in all other cases.
By this construction, F pp = exp(w) is positive.In particular the functional V becomes a Lyapunov functional on X which decreases strictly along non-equilibrium orbits.With respect to the Riemannian metric on X de ned by F pp ; the semi ow (1.1), (1.2) is in fact gradient, or Morse with respect to V: The functional V reveals that the global attractor A f consists entirely of equi- libria E f and heteroclinic or connecting orbits.These orbits, by de nition, limit onto (di erent) equilibria ũ; u for t !+1; t !1; respectively.They can be viewed as intersections of unstable and stable manifolds W u (u) \ W s (ũ): Note that A f = E f u2E f W u (u): (1.17)Although this will not be very visible below, we emphasize the importance of nodal properties in our proof of theorem 1.1.Based on observations for linear equations, they imply that t 7 !z(u 1 (t; ) u 2 (t; )) (1.18) is nonincreasing along solutions u 1 (t; ); u 2 (t; ) of (1.1), (1.2).Here z, the zero number, denotes the number of strict sign changes of x-pro les.The zero number in (1.18) drops strictly whenever a multiple zero of the x-pro le is encountered.Historically, the use of nodal properties dates back as far as Stu36].In Mat82], their importance for in nite dimensional nonlinear dynamics was rst realized.A comprehensive modern account of zero numbers is given in Ang88].
The most striking consequence of nodal properties for our global attractors A f is the Morse-Smale property.The Morse structure is generated by the Lyapunov functional V; as discussed above.By Hen85], Ang86], the intersections between stable and unstable manifolds which make up the global attractors A f are automatically transverse, without further genericity assumption on f or : W u (u)\ W s (ũ); (1.19) without any nondegeneracy assumptions except hyperbolicity of the equilibria E f : Structural stability is the most important consequence of this Morse-Smale property.In fact, let g be C 2 -near f and satisfy dissipation conditions (1.3), (1.4).Let also 2 Q = 0; 1] 2 be near : Then A g = A f ; (1.20) as claimed in (1.7).For reference see P69], PS70], PdM82], and for the in nite dimensional case Oli92].Since the argument is local in f; ; this does not prove our theorem, of course.
2 Proof of theorem 1.1 If all equilibria E f are hyperbolic, then the global attractor is Morse-Smale and therefore structurally stable, as we have seen at the end of the previous section.We give a geometric criterion for hyperbolicity of E f , in lemma 2.1.The criterion is based on a shooting approach to equilibria.In lemma 2.2, we relate global attractors for di erent boundary conditions, by an augmentation argument.Piecing these elements together, we nally prove theorem 1.1 by a homotopy argument which uses the Morse-Smale property.
Our geometric criterion for hyperbolicity is a slight adaptation of an argument which in addition satisfy the boundary conditions l 0 : (1 0 )u 0 p = 0; at x = 0; l 1 : (1 1 )u + 1 p = 0; at x = 1: (2.2) In passing we note that (2.1), together with _ w = f p (x; u; p); are the equations of the characteristics of (1.15).For any real a, let u(x; a); p(x; a) denote the solution of (2.1) with initial condition u(0; a) := 0 a p(0; a) := (1 0 )a (2.3) at x = 0: By normalization (1.10), these solutions are globally de ned.De ne the shooting surface S f 0; 1] IR 2 as S f := f(x; u; p) j u = u(x; a); p = p(x; a); a 2 IRg: (2.4) The sections S ;x f IR 2 of S f for given x are called shooting curves .The shooting curves are planar C 1 Jordan curves, parametrized by the shooting parameter a 2 IR: The set E f of equilibria is given by precisely those values a 2 IR where the shooting curve S ;x f at x = 1 intersects the line l 1 of boundary conditions at x = 1: Lemma 2.1 An equilibrium in E f given by the shooting parameter a is hyperbolic if, and only if, the shooting curve S ;x=1 f intersects the target line l 1 transversely, at the intersection value a: Proof: Consider the equilibrium u(x; a) corresponding to the intersection value a; or vice versa.The partial derivative (u a (x; a); p a (x; a)) is the nontrivial solution of the linearized equation which satis es the homogeneous linear boundary condition l 0 : (Well, we could take constant multiples instead.)Clearly, u is nonhyperbolic if, and only if, this partial derivative also satis es the other boundary condition l 1 ; at x = 1: Reinterpreting geometrically, nonhyperbolicity is then equivalent to the tangent vector of the shooting curve S ;x=1 f : a 7 !(u(1; a); p(1; a)) being parallel to the line l 1 : This is exactly nontransversality of intersection, and the lemma is proved.2 Augmentation works as follows.We append new segments (2.5) 0 ; 1 > 0; to the original x-interval x 2 0; 1]: In the appended intervals, we de ne f by f(x; u; p) := 2 (x)u; (2.6) where 2 (x) := 2 > 0 is constant for x 2 I : We will specify = ( 0 ; 1 ) below.In (2.7), (2.8) on is _ = ( 2 2 (1 ) 2 ); (2.9) by direct calculation.This equation plays a central role in the proof of the following lemma.
