Noname manusc ip No.
(will be inse ed by he edi o )
Ex emal bounded comple e ajec o ies o
nonau onomous eac ion-di usion equa ions wi h
discon inuous o cing e m
Tom´as Ca aballo ·Jos´e A. Langa ·Jos´e
Vale o
Abs ac In his pape we es ablish a s ong compa ison p inciple o a nonau-
onomous di e en ial inclusion wi h a o cing e m o Hea iside ype. Using his
p inciple, we s udy he s uc u e o he global a ac o in bo h he au onomous
and nonau onomous cases. In pa icula , in he las case we p o e ha he pullback
a ac o is con ined be ween wo special bounded comple e ajec o ies, which
play he ole o nonau onomous equilib ia.
Keywo ds: di e en ial inclusions, eac ion-di usion equa ions, pullback a -
ac o s, nonau onomous dynamical sys ems, mul i alued dynamical sys ems, s uc-
u e, compa ison o solu ions.
AMS Subjec Classi ica ion (2010): 35B40, 35B41, 35B51, 35K55, 35K57
1 In oduc ion
Compa ison o solu ions o eac ion-di usion equa ions is a powe ul ool in o de
o s udy he s uc u e o global a ac o s. In pa icula , in he au onomous case i
allows us o es ablish ha he global a ac o is con ined be ween wo s a iona y
solu ions, which a e he maximal and minimal elemen s o he a ac o . Fo a class
o au onomous eac ion-di usion equa ions, such esul was p o ed in [5], [31]. I is
wo h no icing ha a gene al heo y o mono one andom dynamical sys ems was
This wo k has been pa ially suppo ed by Spanish Mini e io de Econom´ıa y Compe i i idad
and FEDER, p ojec s MTM2015-63723-P and MTM2016-74921-P, and by Jun a de Andaluc´ıa
unde P oyec o de Excelencia FQM-1492.
T. Ca aballo and J.A. Langa
Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico
Facul ad de Ma em´a icas, Uni e sidad de Se illa,
C/ Ta ia s/n, 41012-Se illa, Spain
E-mail: [email p o ec ed], [email p o ec ed]
J. Vale o
Cen o de In es igaci´on Ope a i a, Uni e sidad Miguel He n´andez de Elche,
A da. de la Uni e sidad, s/n, 03202-Elche, Spain
E-mail: j [email p o ec ed]
2 Tom´as Ca aballo e al.
i s s udied in [1], [2], [17]. These esul s we e ex ended o mul i alued au onomous
dynamical sys ems in [11].
In nonau onomous p oblems he si ua ion is mo e complica ed because s a ion-
a y solu ions do no exis in gene al bu only in a he pa icula cases, a leas no
in he classical sense. Fo his eason, we need o eplace hem by a special ype
o bounded comple e ajec o ies, which play he ole o “nonau onomous equilib-
ia”. The gene al heo y o o de -p ese ing nonau onomous dynamical sys ems
was s udied in [16], [25]. In his sense, a esul p o ed in [30] (see also [13], [25]
and [29]) is ema kable because a comple e bounded posi i e non-degene a e solu-
ion was cons uc ed o a nonau onomous eac ion-di usion equa ion. Using his
solu ion, a nonau onomous in e al con aining he pullback a ac o is p o ided.
In he mul i alued nonau onomous amewo k, simila esul s we e es ablished in
[12], whe e an o dina y nonau onomus di e en ial inclusion was s udied.
We aim o s udy he s uc u e o a ac o s o he ollowing nonau onomous
di e en ial inclusion





