Manusc ip submi ed o Websi e: h p://AIMsciences.o g
AIMS’ Jou nals
Volume xx, Numbe 0x, xxxxx xxxx pp. 1–xx
MORSE DECOMPOSITION OF GLOBAL ATTRACTORS WITH
INFINITE COMPONENTS
Tom´
as Ca aballo, Juan C. Ja a, Jos´
e A. Langa
Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico,
Uni e sidad de Se illa, Apdo. de Co eos 1160, 41080-Se illa, Spain
Jos´
e Vale o
Cen o de In es igaci´on Ope a i a, Uni e sidad Miguel He n´andez,
A da. de la Uni e sidad, s/n, 03202-Elche, Spain
(Communica ed by Associa e Edi o )
Abs ac . In his pape we desc ibe some dynamical p ope ies o a Mo se
decomposi ion wi h a coun able numbe o se s. In pa icula , we a e able o
p o e ha he g adien dynamics on Mo se se s oge he wi h a sepa a ion
assump ion is equi alen o he exis ence o an o de ed Lyapuno unc ion
associa ed o he Mo se se s and also o he exis ence o a Mo se decomposi ion
- ha is, he global a ac o can be desc ibed as an inc easing amily o local
a ac o s and hei associa ed epelle s.
1. In oduc ion. The asymp o ic beha iou o a sys em o (o dina y o pa ial)
di e en ial equa ions modeling eal phenomena om di e en a eas o Science is
usually desc ibed by he analysis o hei global a ac o s, a compac in a ian
se o he associa ed semig oups a ac ing (uni o mly) bounded se s o wa ds in
ime. This subjec has ecei ed much a en ion h oughou he las decades (see,
o ins ance, [4], [9], [12], [16], [19], [18] o [20]). We ecall now he de ini ion o
global a ac o associa ed o a semig oup.
Fi s , le Xbe a me ic space wi h me ic d:X×X→R+, whe e R+= [0,∞),
and deno e by C(X) he se o con inuous maps om Xin o X. Gi en a subse A⊂
X, he -neighbo hood o Ais he se O(A) := {x∈X:d(x, a)<  o some a∈
A}.
De ini ion 1.1. A amily {T( ) : ≥0} ⊂ C(X)is a semig oup in a comple e
me ic space Xi :
•T(0) = IX, wi h IXbeing he iden i y map in X,
•T( +s) = T( )T(s), o all , s ∈R+,
•R+×X3( , x)7→ T( )x∈Xis con inuous.
2000 Ma hema ics Subjec Classi ica ion. 37B25, 37L99, 35B40, 35B41.
Key wo ds and ph ases. Mo se decomposi ion; in ini e componen s; g adien dynamics; Lya-
puno unc ion; g adien -like semig oup.
Pa ially suppo ed by FEDER and Minis e io de Econom´ıa y Compe i i idad (Spain) unde
g an s MTM2011-22411 and MTM2012-31698, and by Jun a de Andaluc´ıa unde P oyec o de
Excelencia P12-FQM-1492.
1
2 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO
The no ion o in a iance plays a undamen al ole in he s udy o he asymp o ic
beha io o semig oups.
De ini ion 1.2. A subse Ao Xis said in a ian unde he semig oup {T( ) : ≥
0}i T( )A=A o all ≥0.
Gi en A, B ⊂X, he Hausdo semidis ance om A o Bis gi en by
d(A, B) := sup
a∈A
in
b∈Bd(a, b).
De ini ion 1.3. Gi en wo subse s A, B o Xwe say ha Aa ac s Bunde he
ac ion o he semig oup {T( ) : ≥0}i d(T( )B, A) →∞
−→ 0.
We a e now in a posi ion o de ine global a ac o s.
De ini ion 1.4. A subse Ao Xis a global a ac o o a semig oup {T( ) :
≥0}i i is compac , in a ian unde he ac ion o {T( ) : ≥0}and o e e y
bounded subse Bo Xwe ha e ha Aa ac s Bunde he ac ion o {T( ) : ≥0}.
De ini ion 1.5. The semig oup {T( ) : ≥0}is e en ually dissipa i e i o any
bounded se B he e exis s ∗= ∗(B)≥0such ha ∪ ≥ ∗T( )Bis bounded.
Rema k 1.6. I is ob ious ha i T( )possesses a global a ac o , hen i is e en-
ually dissipa i e.
One o he main p ope ies in he s udy o a ac o s is e e ed o he desc ip ion
o hei geome ical in e nal s uc u e. Gene ically, a global a ac o is cha ac e -
ized by a ( ini e o in ini e) numbe o isola ed in a ian se s and he connec ing
o bi s among hem. This ac leads o a Mo se decomposi ion o he global a ac o
in e ms o a amily o a ac ing- epelle pai s (see [8,17,11,14,15]). We now
in oduce his concep .
