An Invariant Set Bifurcation Theory for Nonautonomous Nonlinear Evolution Equations
aa r X i v : . [ m a t h . D S ] J a n An Invariant Set Bifurcation Theory forNonautonomous Nonlinear EvolutionEquations
Xuewei Ju, Ailing Qi
Department of Mathematics, Civil Aviation University of ChinaTianjin 300300, China
Abstract.
In this paper we establish an invariant set bifurcation theory forthe nonautonomous dynamical system ( ϕ λ , θ ) X, H generated by the evolutionequation u t + Au = λu + p ( t, u ) , p ∈ H = H [ f ( · , u )] (0.1)on a Hilbert space X , where A is a sectorial operator, λ is the bifurcationparameter, f ( · , u ) : R → X is translation compact, f ( t, ≡ H [ f ] isthe hull of f ( · , u ). Denote by ϕ λ := ϕ λ ( t, p ) u the cocycle semiflow generatedby the equation. Under some other assumptions on f , we show that as theparameter λ crosses an eigenvalue λ ∈ R of A , the system bifurcates from0 to a nonautonomous invariant set B λ ( · ) on one-sided neighborhood of λ .Moreover, lim λ → λ H X α ( B λ ( p ) ,
0) = 0 , p ∈ P, where H X α ( · , · ) denotes the Hausdorff semidistance in X α (here X α ( α ≥ A ).Our result is based on the pullback attractor bifurcation on the local centralinvariant manifolds M λloc ( · ). Keywords.
Stability of pullback attractors; local invariant manifolds;nonautonomous invariant set bifurcations.
Invariant set bifurcation theory of autonomous dynamical systems has beenextremely well developed [1, 6, 12, 13, 15, 19–23, 25–27]. A relatively simplerbut important case is that of bifurcations from equilibria, including bifurca-tion to multiple equilibria (static bifurcation) and to periodic solutions (Hopf
This work was supported by the National Natural Science Foundation of China[11871368].
E-mail : [email protected] (X.W. Ju), [email protected] (A.L. Qi). e of an autonomous system changes from an attractor to a repelleron the local center manifold of the equilibrium when the bifurcation parameter λ crosses a critical value λ , then the system bifurcates a compact invariantset K which is an attractor of the system restricted to the center manifold.Chow and Hale [6] started to discuss stability and bifurcation phenomena as-sociated with more general invariant sets, e.g. periodic orbits. Using Conleyindex theory, Rybakowski [25] and Li and Wang [15] developed global bifur-cation theorems to discuss bifurcation phenomena of nonlinear autonomousevolution equations.However, except some relatively simple nonautonomous cases, there arefew papers studying the invariant set bifurcation for nonautonomous dynam-ical system. In [14] Langa et al. presented a collection of examples to il-lustrate bifurcation phenomena in nonautonomous ordinary differential equa-tions. In [5] Carvalho et al. studied the structure of the pullback attractor fora nonautonomous version of the Chafee-Infante equation, and investigated thebifurcations that this attractor undergoes as bifurcation parameter varies.Unlike autonomous dynamical systems for which forward dynamics is stud-ied, pullback dynamics is much more natural than the more familiar forwarddynamics for nonautonomous dynamical systems. This makes it very difficultto extend the invariant set bifurcation theory of autonomous systems to nonau-tonomous systems. Our approach in the paper is to treat the nonautonomoussystem as a cocycle semiflow over a suitable base space rather than a process.The biggest advantage of the cocycle semiflow framework is that in many casesthe base spaces are compact, while the default base space R (real number set)for processes is unbounded. Based on the compactness of the base spaces, wecan establish the equivalence between pullback attraction of cocycle semiflowand forward attraction of the associated autonomous semiflow. This devicemakes the dynamics of such a nonautonomous system appear like those of anautonomous system.Without the compactness assumption on the base spaces, the upper semi-continuity of global pullback attractors for nonautonomous systems was ob-tained in Caraballo and Langa [2]. However, compact forward invariant setsof the perturbed systems are required to guarantee the existence of perturbedpullback attractors. In the paper, we suppose that the base spaces of cocyclesemiflows considered are compact, which will require some restrictions on thenonlinearities. As a result, after introducing the notion of (local) pullback at-tractors (see Definition 2.5), we can establish a general result on the stability2f local pullback attractors as the perturbation parameter is varied. Based onthis result, a local pullback attractor bifurcation theory can be developed. Thiscan be regarded as a nonautonomous generalization of autonomous attractorbifurcation theory in [20]. Finally, we study the bifurcation of invariant sets forthe cocycle semiflow ϕ λ generated by the nonautonomous nonlinear evolutionequation (0.1). We first construct a local central invariant manifold M λloc ( · )for ϕ λ with λ near λ . Under further assumptions on f to ensure that 0 is apullback attractor for ϕ λ , we then restrict ϕ λ to M λloc ( · ) and obtain a pullbackattractor bifurcation on M λloc ( · ) as λ crosses λ . It leads to an invariant setbifurcation for ϕ λ . It is worth mentioning that if 0 is not an attractor but arepeller for ϕ λ , our result still holds. Denote by B λ ( · ) the bifurcated invariantset. We further know thatlim λ → λ H X α ( B λ ( p ) ,
0) = 0 , p ∈ P. This paper is organized as follows. In Section 2, we present respectivelysome basic facts in autonomous and nonautonomous dynamical systems whichwill be required in the rest of the work. Section 3 deals with the stability ofpullback attractors as bifurcation parameter varies. In Section 4, we establishan invariant set bifurcation theory for (0.1). We illustrate the main resultswith an example in Section 5. Finally, Section 6 contains the proofs of twopropositions presented earlier in the paper.