∂u
∂ −∂2u
∂x2∈b( )H0(u) + ω( )u, on (0,1) ×(τ, ∞),
u(0, ) = u(1, ) = 0,
u(x, τ) = uτ(x),
(1)
whe e H0is a Hea iside unc ion. P oblems o his ype appea when we ha e a
di e en ial equa ion d i en by a nonlinea unc ion ha ing a discon inui y, which
can be ew i en as a di e en ial inclusion by means o a Hea iside unc ion. Well
known applica ions like combus ion in po ous media [21], he conduc ion o elec-
ical impulses in ne e axons (see [33], [34]) o he su ace empe a u e on Ea h
(see [10], [20]) a e modeled by inclusions o simila ype.
The s uc u e o he global a ac o o p oblem (1) in he au onomous case
has been s udied in de ail in [4]. Ne e heless, se e al challenging p oblems s ill
emain open. Fo models conce ning he clima e on Ea h, some esul s abou
bi u ca ions o s eady s a es we e p o ed in [8], [9].
In he mul i alued amewo k, ha is, when mo e han one solu ion can exis
o he Cauchy p oblem o a di e en ial equa ion, i is no possible o compa e
solu ions wi h o de ed ini ial da a in he same way as in he single- alued case.
Ins ead, we need o es ablish some so o o de ela ionship be ween he se o
solu ions co esponding o he o de ed ini ial condi ions. In his sense, di e en
de ini ions ha e been gi en in he li e a u e. A s ong compa ison p inciple was
de ined and applied o o dina y di e en ial equa ions wi h delays in [11]. A weak
compa ison p inciple was es ablished in [38] o eac ion-di usion equa ions wi h-
ou uniqueness. Also, an in e media e compa ison p inciple was gi en in [14] o
di e en ial inclusions go e ned by subdi e en ial maps.
In his pape we i s ly p o e in he second sec ion a s ong compa ison p in-
ciple o he solu ions o p oblem (1). Mo eo e , we ob ain also s ong compa ison
be ween posi i e solu ions o (1) and i s co esponding au onomous equa ion ( ha
is, o b( ), ω ( )iden ically equal o cons an s). A e ha , in he hi d sec ion,
we use his compa ison p inciple and he abs ac esul s om [11] in o de o
es ablish ha he global a ac o o he au onomous p oblem (1) is con ined be-
ween maximal and minimal s a iona y poin s. Mo eo e , he exp essions o hese
ixed poin s a e known om [4].
Ex emal ajec o ies o equa ions wi h discon inuous o cing e m 3
The main esul s o his pape , conce ning he s uc u e o he pullback a -
ac o o he nonau onomous inclusion (1), a e gi en in he ou h sec ion. We
p o e he exis ence o wo bounded comple e ajec o ies ha gene a e a ime-
depending in e al con aining he pullback a ac o . These solu ions a e s ic ly
posi i e (nega i e) o any ime and any x∈(0,1), ha is, hey a e non-degene a e.
Mo eo e , hey a e he unique non-degene a e bounded comple e ajec o ies o
he p oblem and play he ole o nonau onomous posi i e equilib ia.
2 Compa ison o solu ions
In his sec ion we es ablish a s ong compa ison p inciple o he s ong solu ions
o a nonau onomous di e en ial inclusion in a bounded n-dimensional domain.
Le Ω⊂Rnbe a bounded open subse wi h smoo h bounda y. We conside
he p oblem 




∂u
∂ −∆u ∈b( )H0(u) + ω( )u, on Ω×(τ, ∞),
u|∂Ω = 0,
u(τ, x) = uτ(x),
(2)
whe e b:R→R+, ω :R→R+a e con inuous unc ions such ha
0< b0≤b( )≤b1,0≤ω0≤ω( )≤ω1,
and
H0(u) = 