For abbreviation, let F denote the set of dissipative nonlinearities f; g 2 C 2 with hyperbolic equilibria as speci ed in theorem 1.1.Lemma 2.2 Let f 2 F and consider arbitrary boundary conditions 2 (0; 1) 2 ; in the interior of the closed unit square Q = 0; 1] 2 ; and 2 Q: Then there exists g 2 F such that the global attractors A f ; A g are orbit equivalent, A f = A g (2.10) Documenta Mathematica 1 (1996) 215{228 Proof: We will use augmentation (2.5){(2.9) to construct g on an interval x 2 0 ; 1 + 1 ]: Rescaling x by a scaling factor s := 1=(1 + 0 + 1 ) and adjusting accordingly, rst, we will not lose generality.As a main step, we will then construct a dissipative homotopy from (f; ) to (g; ); by augmentation, such that hyperbolicity of equilibria is preserved throughout the homotopy.In a nal, third step we address the issue of C 2 -regularization of our piecewise de ned nonlinearities, by smoothing.
By Morse-Smale structural stability, the homotopy which preserves hyperbolicity of equilibria then proves the lemma.
We describe our homotopy of attractors in terms of changing the boundaries x = ; 1 + , simultaneously, from their original value = 0 to their nal values = : On these larger x-intervals the nonlinearity f is augmented to f by (2.6).
The boundary conditions = ( ) are adjusted, according to (2.9), in parallel with the homotopy parameter : Note that by a rescaling of x with factor s = 1=(1 + 2 ); this induces a homotopy of global attractors for rescaled nonlinearities in the class F: Clearly, dissipativeness is preserved.In view of Morse-Smale structural stability, it therefore only remains to prove that hyperbolicity of equilibria is preserved throughout the homotopy.Hyperbolicity of equilibria follows from lemma 2.1.Indeed, transversality of the shooting curve S ( );x f ; at x = 1 + ; to the line l( 1 ( )) follows in three steps, using (2.5){(2.9).First, in I 0 = ; 0); the initial line l(~ 0 ) = l( 0 ( )) = S ( ); f gets Documenta Mathematica 1 (1996) 215{228 mapped di eomorphically to the line l( 0 (0)) = l( 0 ) = l 0 : Second, in x 2 0; 1]; we obtain the original f-shooting curve S ;1 f \ l( 1 ): (2.15) Here we use hyperbolicity of E f and lemma 2.1.Third, in I 1 = (1; 1 + ]; the line l 1 = l( 1 ) = l( 1 (0)) and the f-shooting curve S ;1 f get mapped onto S ( );1+ f \ l(~ 1 ); (2.16) by the shooting di eomorphism.Transversality is inherited from (2.15).A nal application of lemma 2.1 proves that hyperbolicity of equilibria is preserved during our homotopy 0 : Of course, rescaling of x does not a ect hyperbolicity.Smoothing the discontinuities of our augmentation of f, at x = 0; 1; we obtain a C 2 -augmentation.Making the x-intervals, where smoothing acts, small enough, we can guarantee transversality (2.16) to hold throughout our homotopy 0 : In particular, all f are Morse-Smale.De ning g as (the rescaled version of) f ; structural stability of Morse-Smale systems nally implies A f = A g : (2.17) In (2.17) we have used that rescaling does not change the orbit type of the global attractor and, simultaneously, transforms ~ = ( ) to the boundary condition by (2.11).This proves the lemma.