De ini ion 1.7. Le {T( ) : ≥0}be a semig oup on X. We say ha an in a ian
se E⊂X o he semig oup {T( ) : ≥0}is an isola ed in a ian se i he e is
an  > 0such ha Eis he maximal in a ian subse o O(E).
De ini ion 1.8. A disjoin amily o isola ed in a ian se s is a amily {M1,· · · , Mn}
o isola ed in a ian se s wi h he p ope y ha
O(Mi)∩ O(Mj) = ∅,1≤i<j≤n,
o some  > 0.
De ini ion 1.9. A global solu ion o a semig oup {T( ) : ≥0}is a con inuous
unc ion ξ:R→Xwi h he p ope y ha T( )ξ(s) = ξ( +s) o all s∈Rand o
all ∈R+. We say ha ξ:R→Xis a global solu ion h ough x∈Xi i is a
global solu ion wi h ξ(0) = x.
I is also well known ha he global a ac o is he union o all bounded comple e
global solu ions o he semig oup T.
De ini ion 1.10. Le {T( ) : ≥0}be a semig oup which possesses a disjoin
amily o isola ed in a ian se s M={M1,· · · , Mn}. A homoclinic s uc u e asso-
cia ed o Mis a subse {Mk1,· · · , Mkp}o M(p≤n) oge he wi h a se o global
solu ions {ξ1,· · · , ξp}such ha
Mkj
→−∞
←− ξj( ) →∞
−→ Mkj+1 ,1≤j≤p,
whe e Mkp+1 := Mk1.
MORSE DECOMPOSITION WITH INFINITE COMPONENTS 3
Rema k 1.11. He e, ξ( ) →±∞
−→ Mmeans ha d(ξ( ), M)→0as → ±∞.
We will s udy he dynamics o he semig oup inside he global a ac o A. We
now de ine gene alized dynamically g adien semig oups (see [6,5]).
De ini ion 1.12. Le {T( ) : ≥0}be a semig oup wi h a global a ac o Aand
a disjoin amily o isola ed in a ian se s M={M1,· · · , Mn}in A. We say ha
{T( ) : ≥0}is a gene alized dynamically g adien semig oup ela i e o Mi :
a) Fo any global solu ion ξ:R→ A he e a e 1≤i, j ≤nsuch ha
Mi
→−∞
←− ξ( ) →∞
−→ Mj.
b) The e is no homoclinic s uc u e associa ed o M.
Rema k 1.13. The concep o gene alized dynamically g adien semig oup is he
same as he concep o g adien -like semig oup as gi en in [1],[5].
To in oduce he no ion o a Mo se decomposi ion o he a ac o Ao a semi-
g oup {T( ) : ≥0}(see [8], [17] o [18]) we p e iously need he no ion o a ac o -
epelle pai . We ecall ha he omega-limi se o B⊂Xis de ined by
ω(B) = ∩ ≥0∪s≥ T(s)B.
De ini ion 1.14. Le {T( ) : ≥0}be a semig oup wi h a global a ac o A. We
say ha a non-emp y subse Ao Ais a local a ac o i he e is an  > 0such ha
ω(O(A)) = A. The epelle A∗associa ed o a local a ac o Ais he se de ined
by
A∗:= {x∈ A :ω(x)∩A=∅}.
The pai (A, A∗)is called an a ac o - epelle pai o {T( ) : ≥0}.
No e ha i Ais a local a ac o , hen A∗is closed and in a ian .
De ini ion 1.15. Gi en an inc easing amily ∅=A0⊂A1⊂ · · · ⊂ An=A, o
n+1 local a ac o s, o j= 1,· · · , n, de ine Mj:= Aj∩A∗
j−1.The o de ed n- uple
M:= {M1, M2,· · · , Mn}is called a Mo se decomposi ion o A.
De ini ion 1.16. We will say ha a semig oup {T( ) : ≥0}wi h a global a ac o
Aand a disjoin amily o isola ed in a ian se s M={M1,· · · , Mn}in Ais a
g adien semig oup wi h espec o M, i he e exis s a con inuous unc ion V:
X→Rsuch ha [0,∞)3 7→ V(T( )x)∈Ris non-inc easing o each x∈X M,
Vis cons an in Mi o each 1≤i≤n, and V(T( )x) = V(x) o all ≥0i and
only i x∈
n
S
i=1
Mi.
Vis called a Lyapuno unc ion ela ed o M.