In this section we introduce some basic definitions and notions [7, 8].Let X be a complete metric space with metric d ( · , · ). Given M ⊂ X , wedenote M and int M the closure and interior of any subset M of X , respectively.A set U ⊂ X is called a neighborhood of M ⊂ X , if M ⊂ int U . For any ρ > X ( M, ρ ) := { x ∈ X : d ( x, M ) < ρ } the ρ -neighborhood of M in X , where d ( x, M ) = inf y ∈ M d ( x, y ).The Hausdorff semidistance in X is defined as H X ( M, N ) = sup x ∈ M d ( x, N ) , ∀ M, N ⊂ X. Let R + = [0 , ∞ ). A continuous mapping S : R + × X → X is called a semiflowon X , if it satisfies 3 ) S (0 , x ) = x for all x ∈ X ; and ii ) S ( t + s, x ) = S ( t, S ( s, x )) for all x ∈ X and t, s ∈ R + .Let S be a given semiflow on X . As usual, we will rewrite S ( t, x ) as S ( t ) x .A set B ⊂ X is called invariant (resp. positively invariant ) under S if S ( t ) B = B (resp. S ( t ) B ⊂ B ) for all t ≥ B and C be subsets of X . We say that B attracts C under S , iflim t →∞ H X ( S ( t ) C, B ) = 0 . Definition 2.1
A compact subset A ⊂ X is called an attractor for S , if itis invariant under S and attracts one of neighborhood of itself. It is well known that if U is a compact positively invariant set of S , then theomega-limit set ω ( U ) := T T ≥ S t ≥ T S ( t ) U is an attractor of S . The definitionof the attraction basin of the attractor and other properties of local attractorscan be found in [9, 18, 25]. A nonautonomous system consists of a “base flow” and a “cocycle semiflow”that is in some sense driven by the base flow.A base flow { θ t } t ∈ R := { θ ( t ) } t ∈ R is a flow on a metric space P such that θ t P = P for all t ∈ R . Definition 2.2 A cocycle semiflow ϕ on the phase space X over θ is acontinuous mapping ϕ : R + × P × X → X satisfying • ϕ (0 , p, x ) = x , • ϕ ( t + s, p, x ) = ϕ ( t, θ s p, ϕ ( s, p, x )) (cocycle property). Remark 2.3
If we replace R + by R in the above definition, then ϕ is called acocycle flow on X . We usually denote ϕ ( t, p ) x := ϕ ( t, p, x ). Then { ϕ ( t, p ) } t ≥ , p ∈ P can beviewed as a family of continuous mappings on X .For convenience in statement, a family of subsets B ( · ) := { B p } p ∈ P of X iscalled a nonautonomous set in X . As usual, we will rewrite B p as B ( p ),called the p - section of B ( · ). We also denote B the union of the sets B ( p ) × { p } ( p ∈ P ), i.e., B = [ p ∈ P B ( p ) × { p } . B is a subset of X × P .A nonautonomous set B ( · ) is said to be closed (resp. open, compact), ifeach section B ( p ) is closed (resp. open, compact) in X . A nonautonomous set U ( · ) is called a neighborhood of B ( · ), if B ( p ) ⊂ int U ( p ) for each p ∈ P .A nonautonomous set B ( · ) is said to be invariant (resp. forward invari-ant ) under ϕ if for t ≥ ϕ ( t, p ) B ( p ) = B ( θ t p ) , p ∈ P. (resp. ϕ ( t, p ) B ( p ) ⊂ B ( θ t p ) , p ∈ P. )Let B ( · ) and C ( · ) be two nonautonomous subsets of X . We say that B ( · ) pullback attracts C ( · ) under ϕ if for any p ∈ P ,lim t →∞ H X ( ϕ ( t, θ − t p ) C ( θ − t p ) , B ( p )) = 0 . Let ϕ be a given cocycle semiflow on X with driving system θ on base space P . The (autonomous) semiflow Φ := { Φ( t ) } t ≥ on Y := P × X , given byΦ( t )( p, x ) = ( θ t p, φ ( t, p ) x ) , t ≥ , is called the skew product semiflow associated to ϕ . The following funda-mental result studies the relationship between the pullback attraction of ϕ andattraction of Φ. The proof is given in Appendixes. Proposition 2.4
Let ( ϕ, θ ) X,P be a nonautonomous system, and let Φ be theskew-product flow associated to ϕ . Let K ( · ) and B ( · ) be two nonautonomoussets. Suppose P and K P := S p ∈ P K ( p ) ⊂ X are both compact. Then K ( · ) pullback attracts B ( · ) through ϕ if and only if K := S p ∈ P K ( p ) × { p } attracts B := S p ∈ P B ( p ) × { p } through Φ . Definition 2.5
Let ( ϕ, θ ) X,P be a nonautonomous system. A nonautonomousset A ( · ) is called a (local) pullback attractor for ϕ if it is compact, invariantand pullback attracts a neighborhood U ( · ) of itself. The local pullback attractor defined here, very similar to the notion of a pastattractor in Rasmussen [24], can be seen as a nature nonautonomous general-ization of the local attractor from the autonomous theory. Similar to the caseof autonomous systems, if U ( · ) is a compact forward invariant set of ϕ , thenthe omega-limit set ω ( U )( · ) defined as ω ( U )( ω ) = \ T ≥ [ t ≥ T ϕ ( t, θ − t ω ) U ( θ − t ω ) , ω ∈ Ω5s a pullback attractor of ϕ . For instance, consider the following simple systemon X = R : x ′ ( t ) = − x + p ( t ) x , p ∈ H [ h ] , (2.1)where h ( t ) = 2 + sin t and H [ h ] is its hull which is the closure for the uniformconvergence topology of the set of t -translates of h . The translation map θ t : H → H given by θ t p ( s ) = p ( t + s ) defines a flow on H . Then the uniquesolution of (2.1) define a cocycle flow on X given by ϕ ( t, p ) x = x ( t, p ; x ).Since 12 ddt x = − x + p ( t ) x ≤ − x − x ) < | x | ≤ /
2. Therefore [ − / , /
2] is a forward invariant set of ϕ and it is pullback attracted by the pullback attractor 0. It is worth notingthat 0 is only a local pullback attractor. Indeed,12 ddt x = − x + p ( t ) x ≥ − x + x > | x | ≥
2. It follows that 0 is only a local pullback attractor of ϕ .In general, it is difficult to define the attraction basin of a pullback attrac-tor. Fortunately, under the assumptions of Proposition 2.4, we can define thepullback attraction basin of a pullback attractor A ( · ). Specifically, we have Definition 2.6
Let ( ϕ, θ ) X,P be a nonautonomous system, and let Φ be theskew-product flow associated to ϕ . Suppose P is compact. Let A ( · ) be a pullbackattractor of ϕ such that A P := S p ∈ P A ( p ) is compact. Let B ( A ) = { ( x, p ) : A attracts ( x, p ) through Φ } be the attractor basin of A under Φ . Then the pullback attraction basin B ( A )( · ) of A ( · ) can be defined as B ( A ) = [ p ∈ P B ( A )( p ) × { p } . We now establish a result on the stability of pullback attractors under a smallperturbation. In fact, we prove a continuity result with respect to the Hausdorffsemidistance.Let X be a Banach space with norm k · k , and let A be a sectorial operatoron X . Pick a number a > σ ( A + aI ) > . A + aI . For each α ≥
0, define the fractional power space as X α = D (Λ α ), which is equipped with the norm k · k α defined by k x k α = k Λ α x k , x ∈ X α . Note that the definition of X α is independent of the choice of the number a . If A has compact resolvent, the inclusion X α ′ ֒ → X α is compact for α ′ > α ≥ ϕ λ ( λ ∈ R ) be a given cocycle semiflow on X with driving system θ on base space P . For δ >