2
In the previous lemma we have shown A( ) = A( ); for attractor classes with ; 2 (0; 1) 2 : (The transformation (1.8) would even allow for ; 2 (0; 1] 2 :) To complete the proof of theorem 1.1, anyway, it remains to address the case of or in the boundary @Q of the square Q = 0; 1]: If g 2 F; 2 @Q; then local structural stability of Morse Smale systems shows that for f := g and any 2 (0; 1) 2 close to we have orbit equivalence A g = A f : Therefore A( ) A( ): To complete the proof of theorem 1.1 it remains to show that, conversely, A( ) A( ); (2.18) for some 2 (0; 1) 2 : By lemma 2.2, claim (2.18) actually holds for all 2 (0; 1) 2 ; 2 Q: This completes our proof of theorem 1.1.

3 Discussion
We begin our discussion with remarks on x-dependent di usion and on another attempt of simplifying our proof, by transformation of x.We then indicate why periodic boundary conditions x 2 S 1 produce a class of Morse-Smale attractors quite di erent from the class A( ) of separated boundary conditions = ( 0 ; 1 ) 2 Q = 0; 1] 2 .
We conclude with a few comments on global attractors in the case of higher space dimension, dim x > 1, and the case of systems, dim u > 1.
A main disadvantage of this rather simple argument is the fact that Neumann as well as Dirichlet boundary conditions = 0; 1 remain unchanged by the transformation ; see (3.3).It is the case 2 @Q, where we really seem to need the augmentation in lemma 2.2.
Of course we could have discussed orbit equivalence of attractors in the class of pairs (f; D), allowing for space{dependent di usion from the very start.Fixing D 1; though, provides a stronger statement in theorem 1.1.Parenthetically we note that introducing D > 0 does not produce any additional global attractors, by the above arguments.As we have argued in the discussion section of FiRo96], we do not expect additional global attractors to arise, even in fully nonlinear, uniformly parabolic, dissipative cases.
Passing to higher-dimensional domains x 2 IR d , with @ smooth and bounded, we may again consider dissipative scalar equations u t = u + f(x; u; ru) (3.4) on , under mixed boundary conditions (1 )u + @ u = 0: (3.5) Now = (x) 2 0; 1] is a given function on @ .A transformation u(x) = A(x)v(x) is still feasible, normalizing 2 (0; 1] to become a uniform Neumann condition 1 for v; see (1.8), (1.9).But we have lost variational structure, nodal properties, and Morse-Smale when passing to (3.4), (3.5).Essentially arbitrary nite-dimensional ows occur in (3.4), see Pol95].Even if we assume the global attractor A f to be structurally stable, there is no reason to believe that its orbit equivalence class is determined by the equilibria, alone.
To include the Dirichlet cases, it is tempting to try and augment , by attaching a collar outside @ , such that boundary conditions on the enlarged region di er from the original ones.A structurally stable attractor A ; f should still be recovered in the enlarged region 0 .If is starshaped with respect to the origin, a homothety 0 = s by a scaling factor s > 1 comes to mind.In the annular region A = 0 n we can determine an eigenfunction for a positive eigenvalue of the Laplacian with boundary conditions (1 )u @ 0 u = 0 on @ (1 )u + @ 0 u = 0 on @ 0 : (3.6)Here 0 denotes the outward normal of A; on @ we have 0 = .In the onedimensional case, this eigenfunction was the crucial shooting augmentation in the "annulus\ A = I 0 I 1 .The task remains open to augment the PDE (3.4) in A in such a "singular way\ that the original attractor A ; f is recovered on 0 with new boundary conditions .For systems, u 2 IR k , a similar problem arises.Even in the case of one-dimensional x, though, is is not yet clear how to properly recover A f on the enlarged interval 0 then.