−1,i u < 0,
[−1,1],i u= 0,
1,i u > 0,
is he Hea iside unc ion.
We ew i e (2) in he abs ac o m
(∂u
∂ +∂ψ(u)−R( , u( )) 30,
u(τ) = uτ,
whe e ∂ψ is he subdi e en ial o he p ope , con ex, lowe semicon inuous unc ion
ψ:L2(Ω)→(−∞,+∞] gi en by
ψ(u)=1
2RΩ|∇u|2dx, i u∈H1
0(Ω),
+∞, o he wise,
∂ψ(u) = ny∈L2(Ω) : y(x) = −∆u(x), a.e. on Ωo,
D(∂ψ)=H2(Ω)∩H1
0(Ω)and o any ∈R,
R( , u) = ny∈L2(Ω):y(x)∈b( )H0(u(x)) +ω( )u(x), a.e. on Ωo.
We no e ha in ou pa icula case he ope a o ∂ψ :H2(Ω)∩H1
0(Ω)→L2(Ω)
is single- alued and linea . In he sequel, as ∂ψ is also he gene a o o a C0-
semig oup, o con enience we shall use he no a ion A=∂ψ. Also, we obse e
ha D(ψ) = L2(Ω).
4 Tom´as Ca aballo e al.
Le us in oduce some no a ion. Th oughou his pape we deno e by k·kX
he no m in he Banach space X, whe eas k·k,(·,·)will be used o he no m
and scala p oduc in he space L2(Ω) (and wi h some abuse o no a ion also in
(L2(Ω))d,d∈N). Also, P(X) will be he se o all non-emp y subse s o Xand
2X=P(X)∪∅. The Hausdo semidis ance om he se C o he se Bis gi en
by
dis (C, B) = sup
y∈C
in
z∈Bky−zk,
whe eas he Hausdo dis ance is de ined by
dis H(C, B) = max{dis (C, B), dis (B, C)}.
Fo C⊂Xan ε-neighbo hood is he se Oε(C) = {z∈X:dis (z, C)< ε}.
Fo a mul i alued map G:X→2Xwe deno e D(G) = {u∈X:G(u)∈
P(X)}. The map Gis called uppe semicon inuous i o any u∈D(G) and any
neighbo hood Oo G(u) he e exis s δ > 0 such ha G( )⊂Oas soon as ku− k<
δ. i is said o be w-uppe semicon inuous i o all  > 0, u ∈D(G), he e is δ > 0
such ha G( )⊂O(G(u)) i ku− k< δ. Any uppe semicon inuous map is
w-uppe semicon inous, he con e se being ue i Ghas compac alues [3, p.45].
We ecall he concep o s ong solu ion o p oblem (2).
De ini ion 1 We say ha he unc ion u∈C([τ, +∞), L2(Ω)) is a s ong solu ion
o (2) i :
1. u(τ) = uτ;
2. Fo any δ > 0, τ+δ < T,u(·) is absolu ely con inuous on [τ+δ, T] and
u( )∈D(A) o a.a. ∈(τ, T);
3. The e exis s a unc ion : [τ, +∞)→L2(Ω) such ha ( )∈R( , u( )), ∈
L2τ, T;L2(Ω) o any T > τ, and
du
d +Au( ) = ( ), o a.a. ∈(τ, +∞),(3)
whe e he equali y is unde s ood in he sense o he space L2(Ω).
Lemma 1 Fo e e y s ong solu ion o (2) he unc ion (·)belongs o L∞τ, T;L2(Ω)
o any T > τ.
P oo The s a emen ollows eadily om u∈C([τ, +∞), L2(Ω)), ∈L2τ, T;L2(Ω)
and he inequali y
| ( , x)| ≤ b( )+ω( )|u( , x)| o a.a. ( , x).
We deno e by :R×R→P(R) he mul i alued unc ion gi en by ( , u) =
b( )H0(u).Then possesses nonemp y, closed, bounded and con ex alues, and
o all ∈R he map ( , ·) : R→P(R) is uppe semicon inuous. Mo eo e , o
any , s ∈R+,u∈R,
dis H( ( , u), (s, u)) = |b( )−b(s)|,
and
sup
y∈ ( ,u)|y|=b( ).
Ex emal ajec o ies o equa ions wi h discon inuous o cing e m 5
The las equali y implies ha , o all y∈L2(Ω) and o a.a. ∈R+,
sup
ξ∈R( ,y)kξk ≤ |Ω|1/2b( ).
The ollowing esul ollows om Lemma 6.28 in [24].
Lemma 2 The map Rsa is ies he ollowing p ope ies:
1. R:R×L2(Ω)→2L2(Ω)has nonemp y, closed, bounded and con ex alues;
2. Fo any ∈R, he map R( , ·) : L2(Ω)→P(L2(Ω)) is w-uppe semicon inuous;
3. Fo all y∈L2(Ω),τ∈R, he map R(·, y):[τ, +∞)→P(L2(Ω)) possesses a
measu able selec ion, ha is, he e exis s a measu able unc ion h: [τ, +∞)→
L2(Ω)such ha h( )∈R( , y) o a.a. > τ.
Theo em 1 Fo any uτ∈L2(Ω), p oblem (2) has a leas one s ong solu ion.
P oo I we ix an in e al [τ, T], he exis ence o a s ong solu ion ollows om
Theo em 6.11 and Lemma 6.16 in [24]. Also, adap ing Lemma 6.31 in [24] o he
nonau onomous case we ob ain ha he conca ena ion o wo s ong solu ions is
again a s ong solu ion, so e e y solu ion in an in e al [τ, T] can be ex ended o
a global one, ha is, de ined o ∈[τ, +∞).
Le us conside he auxilia y p oblem
(du
d +Au( ) = g( ), ∈(τ, T),
u(τ) = uτ,
(4)
whe e g∈L1(τ, T;L2(Ω)).
The con inuous unc ion u: [τ, T]→L2(Ω)is said o be a s ong solu ion
o (4) on [τ, T], i u(·) is absolu ely con inuous on any compac subse o (τ, T),
u( )∈D(A) o a.a. ∈(τ, T)and
du
d +Au( ) = g( ) o a.a. ∈(τ, T).
The con inuous unc ion u: [τ, +∞)→L2(Ω)is called in gene al a s ong
solu ion i i is a s ong solu ion on e e y in e al [τ, T].
P oposi ion 1 ([7, Theo em 3.6] o [6, p.189]) Fo any g(·)∈L2(τ, T;L2(Ω)),uτ∈
L2(Ω), he e exis s a unique s ong solu ion o inclusion (2) on [τ, T]sa is ying
√ du
d ∈L2(τ, T;L2(Ω)), ψ(u(·)) ∈L1(τ, T ).(5)
Also, he map 7→ ψ(u( )) is absolu ely con inuous on [τ+δ, T], o all 0< δ < T −τ.
I , mo eo e , uτ∈ D(ψ), hen du
d ∈L2τ, T;L2(Ω)and 7→ ψ(u( )) is abso-
lu ely con inuous on [τ, T].