I has been p o ed in [1] ha gi en a disjoin amily o isola ed in a ian se s
on he global a ac o M={M1,· · · , Mn} o a semig oup T( ), he dynamical
p ope y o being gene alized dynamically g adien , he exis ence o an associa ed
o de ed amily o local a ac o - epelle s, and he exis ence o a Lyapuno unc-
ional ela ed o M, a e equi alen p ope ies. Many o he a gumen s in [1] make
a p ecise use o he ac ha he numbe o Mo se se s is ini e. The aim o his
pape is o gene alize his esul o he case o a coun able numbe o Mo se se s.
Indeed, he gene al heo y o Mo se decomposi ion o in a ian se s is gene ically
adap ed o he exis ence o a ini e numbe o isola ed Mo se se s. Howe e , i is no
4 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO
unusual o ha e an in ini e numbe o in a ian s in a global a ac o . Fo ins ance,
conside he scala di e en ial equa ion
dy
d = (y)
wi h
(y) = 




−y, i y≤0,
(1 −e−y)sin π
y,i 0 < y ≤1,
1−y, i y≥1.
No e ha he equa ion possesses he ollowing ixed poin s:
y1= 1, y2=1
2, y3=1
3, ..., yk=1
k, ..., y∞= 0,
wi h hei espec i e associa ed uns able mani olds (see De ini ion 4.2)
Wu(1) = 1, Wu1
2= [1
2,1), ..., Wu1
k= [1
k,1
k−1), ..., Wu(0) = 0,
and as global a ac o A= [0,1].In [3] he au ho s s udy a mul i alued e sion o
he well-known Cha ee-In an e equa ion, also leading o a global a ac o wi h an
in ini e numbe o equilib ia, which ac ually has mo i a ed he necessi y o de el-
oping he heo y in his pape . We will conside his applica ion in a subsequen
wo k.
In Sec ion 2we ecall some esul s on he dynamics ela ed o an a ac o - epelle
pai . In Sec ion 3we will gene alize De ini ions 1.12,1.15 and 1.16 o he case o
an in ini e numbe o disjoin isola ed in a ian se s M∞={Mi}∞
i=1 ∪M∞inside
he global a ac o . In Sec ions 4,5and 6we p o e he main esul o his pape ,
he equi alence be ween a gene alized dynamically g adien semig oup e e ed o
M∞wi h a sui able sepa a ion assump ion, he exis ence o an o de ed Lyapuno
unc ion associa ed o M∞, and he exis ence o a Mo se decomposi ion on he
global a ac o . This is done in se e al s eps: i s , we p o e ha he p ope y o
he semig oup o being gene alized dynamically g adien oge he wi h a sepa a ion
assump ion implies ha a Mo se decomposi ion can be cons uc ed; hen we p o e
ha om a Mo se decomposi ion ela ed o M∞an o de ed Lyapuno unc ion
can be de ined; inally, we check ha he exis ence o an o de ed Lyapuno unc ion
implies ha he semig oup is gene alized dynamically g adien semig oup e e ed
o M∞and ha he sepa a ion assump ion holds.
2. P elimina y esul s on a ac o - epelle pai s. The ollowing esul s on
he dynamics on a ac o - epelle pai s a e aken om [1].
We ecall ha local a ac ion o Ain Ais equi alen o local a ac ion in X,
o which we i s ly need he ollowing esul .
Lemma 2.1. Le {T( ) : ≥0}be a semig oup in Xwi h a global a ac o A. I
A⊂ A is a compac in a ian se o {T( ) : ≥0}and he e is an  > 0such
ha Aa ac s O(A)∩ A, hen gi en δ∈(0, ε) he e is a δ0∈(0, δ)such ha
γ+(Oδ0(A)) ⊂ Oδ(A), whe e γ+(Oδ0(A)) = S
x∈Oδ0(A)S
≥0
{T( )x}.
The nex esul gene alizes o semig oups a known esul o g oups gi en in [8]
and shows ha ou de ini ion o local a ac o is equi alen o ha one in [8,17].
MORSE DECOMPOSITION WITH INFINITE COMPONENTS 5
Lemma 2.2. I {T( ) : ≥0}is a semig oup in Xwi h a global a ac o Aand
S( ) := T( )|A, clea ly {S( ) : ≥0}is a semig oup in he me ic space A. I Ais
a local a ac o o {S( ) : ≥0}in he me ic space A( ha is, he e exis s ε > 0
wi h ω(O(A)∩ A) = A), and Kis a compac subse o Asuch ha K∩A∗=∅,
hen Aa ac s K. Fu he mo e Ais a local a ac o o {T( ) : ≥0}in X.