0, denote I λ ( δ ) := ( λ − δ, λ + δ ). Assume that ϕ λ , λ ∈ I λ ( δ ) is a small perturbation of the given flow ϕ λ based on P . Letus make the following assumptions: (H1) : The base space P is compact. (H2) : For every T > B of X , we havelim λ → λ k ϕ λ ( t, p ) x − ϕ λ ( t, p ) x k α = 0 , (3.1)uniformly with respect to ( t, x ) ∈ [0 , T ] × B and p ∈ P .Under the assumptions (H1), (H2) , we can get a result on the stability ofpullback attractors. Theorem 3.1
Let A λ ( · ) := { A λ ( p ) } p ∈ P be an attractor of the cocycle semi-flow ϕ λ which pullback attracts a neighborhood U ( · ) of itself. Let U := [ p ∈ P U ( p ) × { p } and A λ := [ p ∈ P A λ ( p ) × { p } . Assume U is a compact neighborhood of A λ in Y = X × P , then under theassumptions (H1), (H2) , the following statements hold. ( a ) There exists a small δ > such that for each λ ∈ I λ ( δ ) , ϕ λ has apullback attractor A λ ( · ) such that lim λ → λ H X ( A λ ( p ) , [ p ∈H A λ ( p )) = 0 . (3.2)( b ) In addition, if U ( · ) is forward invariant, then lim λ → λ H X ( A λ ( p ) , A λ ( p )) = 0 . (3.3)7 roof. ( a ) By the compactness of U , we know that A λ P := S p ∈ P A λ ( p ) iscompact. Since A λ ( · ) pullback attracts U ( · ) and P is compact, by Proposition2.4, A λ attracts U through Φ λ . Since U is a neighborhood of A λ , one knowsthat A λ is an attractor of Φ λ . By the assumption (H2) , for any compact set B ⊂ X , we have thatlim λ → λ H Y (Φ λ ( t )( x, p ) , Φ λ ( t )( x, p )) = lim λ → λ k ϕ λ ( t, p ) x − ϕ λ ( t, p ) x k α = 0 (3.4)uniformly with respect to t ∈ [0 , T ] and ( x, p ) ∈ B × P . Then by the stability ofthe autonomous attractors [17, Theorem 4.1], there exists a δ > p ∈ P ) such that for each λ ∈ I λ ( δ ) := ( λ − δ, λ + δ ), Φ λ has an attractor A λ contained in U . Moreover,lim λ → λ H Y ( A λ , A λ ) = 0 . (3.5)Write A λ as S p ∈ P A λ ( p ) × { p } , λ ∈ I λ ( δ ). Using Proposition 2.4 again, wehave that A λ ( · ) pullback attracts U ( · ) through ϕ λ , i.e., A λ ( · ) is a pullbackattractor of ϕ λ . (3.2) is a direct consequence of (3.5).To complete the proof of ( b ), we shall prove (6.16) by contradiction. Thus,let us assume that there exist σ > λ j → λ , as j → ∞ , x j ∈ A λ j ( p ) such that d X ( x j , x ) > σ, for all x ∈ A λ ( p ) . (3.6)Note that x j = ϕ λ j ( n, θ − n p ) x nj , for some x nj ∈ A λ j ( θ − n p ) . Similar to the argument in ( a ), we can assume that A λ j ( p ) ⊂ U ( p ), thus x j ∈ U ( p ). By the compactness of U ( p ), there exists a subsequence of x j (stilldenoted by x j ) which converges to some x ∈ U ( p ). Now, for each fixed n wehave x nj ∈ U ( θ − n p ) so that there is a further subsequence of x nj (still denotedby x nj ) which converges to some x n ∈ U ( θ − n p ). On the other hand, for anygiven ν >
0, we can use the assumption (H2) and the continuity of ϕ ( n, θ − n p )to show that for j large enough, d ( ϕ λ j ( n, θ − t p ) x nj , ϕ λ ( n, θ − t p ) x n ) ≤ d ( ϕ λ j ( n, θ − t p ) x nj , ϕ λ ( n, θ − t p ) x nj )+ d ( ϕ λ ( n, θ − t p ) x nj , ϕ λ ( n, θ − t p ) x n ) ≤ ν + ν. Then, for each fixed n ∈ N , x = lim j →∞ x j = lim j →∞ ϕ λ j ( n, θ − n p ) x nj = ϕ λ ( n, θ − n p ) x n . U ( p ) is forward invariant, we have x ∈ \ n ∈ N ϕ λ ( n, θ − n p ) U ( θ − n p ) = A λ ( p ) , which contradicts (3.6). The proof is complete. (cid:3) The main contribution of Theorem 3.1 is the existence of pullback attractor A λ ( · ) for ϕ λ as λ near λ , while the argument of the upper semicontinuity ofpullback attractors is an adaptation of that of [2].The conditions of the following results may be easier to be verified in ap-plications. Corollary 3.2
Let A λ ( · ) := { A λ ( p ) } p ∈ P be an attractor of the cocycle semi-flow ϕ λ and U ⊂ X is a compact forward invariant neighborhood of A λ ( · ) .Then under the assumptions (H1), (H2) , there exists a small δ > such thatfor each λ ∈ I λ ( δ ) , ϕ λ has a pullback attractor A λ ( · ) satisfying lim λ → λ H X ( A λ ( p ) , A λ ( p )) = 0 . Based on the general result of the stability of pullback attractors, in the sectionwe can establish some results on invariant set bifurcation for nonautonomousdynamical systems.
From now on, we assume X is a Hilbert space with inner product ( · , · ). Wewill consider and study invariant set bifurcation of the evolution equation u t + Au = λu + f ( t, u ) (4.1)on X , where λ ∈ R is a bifurcation parameter, the nonlinearity f : R × X α → X is bounded continuous mapping satisfying (F1) f ( t, u ) = o ( k u k α ) , as k u k α → t ∈ R . Moreover, there is β > f ( t, u ) , u ) ≤ − β · κ ( u ) (4.3)for t ∈ R and u ∈ X α , where κ : X → R + is a nonnegative functionsatisfying that κ ( u ) = 0 if and only if u = 0.9enote k ( ρ ) the Lipschitz constant of f ( t, · ) in B X α ( ρ ). Then by (4.2),lim ρ → k ( ρ ) = 0and k f ( t, u ) − f ( t, u ) k ≤ k ( ρ ) k u − u k α , ∀ u , u ∈ B X α ( ρ ) . (4.4)Denote C b ( R , X ) the set of bounded continuous functions from R to X .Equip C b ( R , X ) with either the uniform convergence topology generated bythe metric r ( h , h ) = sup t ∈ R k h ( t ) − h ( t ) k , or the compact-open topology generated by the metric r ( h , h ) = ∞ X n =1 n · max t ∈ [ − n,n ] k h ( t ) − h ( t ) k t ∈ [ − n,n ] k h ( t ) − h ( t ) k . Then C b ( R , X ) is a complete metric space.Let f ( · , u ) ∈ C b ( R , X ) be the function in (4.1). Define the hull of f ( · , u ) asfollows H := H [ f ( · , u )] = { f ( τ + · , u ) : τ ∈ R } C b ( R ,X ) . In application, f ( · , u ) is often taken as a periodic function, quasiperiodic func-tion, almost periodic function, local almost periodic function [7, 16] or uni-formly almost automorphic function [28]. In this case, the hull H is a compact metric space. Accordingly, the translation group θ on H is given by θ t p ( · , u ) = p ( t + · , u ) , t ∈ R , p ∈ H . Instead of (3.2), we will consider the more general cocycle system in X α (where α ∈ [0 , u t + Au = λu + p ( t, u ) , p ∈ H . (4.5) Proposition 4.1 [10] Let A and p be given as above. Assume that p is locallyH ¨ o lder continuous in t . Then for each u ∈ X α , there is a T > t such that (4 . has a unique solution u ( t ) = u λ ( t, t ; u , p ) on [ t , T ) satisfying u ( t ) = e − A ( t − t ) x + Z tt e − A ( t − s ) [ λu ( s ) + p ( s, u ( s ))] ds, t ∈ [ t , T ) . (4.6)For convenience, from now on we always assume that the unique solution(4 .