6 Tom´as Ca aballo e al.
Co olla y 1 E e y s ong solu ion u(·) o p oblem (2) sa is ies
u∈L2τ, T;H1
0(Ω),(6)
√ du
d ∈L2(τ, T;L2(Ω)),
du
d ∈L2(τ, T;H−1(Ω)),
o all T > τ. Mo eo e :
1. The map 7→ ku( )k2is absolu ely con inuous on e e y in e al [τ, T]and
d
d ku( )k2= 2 du
d , u ( ) o a.a. ∈(τ, T).(7)
2. The map 7→ k∇u( )k2is absolu ely con inuous on e e y in e al [τ+δ, T]wi h
0< δ < T −τ,
d
d k∇u( )k2= 2 du
d ,−∆u ( ), o a.a. ∈(τ, T),(8)
and
u∈C([τ+δ, +∞), H1
0(Ω)).(9)
3. I uτ∈H1
0(Ω), hen du
d ∈L2τ, T;L2(Ω)and 7→ k∇u( )k2is absolu ely
con inuous on e e y in e al [τ, T].Also,
u∈C([τ, +∞), H1
0(Ω)).(10)
P oo Equali y (7) ollows om [15, p.285] and he es o p ope ies, excep (8)-
(10), a e a consequence o P oposi ion 1. Fo he equali y (8) see [6, p.189]. I uτ∈
H1
0(Ω), as 7→ u( )is weakly con inuous wi h espec o H1
0(Ω), 7→ ku( )kH1
0(Ω)
is con inuous and H1
0(Ω)is a Hilbe space, p ope y (10) ollows. The p oo o
(9) is analogous.
Ou aim is o p o e he ollowing compa ison p inciple.
De ini ion 2 The s ong solu ions o p oblem (2) sa is y a s ong compa ison
p inciple i o any ini ial da a uτ≤ τ he e exis s ong solu ions u(·), (·)such
ha u(τ)=uτ, (τ)= τand
u( )≤ ( ),
u( )≤ ( ),∀ ≥τ,
whe e u(·), (·)a e a bi a y s ong solu ions o p oblem (2) such ha u(τ)=
uτ, (τ)= τ.
Ex emal ajec o ies o equa ions wi h discon inuous o cing e m 7
In pa icula , his de ini ion implies ha o e e y ini ial da a he e exis a
maximal and a minimal s ong solu ion.
Le us conside he ollowing pa abolic p oblems





∂u
∂ −∆u =b( ) ε(u) + ω( )u, on Ω×(τ, +∞),
u|∂Ω = 0,
u(τ, x) = uτ(x),
(11)





∂u
∂ −∆u =b( ) ε(u) + ω( )u, on Ω×(τ, +∞),
u|∂Ω = 0,
u(τ, x) = uτ(x),
(12)
whe e ε, ε∈C1(R), 0
ε(u), 0
ε≥0,  0
ε(u), 0
ε(u)≤Cε o all u, and
ε(u) = 


−1,i u≤ −ε,
−1≤ ε(u)≤1,i −ε < u < 0,
1,i u≥0,
ε(u) = 


−1,i u≤0,
−1≤ ε(u)≤1,i 0 < u < ε,
1,i u≥ε.
I is s aigh o wa d ha
ε(u)≥sup
y∈H0( )
yi u≥ , (13)
ε(u)≤in
y∈H0( )yi u≤ .
Wi h ob ious li le changes, we can ex end he de ini ion o s ong solu ions o
p oblems (11)-(12). Le us show ha hese p oblems ha e a unique s ong solu ion.
Le us jus conside p oblem (11).
F om now on, o ∈L2(Ω), we deno e by ε( ) ( ε( )) he elemen y∈
L2(Ω))such ha y(x)= ε( (x)) (= ε( (x))) o a.a. x. In he same way, o
h∈L1(τ, T;L2(Ω)) we deno e h( ):= h( ,·)∈L2(Ω)).
We know om [15, p.283] ha o any uτ∈L2(Ω) he e exis s a unique
weak solu ion o p oblem (11), which means ha u∈C([0,+∞), L2(Ω)), u∈
L2(τ, T;H1
0(Ω)), o all T > τ, and
d
d (u( ), )−h∆u, i=( ε(u( )), ),∀ ∈H1
0(Ω),
whe e he equali y is unde s ood in he sense o dis ibu ions on e e y in e al
(τ, T)and h·,·i is pai ing be ween H−1(Ω) and H1
0(Ω).
Lemma 3 Fo any uτ∈L2(Ω) he e exis s a unique s ong solu ion uε(·)o p oblem
(11), which coincides wi h he unique weak solu ion o (11).
8 Tom´as Ca aballo e al.
P oo Le uε(·)be he unique weak solu ion o p oblem (11). I we pu gε( ) =
ε(uε( )) + ω( )uε( ), hen gε∈L2(τ, T;L2(Ω)), o all T > τ, and we can conside
he linea p oblem