We now desc ibe he dynamics on an a ac o - epelle pai .
Lemma 2.3. Le {T( ) : ≥0}be a semig oup in Xwi h a global a ac o Aand
(A, A∗)an a ac o - epelle o {T( ) : ≥0}. Then:
(i)I ξ:R→Xis a global bounded solu ion o {T( ) : ≥0} h ough x /∈A∪A∗,
hen ξ( ) →∞
−→ Aand ξ( ) →−∞
−→ A∗.
(ii)A global solu ion ξ:R→Xo {T( ) : ≥0}wi h he p ope y ha
ξ( )∈ Oδ(A∗) o all ≤0 o some δ > 0such ha Oδ(A∗)∩A=∅mus sa is y
d(ξ( ), A∗) →−∞
−→ 0.
(iii)I x∈X A, hen T( )x →∞
−→ A∪A∗.
Pa (i) o he p e ious lemma is p o ed in Theo em 1.4 in [17]. Pa s (ii) and
(iii) can be ound in [1].
3. Gene alized dynamically g adien semig oups. In his sec ion we will in-
oduce he concep s o gene alized dynamically g adien semig oups and Mo se
decomposi ion o a coun able se o isola ed in a ian se s.
De ini ion 3.1. A disjoin (coun able) amily o in a ian se s is a amily M∞=
{Mi}∞
i=1 ∪M∞o in a ian se s wi h he p ope y ha , gi en j∈N, he e exis s δj
such ha
Oδj(Mj)∩ Oδj(Mi) = ∅, o all i6=j,i∈N∪ {∞}.(3.1)
De ini ion 3.2. Le {T( ) : ≥0}be a semig oup which possesses a disjoin
amily o in a ian se s M∞={Mi}∞
i=1 ∪M∞wi h Mjisola ed o each j∈N. A
homoclinic s uc u e associa ed o M∞is a ini e subse {Mk1,· · · , Mkp}o M∞
oge he wi h a se o global solu ions {ξ1,· · · , ξp}such ha
Mkj
→−∞
←− ξj( ) →∞
−→ Mkj+1 ,1≤j≤p,
whe e Mkp+1 := Mk1.
Rema k 3.3. The se M∞is no assumed o be isola ed. The eason is ha ypically
in applica ions M∞is an accumula ion se o he sequence Mnas n→ ∞. Hence,
i is no isola ed. This is he case in he example gi en in he in oduc ion, and
also, o ins ance, in he applica ion o mul i alued semi lows in [3].
De ini ion 3.4. Le {T( ) : ≥0}be a semig oup wi h a global a ac o Aand
a disjoin amily o in a ian se s M∞={Mi}∞
i=1 ∪M∞in Awi h Mjisola ed
o each j∈N. We say ha {T( ) : ≥0}is a gene alized dynamically g adien
semig oup ela i e o M∞i o any global solu ion ξ:R→ A such ha ξ( 0)6∈ Mk,
o some 0∈Rand any k∈N∪ ∞, i holds ha
Mj
→−∞
←− ξ( ) →∞
−→ Mi, o 1≤i<j≤ ∞.(3.2)
Rema k 3.5. I is ob ious ha condi ion (3.2) implies he ollowing p ope ies:
•Fo any global solu ion ξ:R→ A he e a e 1≤i, j ≤ ∞ such ha
Mj
→−∞
←− ξ( ) →∞
−→ Mi.

6 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO
•The e is no homoclinic s uc u e associa ed o M∞.
When he numbe o se s Miis ini e, hen i is p o ed in [2] ha hese las wo
p ope ies imply (3.2) o a sui able ea angemen o he se s. In ac , De ini ion
3.4 is he way in which i is de ined a Mo se decomposi ion o a global a ac o in
[18].
No e ha , in pa icula , (3.2) implies ha he e is no global solu ion ξ( ) : R→ A
wi h ξ( 0)6∈ M1 o some 0∈Rsuch ha
lim
→−∞ d(ξ( ), M1)=0.
The ollowing lemma implies ha an isola ed in a ian se inside a global a ac-
o is compac .
Lemma 3.6. Le Mbe an isola ed in a ian se which is ela i ely compac . Then
Mis compac .
P oo . We need o p o e ha Mis closed. Le yn→y, whe e yn∈M. By he
con inui y o Twe ha e ha T( )yn→T( )y o any > 0. Hence, T( )y∈M.
Thus, T( )M⊂M o all ≥0. On he o he hand, as Mis in a ian , o any
> 0 he e exis s zn∈Msuch ha T( )zn=yn. Since Mis ela i ely compac ,
passing o a subsequence we ha e zn→z∈M, and hen T( )z=y. The e o e,
M⊂T( )M o all > 0. I ollows ha Mis in a ian . As Mis an isola ed
in a ian se , we ge M=M.