6) is globally defined. Define ϕ λ ( t, p ) u := u λ ( t, u, p ) , u ∈ X α , p ∈ H . Then ϕ λ is a cocycle semiflow on X α driven by the base flow θ on H . Notethat for each p ∈ H , u ( t ) is a p -solution of ϕ λ on an interval J if and only if it solves the equation (4.5) on J . 10 .2 Local invariant manifolds Let λ ∈ R be an isolated eigenvalue of A . Suppose that (F2) there is a η > σ ( A ) ∩ { z ∈ C : λ − η < Re z < λ + η } = λ . Denote A λ := A − λ . Then for λ ∈ I λ ( η/
4) := ( λ − η/ , λ + η/ σ ( A λ ) has a decomposition σ ( A λ ) = σ c ∪ σ + ∪ σ − , where σ c = { λ − λ } , σ + = σ ( A λ ) ∩ { Re λ > } and σ − = σ ( A λ ) ∩ { Re λ < } . Accordingly, the space X has a direct sum decomposition: X = X c ⊕ X + ⊕ X − .Denote X ± = X + ⊕ X − and X αi := X i ∩ X α , i = c, + , − , ± . Note that X αc is finite dimensional.Under the assumptions on A and f , we can construct a local invariantmanifold for ϕ λ , λ ∈ I λ ( η/ Proposition 4.2
Suppose the assumptions (F1), (F2) hold. Then there ex-ists ̺ > such that the cocycle semiflow ϕ λ , λ ∈ I λ ( η/ has a local invariantmanifold M λloc ( · ) := {M λloc ( p ) } p ∈H in X α which is represented as M λloc ( p ) = { y + ξ λp ( y ) : y ∈ B X αc ( ̺ ) } , where ξ · p ( · ) : I λ ( η/ × B X αc ( ̺ ) → X α ± is a Lipschitz continuous mappingsatisfying that ξ λp (0) = 0 and k ξ λp ( y ) − ξ λp ( z ) k α ≤ L k y − z k α (4.7) and k ξ λ p ( y ) − ξ λ p ( y ) k α ≤ L | λ − λ | , (4.8) where L > is independent of p ∈ P and λ ∈ I λ ( η/ , and L > isindependent of p ∈ P and y ∈ B X αc ( ̺ ) . The long proof of the above proposition is given in Appendixes.11 .3 Invariant set bifurcation
Firstly, let us restrict the equation (4.5) on the invariant manifold M λloc ( · ), λ ∈ I λ ( η/ y t + ( λ − λ ) y = p ( t, y + ξ λθ t p ( y )) , y ∈ B X αc ( ̺ ) , p ∈ H . (4.9)Denote φ λ the cocycle flow on B X αc ( ̺ ) with driving system θ on the base space H generated by (4.9).We first say that the condition (H2) (in Section 3) holds for the cocycleflow φ λ , λ ∈ I λ ( η/ Lemma 4.3
For every
T > , we have lim λ → λ k φ λ ( t, p ) y − φ λ ( t, p ) y k α = 0 , (4.10) uniformly with respect to ( t, y ) ∈ [0 , T ] × B X αc ( ̺ ) and p ∈ P . Proof.
For λ ∈ I λ ( η/ y λ ( t ) := φ λ ( t, p ) y and v ( t ) = y λ ( t ) − y λ ( t ),then v satisfies v t + ( λ − λ ) y λ = p ( t, y λ + ξ λθ t p ( y λ )) − p ( t, y λ + ξ λ θ t p ( y λ )) . (4.11)Note that k y λ k ≤ ρ and k p ( t, y λ + ξ λθ t p ( y λ )) − p ( t, y λ + ξ λ θ t p ( y λ )) k≤ k ( ρ ) (cid:0) ( L + 1) k v k α + L | λ − λ | (cid:1) ≤ C ′ (cid:0) k v k + ( λ − λ ) (cid:1) for some constant C ′ , (4.12)where ρ > u ∈ M λloc ( · ), which is independent of λ by (4.6).Taking the inner product of the equation (4.11) with v and using (4.12) toobtain that there is a constant C > λ such that ddt k v k ≤ C (cid:0) k v k + ( λ − λ ) (cid:1) . Applying the classical Gronwall lemma to get that k v ( t ) k ≤ (cid:0) e Ct − (cid:1) ( λ − λ ) , which completes the proof. (cid:3) Lemma 4.4
Under the assumptions (F1), (F2) , y = 0 is locally asymptoti-cally stable for φ λ . Therefore is a pullback attractor of φ λ . roof. Since X αc is finite dimensional, all the norms on X αc are equivalent.Hence for convenience, we equip X αc the norm k · k of X in the followingargument.Note that φ λ is generated by the equation y t = p ( t, y + ξ λ θ t p ( y )) , y ∈ B X αc ( ̺ ) , p ∈ H . (4.13)Taking the inner product of the equation (4.13) with y + ξ λ θ t p ( y ) in X , usingthe fact ( y, ξ λ θ t p ( y )) = 0 and the assumption (F1) , it yields12 ddt k y k = (cid:0) p (cid:0) t, y + ξ λ θ t p ( y ) (cid:1) , y + ξ λ θ t p ( y ) (cid:1) ≤ − β · κ (cid:0) y + ξ λ θ t p ( y ) (cid:1) . (4.14)It is clear that κ (cid:0) y + ξ λ θ t p ( y ) (cid:1) = 0 if and only if y = 0. Therefore lim t →∞ k y k =0 . The proof is complete. (cid:3)
Henceforth we will suppose that (F3)
The hull H is a compact metric space.We then obtain a pullback attractor bifurcation theory for φ λ as λ crosses λ . Theorem 4.5
Under the assumptions (F1), (F2) and (F3) , the cocycle semi-flow φ λ bifurcates from (0 , λ ) a pullback attractor A λ ( · ) for λ > λ , and lim λ → λ +0 H X αc ( A λ ( p ) , { } ) = 0 . (4.15) Proof.