For 6 = 1, the state space of our system (3.7),(3.8) is u = (u 0 ; ; u n ) 2 X = IR n+1 .A natural dissipation condition is u i f i (u i ; u i ; u i ) < 0 (3.11) for all i = 0; ; n, provided ju i j C.Here C is a large constant.Under boundary conditions (3.8) with j 0 j; j 1 j 1; (3.12) condition (3.11) ensures that kuk := max ju i j decreases to level C or below, eventually.If 0 or 1 violate condition (3.12), then max ju i j may grow inde nitely on the boundary, in spite of dissipation condition (3.11).Therefore we restrict attention to the region (3.12).
For Neumann condition = 1, it was argued in FiRo96], theorem 8.2, that the attractor class A dis ( ) for Jacobi systems (3.7), (3.8) coincides with the PDE attractor class A con ( ) := A( ) of our present theorem 1.1.Here A dis ( ) ranges over all Jacobi systems of any dimension n.For 0 1, we expect similar arguments to provide A dis ( ) = A con ( ) (3.13) to be -independent, by theorem 1.1.
Augmentation to i 2 f 2; 1; ; n + 1; n + 2g also seems a viable approach to discrete attractor classes.Consider the new left boundary u 2 ; u 1 , for example.
Let A n ( ) denote the attractor class for Jacobi systems (3.7), (3.8), this time with xed dimension n + 1.In view of theorem 1.1 and (3.13) it seems natural to ask whether A n ( ) can be independent of , at least for 0 1.More daringly: let A con n denote the set of attractor classes in A con of dimension at most n+1.Is it true, for 0 1 and at least for large n, that A n ( ) = A con n ? (3.17) In particular A n ( ) would not depend on , of course.
Aside from these constraints, and are arbitrary.Replacing i by n i we also observe symmetry for all , A n ( 0 ; 1 ) = A n ( 1 ; 0 ): Note that independence of A n+1 ( ) from might break down, at least for 0 & 1.In that case, the boundary condition (3.14) collapses to u 1 = 0, formally.This is equivalent to the Dirichlet attractor class A n (0; 1 ) of Jacobi systems in one lower dimension.
As a nal remark, we emphasize that periodic boundary conditions x 2 S 1 generate sets A con (per); A n (per) of attractor classes which are much richer than their colleagues A con ( ) = A con (sep) living in separated boundary conditions.In fact, the Neumann class can be shown to be contained in the periodic class = (1; 1), by re ection through the boundary and smoothing: A con (sep) A con (per); (3.23) again by theorem 1.1.As remarked in AF88], even for nonlinearities f = f(u; p) independent of x, time periodic rotating waves can arise in A con (per), which simply do not possess any counterpart in the gradient case A con (sep).In particular, Lyapunov functionals like V fail.A similar remark applies to the spatially discrete case A n (per) of cyclic Jacobi systems i (mod (n + 1)).Since re ection through the boundary for Neumann condition yields only an embedding A n (1; 1) 6 A 2n+1 (per); the characterization of attractor classes in the case of periodic boundary conditions remains wide open.

I
the shooting equation (2.1) becomes the hyperbolic linear equation _ ; p)-space, this linear equation induces a ow on the lines of boundary conditions l( ) : (1 )u p = 0:(2.8)Here = 1 carries the plus-sign, whereas = 0 requires a minus.The boundary condition parameters (x) are now considered to depend on x 2 I ; with their values at x = taken from the original boundary conditions (1.2).The ow induced by (3.14) Adding the two equations with real coe cients ; we obtain the right hand side of an augmentation _ 1 6 = 1.Similarly, for 0 ; 1 6 = 0; 1 and all we obtain A n ( ) A n+2 ( )(3.22) from (3.19), (3.20).