As a consequence o he i s s a emen in Lemma 2.3 we ob ain he ollowing.
Co olla y 3.7. I {T( ) : ≥0}is a semig oup in Xwi h a global a ac o A
and (A, A∗)is an a ac o - epelle pai o {T( ) : ≥0}, hen {T( ) : ≥0}
is a gene alized dynamically g adien semig oup associa ed o he disjoin amily o
isola ed in a ian se s {A, A∗}.
No e ha (3.1) implies
Mi∩M∞=∅, o each i∈N.(3.3)
Lemma 3.8. Condi ion (3.2) implies ha he e is no global solu ion ξ:R→ A
wi h ξ( 0)∈ A M∞ o some 0∈Rsuch ha
lim
→+∞d(ξ( ), M∞)=0.(3.4)
P oo . I is ob ious, as in (3.2), ha he index icanno be ∞.
Lemma 3.9. Le M∞={Mi}∞
i=1 ∪M∞be compac in a ian se s such ha Mj∩
Mi=∅ o i6=j,i, j ∈N∪ ∞, and also suppose ha he in a ian compac se
M∞⊂ A is such ha
lim
i→∞ d(Mi, M∞)=0.(3.5)
Then M∞is a disjoin amily o in a ian se s.
P oo . Take j∈Na bi a y. We ha e o check (3.1). The e exis s δ1>0 such ha
Oδ1(Mj)∩ Oδ1(M∞) = ∅.
In iew o (3.5) he e is N > j such ha
Mi⊂ Oδ1
2
(M∞) i i > N.
MORSE DECOMPOSITION WITH INFINITE COMPONENTS 7
Hence,
Oδ1(Mj)∩ Oδ1
2
(Mi) = ∅i i > N.
Ob iously, he e exis s δ2>0 o which
Oδ2(Mj)∩ Oδ2(Mi) = ∅ o 1 ≤i≤N,i6=j.
Then he esul ollows o δj= min{δ1/2, δ2}.
We can now in oduce he concep o a Mo se decomposi ion e e ed o M∞.
De ini ion 3.10. Gi en an inc easing amily ∅=A0⊂A1⊂ · · · ⊂ An⊂ · · · A∞=
Ao local a ac o s, o j∈Nde ine Mj:= Aj∩A∗
j−1,M∞=∩∞
j=0A∗
j.The o de ed
coun able se M∞:= {Mi}∞
i=1 ∪M∞is called a Mo se decomposi ion o A.
The ollowing p ope ies o he se s Mj ollow.
Lemma 3.11. M∞∩Aj=∅ o any j∈N.Hence, M∞⊂A∞ ∪∞
j=1 Ajand
M∞∩Mj=∅ o all j∈N.
P oo . Le y∈M∞. Then y∈A∗
j, o any j∈N, implies y6∈ Aj o all j∈N.
Lemma 3.12. The se s Mj,j∈N∪ ∞, a e compac .
P oo . Since Mj⊂ A, hey a e ela i ely compac . Also, as Mja e he in e sec ion
o closed se s, hey a e closed.
We can also gi e he ollowing cha ac e iza ion.
P oposi ion 3.13. Le {T( ) : ≥0}be a semig oup wi h he global a ac o A
and M∞={Mi}∞
i=1 ∪M∞a Mo se decomposi ion o Awi h he amily ∅=A0⊂
A1⊂ · · · ⊂ A∞=Ao local a ac o s. Then,
∞
j=0
(Aj∪A∗
j)=(
∞
[
j=1
Mj)∪M∞.
P oo . I z∈
∞
S
j=1
Mj, le k∈Nbe such ha z∈Mk=Ak∩A∗
k−1. Hence
z∈Ak⊂Ak+1 ⊂ · · · ⊂ A∞and z∈A∗
k−1⊂A∗
k−2⊂ · · · ⊂ A∗
0. Thus
z∈(
∞
j=k
Aj)∩(
k−1
j=0
A∗
j)⊂

∞
j=k
(Aj∪A∗
j)
∩

k−1
j=0
(Aj∪A∗
j)
=
∞
j=0
(Aj∪A∗
j),
p o ing ha
∞
S
j=1
Mj⊂
∞
T
j=0
(Aj∪A∗
j).I z∈M∞, hen z∈
∞
T
j=0
A∗
j⊂
∞
T
j=0
(Aj∪A∗
j).