Recall from Lemma 4.4 that 0 is a pullback attractor for φ λ andit pullback attracts B X αc ( ̺ ) for sufficiently small ̺ >
0. The bounded setB X αc ( ̺ ) ⊂ X αc is compact due to X αc being finite dimensional. Moreover,B X αc ( ̺ ) is forward invariant under φ λ . Then by Corollary 3.2, there is a η ′ ∈ (0 , η/
8) such that for each λ ∈ I λ ( η ′ ), the cocycle semiflow φ λ has apullback attractor A λ ( · ) and (4.23) holds.In the following, we prove that 0 / ∈ A λ ( · ) for λ ∈ I + λ ( η ′ ) := ( λ , λ + η ′ ),which completes the proof.Let λ ∈ I + λ ( η ′ ) be fixed, and let w ( t ) = y ( − t ). Then w ( t ) satisfies w t − ( λ − λ ) w = − p ( − t, w + ξ λθ − t p ( w )) . (4.16)Taking the inner product of the equation (4.16) with w in X α , we have12 ddt k w k − ( λ − λ ) k w k = − ( p ( t, w + ξ λθ − t p ( w )) , w ) . (4.17)13ince k p ( t, u ) k ≤ k ( k u k α ) k u k α and k ξ λθ − t p ( w ) k α ≤ L k w k α , we have k p ( − t, w + ξ λθ − t p ( w )) k ≤ k ( k w + ξ λθ − t p ( w ) k α ) k w + ξ λθ − t p ( w ) k α ≤ k ( k w + ξ λθ − t p ( w ) k α ) ( k w k α + L k w k α ) ≤ k ( k w + ξ λθ − t p ( w ) k α ) · (1 + L ) k w k α ≤ [(1 + L ) ck ( k w + ξ λθ − t p ( w ) k α )] · k w k≤
12 ( λ − λ ) k w k , for sufficiently small k w k α , (4.18)where c > α . We get from (4.17) and (4.18) that ddt k w k ≤ − ( λ − λ ) k w k for sufficiently small k w k α , which shows for fixed λ ∈ I + λ ( η ′ ), 0 locally asymp-totically stable for the cocycle flow generated by the equation (4.16). In otherwords, 0 is a repeller of φ λ when λ ∈ I + λ ( η ′ ) and repels a neighborhood of 0 in X αc . This implies that 0 / ∈ A λ ( · ), λ ∈ I + λ ( η ′ ). The proof is complete. (cid:3) We are now in position to give and prove the main result of this paperconcerning the invariant set bifurcation of ϕ λ . Theorem 4.6
Under the assumptions (F1), (F2) and (F3) , the cocycle semi-flow ϕ λ bifurcates from (0 , λ ) an invariant compact set B λ ( · ) for λ > λ , andfor each p ∈ P , lim λ → λ +0 H X ( B λ ( p ) , { } ) = 0 . (4.19) Proof.
Let A λ ( · ) be the bifurcated attractor obtained in Theorem 4.5. Define B λ ( · ) by B λ ( p ) = { y + ξ λp ( y ) : y ∈ A λ ( p ) } , p ∈ H . (4.20)We know from Theorem 4.5 that 0 / ∈ B λ ( · ) and B λ ( · ) ⊂ M λloc ( · ). Basedon the compactness of A λ ( p ) and the continuity of ξ λp ( y ) in y , we can directlyderive the compactness of B λ ( p ). So B λ ( · ) is compact.We claim that B λ ( · ) is invariant under ϕ λ . Indeed, let p ∈ P and y + ξ λp ( y ) ∈ B λ ( p ). Since φ λ ( t, p ) y ∈ A λ ( θ t p ), t ≥
0, by the invariance of M λloc ( · ), we have ϕ λ ( t, p )( y + ξ λp ( y )) = φ λ ( t, p ) y + ξ λθ t p ( φ λ ( t, p ) y ) ∈ B λ ( θ t p ) , which shows ϕ λ ( t, p ) B λ ( p ) ⊂ B λ ( θ t p ) , t ≥ .
14n the other hand, for any y + ξ λθ t p ( y ) ∈ B λ ( θ t p ), t ≥
0. Using the invarianceof A λ ( · ) and M λloc ( · ), there is a y ′ ∈ A λ ( p ) such that y = φ λ ( t, p ) y ′ . Then y + ξ λθ t p ( y ) = φ λ ( t, p ) y ′ + ξ λθ t p ( φ λ ( t, p ) y ′ )= ϕ λ ( t, p )( y ′ + ξ λθ t p ( y ′ )) ∈ ϕ ( t, p ) B λ ( p ) , which shows B λ ( θ t p ) ⊂ ϕ λ ( t, p ) B λ ( p ) , t ≥ . Therefore B λ ( · ) is invariant under ϕ λ .Finally, (4.19) is an immediately consequence of (4.23) and (4.7). (cid:3) We now give a result which parallels Theorem 4.6.
Corollary 4.7
Let the assumptions (F1),(F2),(F3) hold, but replace (4.3) by the assumption that ( f ( t, u ) , u ) ≥ β · κ ( u ) . Then the cocycle semiflow ϕ λ bifurcates from (0 , λ ) an invariant compact set B λ ( · ) for λ < λ , and for each p ∈ P , lim λ → λ − H X ( B λ ( p ) , { } ) = 0 . (4.21) Proof.
Let λ ∈ I λ ( η/ z t − ( λ − λ ) z = − p ( − t, z + ξ λθ − t p ( z )) , z ∈ B X αc ( ̺ ) , p ∈ H . (4.22)Denote by φ − λ be the cocycle flow generated by (4.22). Then φ − λ be the inverseflow of φ λ .Repeating the argument of Lemma 4.3, Lemma 4.4 and Theorem 4.5 (re-placing φ λ by φ − λ ) to show φ − λ bifurcates from (0 , λ ) a pullback attractor R λ ( · )for λ < λ , and lim λ → λ − H X αc ( R λ ( p ) , { } ) = 0 . (4.23)It is clear that R λ ( · ) is also an invariant set of φ λ . Define a set B λ ( · ) by B λ ( p ) = { y + ξ λp ( y ) : y ∈ R λ ( p ) } , p ∈ H . Similar to Theorem 4.6, we can show B λ ( · ) is an invariant set of ϕ λ and (4.21)holds. (cid:3) An example
Consider the nonautonomous system: ( u t − ∆ u = λu ± h ( t ) u , t > , x ∈ Ω; u = 0 , t > , x ∈ ∂ Ω , (5.1)where Ω is a bounded domain in R with smooth boundary, h is a functionsuch that h ( t ) ≥ δ > δ > A the operator − ∆ associated with the homogeneous Dirich-let boundary condition. Then A is a sectorial operator on X = L (Ω) withcompact resolvent, and D ( A ) = H (Ω) T H (Ω). Note that A has eigenvalues0 < µ < µ < · · · < µ k < · · · . Denote V = H (Ω). By ( · , · ) and | · | we denotethe usual inner product and norm on H , respectively. The inner product andnorm on V , denoted by (( · , · )) and k · k , respectively, are defined as(( u, v )) = Z Ω ∇ u · ∇ v d x, k u k = (cid:18)Z Ω |∇ u | d x (cid:19) / for u, v ∈ V .The system (5 .