Con e sely, we ake z∈
∞
T
j=0
(Aj∪A∗
j).I z∈
∞
T
j=0
A∗
j, hen z∈M∞. O he wise,
z∈Aj o some j∈N. Deno e I:= {i1, i2,· · · , ik, . . . }and J:= {j1, j2,· · · , jl, .. . }
such ha I∪J=Z+wi h I∩J=∅and z∈Ai o all i∈Iand z∈A∗
j o all j∈J.
Clea ly, i i:= min I, necessa ily I={j≥i}and J={0,1,· · · , i−1}, consequen ly
z∈Aiand z∈A∗
i−1. So, z∈Ai∩A∗
i−1=Mi, om which
∞
T
j=0
(Aj∪A∗
j)⊂
∞
S
j=1
Mj.
8 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO
4. Cons uc ion o a Mo se decomposi ion om he dynamics on M∞.
In his sec ion we desc ibe he cons uc ion o a Mo se decomposi ion o he global
a ac o A ela i e o he disjoin amily o in a ian se s M∞={Mi}∞
i=1 ∪M∞
in Asuch ha Mjis isola ed i j∈Nand sa is ying (3.2). By Lemma 3.8 we ha e
ha (3.4) does no hold.
The ollowing lemma will play an impo an ole in wha ollows.
Lemma 4.1. Le {T( ) : ≥0}be a semig oup wi h a global a ac o Aand he
disjoin amily o in a ian se s M∞={Mi}∞
i=1 ∪M∞={M1, . . . , Mn, . . . ;M∞}in
Abe such ha Mja e isola ed o j∈N. Assume ha Tis gene alized dynamically
g adien ela i e o M∞. Then, M1is a local a ac o o {T( ) : ≥0}.
P oo . We i s ly p o e ha o all δ∈(0, δ1) he e exis s δ0∈(0, δ) such ha
γ+(Oδ0(M1)) ⊂ Oδ(M1),
whe e δ1sa is ies Oδ1(M1)∩ Oδ1(Mi) = ∅ o i > 1 o i=∞.
I no , he e exis 0 < δ < δ1and sequences { k}k∈No posi i e imes and {xk}k∈N
o poin s in Xsuch ha o all k
d(xk, M1)<1
k,
d(T( k)xk, M1) = δ
and
d(T( )xk, M1)< δ o ∈[0, k).
Thus, i we de ine, o each k, ξk( ) := T( + k)xk o ∈[− k,∞),as k→
k→∞ ∞,
we conclude ha he e exis s a global solu ion ξ:R→X o T(·) such ha ξk→
k→∞ ξ
uni o mly in compac se s o imes (see [7, Lemma 3.1]). Then, d(ξk( ), M1)≤δ o
∈[− k,0] implies
d(ξ( ), M1)≤δ < δ1 o ≤0.
Bu by (3.2) we ha e ξ( )→Mj,wi h j > 1, as → −∞, a con adic ion.
M1is he maximal in a ian se in Oε(M1) o some ε > 0. Thus, o δ <
min{ε, δ1} ake δ0∈(0, δ) such ha
γ+(Oδ0(M1)) ⊂ Oδ(M1),
so ha
ω(Oδ0(M1)) ⊂ Oδ(M1)⊂ Oε(M1),
and hen, as ω(Oδ0(M1)) is in a ian ,
ω(Oδ0(M1)) ⊂M1.
The o he inclusion is i ial, so ha M1is a local a ac o .
Fo M1a local a ac o , le M∗
1={x∈ A :ω(x)∩M1=∅}be i s associa ed
epelle , so each Mi, wi h i≥2,is con ained in M∗
1and mo e gene ally he o bi
ξ(R) o any global solu ion ξ:R→ A ha con e ges o Mi,i≥2, when →+∞,is
con ained in M∗
1. Conside ing he es ic ion {T1( ) : ≥0}o {T( ) : ≥0} o M∗
1
we ha e ha {T1( ) : ≥0}sa is ies (3.2) in he space M∗
1wi h he in a ian se s
{Mi}∞
i=2 ∪M∞and we may assume, by he las lemma, ha M2is a local a ac o
o he semig oup {T1( ) : ≥0}in M∗
1. I M∗
2,1is he epelle associa ed o he local
a ac o M2 o {T1( ) : ≥0}in M∗
1we may p oceed and conside he es ic ion
MORSE DECOMPOSITION WITH INFINITE COMPONENTS 9
{T2( ) : ≥0}o he semig oup {T1( ) : ≥0} o M∗
2,1and hen {T2( ) : ≥0}
sa is ies (3.2) in M∗
2,1wi h he associa ed in a ian se s {Mi}∞
i=3 ∪M∞.
Se ing A=: M∗
0,−1and M∗
1,0:= M∗
1, o j≥1 we ha e ha Mjis a local
a ac o o he es ic ion o {T( ) : ≥0} o M∗
j−1,j−2whose epelle will be
indica ed by M∗
j,j−1.
De ini ion 4.2. Le {T( ) : ≥0}be a semig oup. The uns able se o an in a ian
se Mis de ined by
Wu(M) := {z∈X: he e is a global solu ion ξ:R→X
such ha ξ(0) = zand lim
→−∞ d(ξ( ), M)=0}.
De ine A0:= ∅,A1:= M1and o j= 2,3,· · · ,
Aj:= Aj−1∪Wu(Mj) =
j
[
i=1
Wu(Mi).(4.1)
Also, A∞=A.
I is clea ha A=∪∞
i=1Wu(Mi)∪Wu(M∞).
Lemma 4.3. Assume he condi ions o Lemma 4.1. Then M∞=∩∞
j=0A∗
j.
P oo . Le z∈M∞. Then as M∞is in a ian , ω(z)⊂M∞. Then zcanno be in
Wu(Mj) o j∈N, as in such a case by (3.2) we would ha e ω(z)∩Mi6=∅ o
some i≤j, a con adic ion. Thus, by (4.1) we ha e ha z6∈ Aj o j∈N. Hence,
ω(z)∩Aj=∅, so ha z∈ ∩∞
j=0A∗
j.
Con e sely, le z∈ ∩∞
j=0A∗
j. Then ω(z)∩Aj=∅ o all j∈N. I z6∈ M∞, we
ake a global solu ion ξ(·) such ha ξ(0) = z. Then by condi ion (3.2) we ha e ha
ξ( )→Mias →+∞ o some i∈N. Bu hen ω(z)∩Ai6=∅, a con adic ion.
Lemma 4.4. Assume he condi ions o Lemma 4.1. Then he se s Mj,j∈N∪ ∞,
a e compac .
P oo . We no e ha Mj⊂ A implies by Lemma 3.6 ha he se s Mja e compac
i j∈N. Also, Lemma 4.3 implies ha M∞is closed, and hen M∞⊂ A implies
ha i is compac .
Lemma 4.5. Assume he condi ions o Lemma 4.1. Suppose ha , gi en j∈N,
he e exis s δjsuch ha
Wu(Mj)∩ Oδj(
∞
[
i=j+1
Mi∪M∞)) = ∅.(4.2)
Then,
Aj∩ Oδj((
∞
[
i=j+1
Mi)∪M∞) = ∅.(4.3)
P oo . Fo j= 1 he esul ollows since A1=M1=Wu(M1). Suppose (4.3) is
ue o j−1 and we will show i o j. I no , he e exis s a sequence {xk}k∈Nin
Ajsuch ha o all k
d(xk,(
∞
[
i=j+1
Mi)∪M∞)<1
k.
As Aj:= Aj−1∪Wu(Mj) and we ha e (4.3) o j−1, hen xk∈Wu(Mj), om
which, by hypo hesis, we ge a con adic ion.
16 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO
We can now conclude ou main heo em.
Theo em 6.7. Le {T( ) : ≥0}be a semig oup wi h global a ac o Aand
conside a disjoin amily o in a ian se s M∞={Mi}∞
i=1 ∪M∞in Asuch ha
Mja e isola ed o j∈N. Then, he ollowing condi ions a e equi alen :
1. {T( ) : ≥0}is a gene alized g adien semig oup wi h espec o M∞in he
sense o he De ini ion 5.1 and M∞is o de ed wi h espec o he espec i e
Lyapuno unc ion.
2. {T( ) : ≥0}is a gene alized dynamically g adien semig oup wi h espec o
M∞(as in De ini ion 3.4) sa is ying (4.2).
3. M∞is a Mo se decomposi ion o A.
P oo . I is a s aigh o wa d consequence o Theo em 4.8 and P oposi ions 5.3,6.3
and 6.6.
Co olla y 6.8. Le {T( ) : ≥0}be a semig oup wi h global a ac o Aand
conside a disjoin amily o in a ian se s M∞={Mi}∞
i=1 ∪M∞in Asuch ha
Mja e isola ed o j∈N. Assume ha M∞is a Mo se decomposi ion o A. Then
A=∪∞
j=1Wu(Mj)∪Wu(M∞).
P oo . In iew o Theo em 6.7,{T( ) : ≥0}is a gene alized g adien semig oup
wi h espec o M∞in he sense o he De ini ion 5.1 and M∞is o de ed wi h
espec o he Lyapuno unc ion. Hence, he esul ollows om P oposi ion 6.5.