1) can be written into an abstract equation on X : u t + Au = λu ± h ( t ) u . Define the hull H := H [ h ( · ) u ]. By the assumption on h , it is clear that( p ( t, u ) , u ) ≥ δ Z Ω u dx, p ∈ H . Consider the cocycle system: u t + Au = λu ± p ( t, u ) , p ∈ H . (5.2)Denote ϕ ± λ := ϕ ± λ ( t, p ) u the cocycle semiflow on H (Ω) driven by the base flow(translation group) θ on H .Since all the hypotheses in the main theorem above are fulfilled, we obtainsome interesting results concerning the dynamics of the perturbed system. Inparticular, Theorem 5.1
Suppose H is compact. Then the cocycle semiflow ϕ − λ (resp. ϕ + λ ) bifurcates from (0 , µ k ) , k = 1 , , · · · an invariant compact set B − λ ( · ) for λ > λ (resp. B + λ ( · ) for λ < λ ) and for each p ∈ P , lim λ → λ +0 H X ( B − λ ( p ) , { } ) = 0 . (cid:0) resp. lim λ → λ − H X ( B + λ ( p ) , { } ) = 0 . (cid:1) Appendixes ϕ and the attraction of Φ Proof of Proposition 2.4. Necessity:
By the compactness of P , one findsthat lim t →∞ H Y (Φ( t ) B , P × K P ) = lim t →∞ H X ( ϕ ( t, p ) B ( p ) , K P ) ≤ lim t →∞ sup p ∈ P H X ( ϕ ( t, p ) B ( p ) , K P )= lim t →∞ sup p ∈ P H X ( ϕ ( t, θ − t p ) B ( θ − t p ) , K P )=0 . This means the compact set P × K P attracts B through Φ. Therefore theomega-limit set ω ( B ) of B exists and attracts B .In the following, we prove ω ( B ) ⊂ K , which completes the necessity. Forthis purpose, define a nonautonomous set ˜ B ( · ) as follows˜ B ( p ) := [ s ≥ ϕ ( s, θ − s p ) B ( θ − s p ) , p ∈ P. It is clear that B ( · ) ⊂ ˜ B ( · ). We first say ˜ B ( · ) is forward invariant. Indeed, forany t ≥ p ∈ P , ϕ ( t, p ) ˜ B ( p ) = ϕ ( t, p ) [ s ≥ ϕ ( s, θ − s p ) B ( θ − s p ) ⊂ [ s ≥ ϕ ( t, p ) ◦ ϕ ( s, θ − s p ) B ( θ − s p )= [ s ≥ ϕ ( t + s, θ − ( t + s ) ◦ θ t p ) B ( θ − ( t + s ) ◦ θ t p ) ⊂ [ s ≥ ϕ ( s, θ − s ◦ θ t p ) B ( θ − s ◦ θ t p ) = ˜ B ( θ t p ) . (6.1)So ˜ B ( · ) is forward invariant, which implies the omega-limit set ω ( ˜ B )( · ) of ˜ B ( · )17s the maximal invariant set in ˜ B ( · ). Furthermore, for each p ∈ P , ω ( ˜ B )( p ) = \ τ ≥ [ t ≥ τ ϕ ( t, θ − t p ) ˜ B ( θ − t p )= \ τ ≥ [ t ≥ τ ϕ ( t, θ − t p ) [ s ≥ ϕ ( s, θ − ( s + t ) p ) B ( θ − ( s + t ) p )= \ τ ≥ [ t ≥ τ ϕ ( t, θ − t p ) [ s ≥ ϕ ( s, θ − ( s + t ) p ) B ( θ − ( s + t ) p )= \ τ ≥ [ t ≥ τ [ s ≥ ϕ ( t, θ − t p ) ◦ ϕ ( s, θ − ( s + t ) p ) B ( θ − ( s + t ) p )= \ τ ≥ [ t ≥ τ [ s ≥ ϕ ( t + s, θ − ( s + t ) p ) B ( θ − ( s + t ) p )= \ τ ≥ [ t ≥ τ ϕ ( t, θ − t p ) B ( θ − t p )= ω ( B )( p ) , where the third “=” holds since for each fixed t ≥ p ∈ P , ϕ ( t, θ − t p )is a continuous map on X . It follows that ω ( B )( · ) is the maximal forwardinvariant set in ˜ B ( · ). Therefore C := S p ∈ P (cid:0) { p } × ω ( B )( p ) (cid:1) is the maximalinvariant set in ˜ B := S p ∈ P (cid:0) { p } × ˜ B ( p ) (cid:1) . By the forward invariance of ˜ B ( · ), ϕ ( t ) ˜ B = ϕ ( t ) [ p ∈ P (cid:0) { p } × ˜ B ( p ) (cid:1) ⊂ [ p ∈ P ϕ ( t ) (cid:0) { p } × ˜ B ( p ) (cid:1) = [ p ∈ P (cid:0) { θ t p } × ϕ ( t, p ) ˜ B ( p ) (cid:1) ⊂ (by (6.17)) ⊂ [ p ∈ P (cid:0) { θ t p } × ˜ B ( θ t p ) (cid:1) = ˜ B , t ≥ , i.e. ˜ B is positively invariant under ϕ . Then ω ( ˜ B ) is also the maximal invariantset in ˜ B . Therefore ω ( B ) ⊂ ω ( ˜ B ) = C . (6.2)Finally, by the assumption that K ( · ) attracts B ( · ), one knows that ω ( B )( · ) ⊂ K ( · ), and thus C ⊂ K , which shows ω ( B ) ⊂ K . ufficiency: In a very similar way as above, we can prove the sufficiency.By the compactness of P ,lim t →∞ H X ( ϕ ( t, θ − t p ) B ( θ − t p ) , K P )] ≤ lim t →∞ sup p ∈ P H X ( ϕ ( t, p ) B ( p ) , K P )= lim t →∞ sup p ∈ P H Y (Φ( t ) B , P × K P )= lim t →∞ H Y (Φ( t ) B , P × K P )=0 , which implies ω ( B )( · ) exists and pullback attracts B ( · ).To complete the proof, it suffices to show ω ( B )( · ) ⊂ K ( · ). We first definea set ˆ B = [ s ≥ Φ( s ) B . Then ˆ B is positively invariant and ω ( ˆ B ) = ω ( B ) . This implies that ω ( B ) is the maximal invariant set in ˆ B . Write ω ( B ) := S p ∈ P { p } × C ( p ), then C ( · ) is the maximal invariant set in ˆ B ( · ), where ˆ B ( · ) isthe set defined by ˆ B := S p ∈ P { p } × ˆ B ( p ). By the positive invariance of ˆ B , onealso knows that ˆ B ( · ) is forward invariant. This implies ω ( ˆ B )( · ) is the maximalinvariant set in ˆ B ( · ). We then have that ω ( B )( · ) ⊂ ω ( ˆ B )( · ) = C ( · ). We learnfrom the condition ω ( B ) ⊂ K that C ( · ) ⊂ K ( · ). In summary, ω ( B )( · ) ⊂ K ( · ),which completes the sufficiency. (cid:3) Let
M >
0. For µ ≥
0, define a Banach space as X µ = (cid:26) u ∈ C ( R ; X α ) : sup t ∈ R e − µ | t | k x ( t ) k α ≤ M (cid:27) , which is equipped with the norm k · k X µ , k x k X µ = sup t ∈ R e − µ | t | k x ( t ) k α , ∀ x ∈ X αµ . Let A λ = A − λ . Write σ ( A λ ) = σ − ∪ σ c ∪ σ + , where σ c = { λ − λ } ,σ − = σ ( A λ ) ∩ { Re λ < } , σ + = σ ( A λ ) ∩ { Re λ > } . X has a direct sum de-composition: X = X − ⊕ X c ⊕ X + . Denote X ± := X − S X + . Note that each X i , i = − , + , ± , c is independent of λ . LetΠ i : X → X i , i = − , + , ± , c be the projection from X to X i . Denote A λi = A λ | X i . By the assumption (F2) ,we deduce that if λ ∈ ( λ − η/ , λ + η/
4) then for α ∈ [0 , k A α e − A λ − t k ≤ e η t , k e − A λ − t k ≤ e − η t , t ≤ , (6.3) k A α e − A λ + t Π + A − α k ≤ e − η t , k A α e − A λ + t k ≤ t − α e − η t , t > , (6.4) k A α e − A λc t k ≤ e η | t | , k e − A λc t k ≤ e η | t | , t ∈ R . (6.5) Proof of Proposition 4.2.
Let χ : R → R be a smooth function suchthat χ ( z ) = ( , | z | ≤ / , | z | ≥ . For ρ >
0, one can then define a smooth function such that p ρ ( t, u ) = χ (cid:18) k u k α ρ (cid:19) p ( t, u ) . Select suitable χ such that k p ρ ( t, u ) − p ρ ( t, v ) k ≤ k ( ρ ) k u − v k , (6.6)where k ( ρ ) is the local Lipschitz constant of f given in (4.4). Instead of (4.5),we consider the truncated system u t + Au = λu + p ρ ( t, u ) , p ∈ H . (6.7)Suppose that ρ is so small that M ρ := k ( ρ ) Z ∞ (cid:0) τ − α (cid:1) e − η τ dτ < . (6.8)Let u ∈ X η/ . By simple computations, we know that u is the solution of(6.7) if and only if it solves the integral equation u ( t ) = e − A λc t Π c u (0) + Z t e − A λc ( t − τ ) Π c p ρ ( τ, u ( τ )) dτ + Z t −∞ e − A λ + ( t − τ ) Π + p ρ ( τ, u ( τ )) dτ − Z ∞ t e − A λ − ( t − τ ) Π − p ρ ( τ, u ( τ )) dτ. (6.9)20ake a ˜ ̺ > ̺ ≤ (1 − M ρ ) M. (6.10)Let p ∈ H and λ ∈ I λ ( η/
8) be fixed. For each y ∈ B X αc ( ˜ ̺ ), one can use therighthand side of equation (6 .
9) to define a contraction mapping T := T y on X η/ as follows: T u ( t ) = e − A λc t y + Z t e − A λc ( t − τ ) Π c p ρ ( τ, u ( τ )) dτ + Z t −∞ e − A λ + ( t − τ ) Π + p ρ ( τ, u ( τ )) dτ − Z ∞ t e − A λ − ( t − τ ) Π − p ρ ( τ, u ( τ )) dτ. We first verify that T maps X η/ into itself.For notational convenience, we write0 ∧ t = min { , t } , ∨ t = max { , t } , for t ∈ R . Let u ∈ X η/ . By (6.3)-(6.5) and (6.6) we have kT u ( t ) k α ≤ e η | t | k y k α + Z ∨ t ∧ t e η | t − τ | k ( ρ ) k u ( τ ) k α dτ + Z t −∞ ( t − τ ) − α e − η ( t − τ ) k ( ρ ) k u ( τ ) k α dτ + Z ∞ t e η ( t − τ ) k ( ρ ) k u ( τ ) k α dτ. (6.11)It is trivial to verify that e − η | t | Z ∨ t ∧ t e η | t − τ | k ( ρ ) k u ( τ ) k α dτ = Z ∨ t ∧ t e − η | t − τ | (cid:2) e − η | τ | k ( ρ ) k u ( τ ) k α (cid:3) dτ. (6.12)Observing that e − η | t | = e − η | ( t − τ )+ τ | ≤ e η | t − τ | e − η | τ | ,
21y (6.10), (6.11) and (6.12) we find that e − η | t | kT x ( t ) k α ≤ e − η | t | k y k α + Z ∨ t ∧ t e − η | t − τ | (cid:2) e − η | τ | k ( ρ ) k u ( τ ) k α (cid:3) dτ + Z t −∞ ( t − τ ) − α e η | t − τ | e − η ( t − τ ) (cid:2) e − η | τ | k ( ρ ) k u ( τ ) k α (cid:3) dτ + Z ∞ t e η | t − τ | e η ( t − τ ) (cid:2) e − η | τ | k ( ρ ) k u ( τ ) k α (cid:3) dτ = e − η | t | k y k α + Z ∨ t ∧ t e − η | t − τ | (cid:2) e − η | τ | k ( ρ ) k u ( τ ) k α (cid:3) dτ + Z t −∞ ( t − τ ) − α e − η ( t − τ ) (cid:2) e − η | τ | k ( ρ ) k u ( τ ) k α (cid:3) dτ + Z ∞ t e η ( t − τ ) (cid:2) e − η | τ | k ( ρ ) k u ( τ ) k α (cid:3) dτ ≤ k y k α + M ρ k u k X η/ ≤ M, ∀ t ∈ R . (6.13)Hence T u ∈ X η/ .Next, we check that T is contractive. Indeed, in a quite similar fashion asabove, it can be shown that for any u, u ′ ∈ X η/ , e − η | t | kT u ( t ) − T u ′ ( t ) k α ≤ k ( ρ ) Z ∨ t ∧ t e − η | t − τ | (cid:16) e − η | τ | k u ( τ ) − u ′ ( τ ) k α (cid:17) dτ + k ( ρ ) Z t −∞ ( t − τ ) − α e − η ( t − τ ) (cid:16) e − η | τ | k u ( τ ) − u ′ ( τ ) k α (cid:17) dτ + k ( ρ ) Z ∞ t e η ( t − τ ) (cid:16) e − η | τ | k u ( τ ) − u ′ ( τ ) k α (cid:17) dτ ≤ (cid:18) k ( ρ ) Z ∞ (cid:0) τ − α (cid:1) e − η τ dτ (cid:19) k u − u ′ k U η/ := M ρ k u − u ′ k X η/ , ∀ t ∈ R . (6.14)Thus kT u − T u ′ k X η/ ≤ M ρ k u − u ′ k X η/ . The conditon (6.8) then asserts that T is contractive.Thanks to the Banach fixed-point theorem, T has a unique fixed point γ yp,λ ∈ X η/ which is precisely a full solution of (4 .
5) with Π c γ yp,λ (0) = y and22olves the integral equation γ yp,λ ( t ) = e − A λc t y + Z t e − A λc ( t − τ ) Π c p ρ ( τ, γ yp,λ ( τ )) dτ + Z t −∞ e − A λ + ( t − τ ) Π + p ρ ( τ, γ yp,λ ( τ )) dτ − Z ∞ t e − A λ − ( t − τ ) Π − p ρ ( τ, γ yp,λ ( τ )) dτ. (6.15)For y, z ∈ B X αc ( ˜ ̺ ) and t ∈ R , similar to (6.14), by (6 .