Acknowledgemen . We would like o hank he e e ees o hei help ul com-
men s and sugges ions which allowed us o imp o e he p esen a ion o he pape .
REFERENCES
[1] A ag˜ao-Cos a, E.R., Ca aballo, T., Ca alho, A.N., & Langa, J.A., S abili y o g adien
semig oups unde pe u ba ion, Nonlinea i y, 24 (2011), 2099–2117 .
[2] A ag˜ao-Cos a, E.R., Ca aballo, T., Ca alho, A.N., & Langa, J.A., Con inui y o Lyapuno
unc ions and o ene gy le el o a gene alized g adien sys em, Topological Me hods Nonl.
Anal.,39 (2012), 57–82.
[3] J. M. A ie a, A. Rod ´ıguez-Be nal, J. Vale o, Dynamics o a eac ion-di usion equa ion wi h
a discon inuous nonlinea i y, In e na . J. Bi u . Chaos Appl. Sci. Eng g.,16 (10) (2006),
2695-2984.
[4] A. V. Babin and M. I. Vishik, A ac o s in E olu iona y Equa ions S udies in Ma hema ics
and i s Applica ions 25, No h-Holland Publishing Co., Ams e dam (1992).
[5] A. N. Ca alho and J. A. Langa, An ex ension o he concep o g adien semig oups which
is s able unde pe u ba ion, J. Di e en ial Equa ions, 246 (2009), 2646–2668.
[6] Ca alho, A.N., Langa, J.A., & Robinson, J.C., A ac o s o in ini e-dimensional non-
au onomous dynamical sys ems, Applied Ma hema ical Se ies 182, Sp inge , New Yo k (2013).
[7] A.N. Ca alho, J.A. Langa, J.C. Robinson, A. Su´a ez, Cha ac e iza ion o non-au onomous
a ac o s o a pe u bed in ini e-dimensional g adien sys em, J. Di e en ial Equa ions,236
(2007), 570–603.
[8] C. Conley, Isola ed in a ian se s and he Mo se index, CBMS Regional Con e ence Se ies in
Ma hema ics, 38. Ame ican Ma hema ical Socie y, P o idence, R.I. (1978).
[9] J. K. Hale, Asymp o ic Beha io o Dissipa i e Sys ems, Ma hema ical Su eys and Mono-
g aphs Numbe 25 (Ame ican Ma hema ical Socie y, P o idence, RI) (1988).
[10] D. Hen y, Geome ic Theo y o Semilinea Pa abolic Equa ions, Lec u e No es in Ma hema -
ics 840, Sp inge -Ve lag, Be lin, (1981).
[11] M. Hu ley, Chain ecu ence, semi lows and g adien s, J. Dyn. Di . Equa ions, 7(1995),
437–456.

MORSE DECOMPOSITION WITH INFINITE COMPONENTS 17
[12] O. A. Ladyzhenskaya, A ac o s o Semig oups and E olu ion Equa ions, Camb idge Uni-
e si y P ess, Camb idge (1991).
[13] D.E. No on, The undamen al heo em o dynamical sys ems, Commen . Ma h., Uni . Ca -
olinae, 36 (3) (1995), 585–597.
[14] M. Pa ao, Mo se decomposi ion o semi lows on opological spaces, J. Dyn. Di . Equa ions,
19 (1) (2007), 181–198.
[15] M. Pa ao and Luiz A.B. San Ma in, Semi lows on opological spaces: chain ansi i i y and
semig oups, J. Dyn. Di . Equa ions, 19 (1) (2007), 155–180.
[16] Robinson, J.C., In ini e-Dimensional Dynamical Sys ems. Camb idge Uni e si y P ess, Cam-
b idge, England (2001).
[17] K. P. Rybakowski, The homo opy index and pa ial di e en ial equa ions, Uni e si ex ,
Sp inge -Ve lag (1987).
[18] Sell, G.R. & You, Y., Dynamics o e olu iona y equa ions. Applied Ma hema ical Sciences
Vol. 143, Sp inge -Ve lag, New Yo k (2002).
[19] R. Temam, In ini e-Dimensional Dynamical Sys ems in Mechanics and Physics, Sp inge -
Ve lag, Be lin (1988; second edi ion 1996).
[20] Vishik, M.I., Asymp o ic beha iou o solu ions o e olu iona y equa ions, Camb idge Uni-
e si y P ess, Camb idge, England (1992).
E-mail add ess, T. Ca aballo: [email p o ec ed]
E-mail add ess, J.C. Ja a: [email p o ec ed]
E-mail add ess, J.A. Langa: [email p o ec ed]
E-mail add ess, J. Vale o: [email p o ec ed]