15) we have e − η | t | k γ yp,λ ( t ) − γ zp,λ ( t ) k α ≤ e − η | t | k y − z k α + k ( ρ ) Z ∨ t ∧ t e − η | t − τ | (cid:0) e − η | τ | k γ yp,λ ( τ ) − γ zp,λ ( τ ) k α (cid:1) dτ + k ( ρ ) Z t −∞ ( t − τ ) − α e − η ( t − τ ) (cid:0) e − η | τ | k γ yp,λ ( τ ) − γ zp,λ ( τ ) k α (cid:1) dτ + k ( ρ ) Z ∞ t e η ( t − τ ) (cid:0) e − η | τ | k γ yp,λ ( τ ) − γ zp,λ ( τ ) k α (cid:1) dτ ≤ k y − z k α + M ρ k γ yp,λ − γ zp,λ k X η/ . Hence k γ yp,λ − γ zp,λ k X η/ ≤ − M ρ k y − z k α , which implies that k γ yp,λ (0) − γ zp,λ (0) k α ≤ − M ρ k y − z k α . (6.16)For each p ∈ H and λ ∈ I λ ( η/ X αc to X αus as ξ λp ( y ) := Z −∞ e A λ + τ Π + p ρ ( τ, γ yp,λ ( τ )) dτ − Z ∞ e A λ − τ Π − p ρ ( τ, γ yp,λ ( τ )) dτ, y ∈ B X αc ( ˜ ̺ ) . (6.17)Setting t = 0 in (6.15) leads to γ yp,λ (0) = y + ξ λp ( y ) , y ∈ B X αc ( ˜ ̺ ) . (6.18)We conclude from (6.16), (6.17) and (6.18) that ξ λp ( · ) : B X αc ( ˜ ̺ ) → X αus is aLipschitz continuous mapping uniformly on p and λ . More specifically, let L := 11 − M ρ + 1 . y, z ∈ B X αc ( ˜ ̺ ), k ξ λp ( y ) − ξ λp ( z ) k α ≤ k γ yp,λ (0) − γ zp,λ (0) k α + k y − z k α ≤ L k y − z k α . Since γ yp,λ ≡ ξ λp (0) ≡
0, and thuslim k y k α → k ξ λp ( y ) k α = 0uniformly on p ∈ H and λ ∈ I λ ( η/ ̺ > k y + ξ λp ( y ) k ≤ ρ , y ∈ B X αc ( ̺ ) . Define for each p ∈ H the p -section as M λloc ( p ) = { y + ξ λp ( y ) : y ∈ B X αc ( ̺ ) } . By the definition of p ρ , M λloc ( · ) := {M λloc ( p ) } p ∈H is a local invariant manifoldof the cocycle semiflow ϕ λ , λ ∈ I λ ( η/
8) generated by (4.5). And for each p ∈ H , the section M λloc ( p ) is homeomorphic to B X αc ( ̺ ).In the last part, we show ξ · p ( y ) : I λ ( η/ → X αus is Lipschitz uniformly on y ∈ B X αc ( ̺ ) and p ∈ P . Indeed, for λ , λ ∈ I λ ( η/
8) with λ ≤ λ , we havefor t ∈ R that k e − A λ c t − e − A λ c t k ≤ k e − A λ c t k · (cid:12)(cid:12) − e − ( λ − λ ) t (cid:12)(cid:12) ≤ e η | t | · (cid:12)(cid:12) − e − ( λ − λ ) t (cid:12)(cid:12) . Then for t ∈ R , e − η | t | Z t k e − A λ c ( t − τ ) p ( τ, γ yp,λ ( τ )) − e − A λ c ( t − τ ) p ρ ( τ, γ yp,λ ( τ )) k dτ ≤ Z t e − η | t − τ | k ( ρ ) (cid:16) e − η | τ | k γ yp,λ ( τ ) − γ yp,λ ( τ ) k α (cid:17) dτ + Z t e − η | t − τ | · k ( ρ ) (cid:12)(cid:12) − e − ( λ − λ )( t − τ ) (cid:12)(cid:12) · (cid:16) e − η | τ | k γ yp,λ ( τ ) k α (cid:17) dτ ≤ k ( ρ ) Z t e − η | t − τ | (cid:16) e − η | τ | k γ yp,λ ( τ ) − γ yp,λ ( τ ) k α (cid:17) dτ + k ( ρ ) M Z t e − η | t − τ | (cid:12)(cid:12) − e − ( λ − λ )( t − τ ) (cid:12)(cid:12) dτ. (6.19)24e can apply very similar arguments as above to get that e − η | t | Z t −∞ k e − A λ s ( t − τ ) p ρ ( τ, γ yp,λ ( τ )) − e − A λ s ( t − τ ) p ρ ( τ, γ yp,λ ( τ )) k dτ ≤ k ( ρ ) Z t −∞ ( t − τ ) α e − η ( t − τ ) (cid:16) e − η | τ | k γ yp,λ ( τ ) − γ yp,λ ( τ ) k α (cid:17) dτ + k ( ρ ) M Z t −∞ ( t − τ ) α e − η ( t − τ ) (cid:12)(cid:12) − e − ( λ − λ )( t − τ ) (cid:12)(cid:12) dτ (6.20)and e − η | t | Z ∞ t k e − A λ u ( t − τ ) p ( τ, γ yp,λ ( τ )) − e − A λ u ( t − τ ) p ( τ, γ yp,λ ( τ )) k dτ ≤ k ( ρ ) Z ∞ t e η ( t − τ ) (cid:16) e − η | τ | k γ yp,λ ( τ ) − γ yp,λ ( τ ) k α (cid:17) dτ + k ( ρ ) M Z ∞ t e η ( t − τ ) (cid:12)(cid:12) − e − ( λ − λ )( t − τ ) (cid:12)(cid:12) dτ. (6.21)By (6.19), (6.20) and (6.21), we derive that e − η | t | k γ yp,λ ( t ) − γ yp,λ ( t ) k α ≤ e − η | t | Z t k e − A λ c ( t − τ ) p ( τ, γ yp,λ ( τ )) − e − A λ c ( t − τ ) p ( τ, γ yp,λ ( τ )) k dτ + e − η | t | Z t −∞ k e − A λ s ( t − τ ) p ( τ, γ yp,λ ( τ )) − e − A λ s ( t − τ ) p ( τ, γ yp,λ ( τ )) k dτ + e − η | t | Z ∞ t k e − A λ u ( t − τ ) p ( τ, γ yp,λ ( τ )) − e − A λ u ( t − τ ) p ( τ, γ yp,λ ( τ )) k dτ ≤ k ( ρ ) Z ∞ (2 + t − α ) e − η t dt · sup t ∈ R e − η | t | k γ yp,λ ( t ) − γ yp,λ ( t ) k α + k ( ρ ) M Z ∞ (2 + t − α ) e − η t (cid:0) e ( λ − λ ) t − (cid:1) dt. (6.22)It follows that k ξ λ p ( y ) − ξ λ p ( y ) k α = k u λ (0) − u λ (0) k α ≤ sup t ∈ R e − η | t | k γ yp,λ ( t ) − γ yp,λ ( t ) k α ≤ k ( ρ ) M − M ρ Z ∞ (2 + t − α ) e − η t (cid:0) e ( λ − λ ) t − (cid:1) dt ≤ k ( ρ ) M − M ρ Z ∞ (2 + t − α ) t e − [ η − ( λ − λ )] t dt · | λ − λ | , e ( λ − λ ) t − Z ∞ (2 + t − α ) t e − [ η − ( λ − λ )] t dt = Z ∞ (2 t + t − α ) e − [ η − ( λ − λ )] t dt converges. Therefore ξ λ p ( y ) − ξ λ p ( y ) k ≤ L | λ − λ | , where L := k ( ρ ) M − M ρ Z ∞ (2 t + t − α ) e − [ η − ( λ − λ )] t dt, and thus ξ · p ( y ) is Lipschitz continuous on I λ ( η/
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