∂z
∂ −∆z =gε( ),on Ω×(τ, +∞),
z|∂Ω = 0,
z(τ, x) = zτ(x).
(14)
On he one hand, uε(·)is he unique weak solu ion o p oblem (14). On he o he
hand, i ollows om P oposi ion 1 ha p oblem (14) possesses a unique s ong
solu ion z(·). I we we e able o show ha z(·)is also a weak solu ion o (14), hen
we would ob ain ha z=uε, so uε(·)would be a s ong solu ion o (11). Indeed.
In iew o P oposi ion 1 we ob ain ha z∈L2(τ, T;H1
0(Ω)) o all T > τ. F om
he equali y in (14) we in e hen ha dz
d ∈L2(τ, T;H−1(Ω)), which implies by
[28, Lemma 7.4] ha
dz
d , −h∆z, i=(gε( ), ),∀ ∈H1
0(Ω).
Hence, by [32, p. 250] we ha e
d
d (z, )−h∆z, i=(gε( ), ),
so z(·)is a weak solu ion o (14). Thus, z=uεis a s ong solu ion o p oblem
(11).
I emains o check uniqueness. Le (·)be an a bi a y s ong solu ion o (11).
Then, i is a s ong solu ion o p oblem (14) wi h gε( ) = ε( ( )) + ω( ) ( ). By
he p e ious a gumen (·)is a weak solu ion o (14) and hen a weak solu ion o
(11) as well. The e o e, is equal o uε, he unique weak solu ion o (11). Thus,
=z.
Rema k 1 The unc ion hε( ) = b( ) ε(uε( ))+ω( )uε( )belongs o L∞τ, T;L2(Ω)
o any T > τ.
Co olla y 2 Fo any uτ∈L2(Ω) he unc ion uε∈C([τ, +∞), L2(Ω))is a s ong
solu ion o p oblem (11) i and only i i is a weak solu ion.
I is well known [27, Chap e 7] ha ope a o −A=∆u :D(A) = H2(Ω)∩
H1
0(Ω)→L2(Ω)is he gene a o o a s ongly con inuous semig oup o bounded
linea ope a o s S( ):L2(Ω)→L2(Ω), ≥0, which will be deno ed in he sequel
by S( )=e−A . Mo eo e , i is a semig oup o con ac ions, ha is, 

e−A 

≤1.
Fo e e y x∈D(A) he unc ion u( )=e−A xis he unique classical solu ion (see
he de ini ion below) o he p oblem
(du
d +Au( ) = 0, > 0,
u(0)=x.
(15)
Also, he semig oup e−A is posi i e o all ≥0 [13, Chap e 12].
Le us de ined he concep o mild solu ion o he inhomogeneous p oblem (4).
Ex emal ajec o ies o equa ions wi h discon inuous o cing e m 9
De ini ion 3 Le uτ∈L2(Ω)and g∈L1
loc(τ, +∞;L2(Ω)). Then he unc ion
u∈C([τ, +∞), L2(Ω)) is called a mild solu ion o p oblem (4) i
u( )=e−A( −τ)uτ+Z
τ
e−A( −s)g(s)ds,τ≤ < ∞.(16)
I is called a classical solu ion i u(·)is con inuously di e en iable on (τ, +∞),
u( )∈D(A) o any ∈(τ, +∞),u(τ)=xand he equali y in (4) is sa is ied o
e e y ∈(τ, ∞).
Fo e e y uτ∈L2(Ω)and g∈L1
loc(τ, +∞;L2(Ω)) he e exis s a unique mild
solu ion o p oblem (4). Mo eo e , i uτ∈D(A)and gis con inuously di e en iable
on [τ, +∞), hen he mild solu ion is he unique classical solu ion [27, p.107].
We can also de ine mild solu ions o p oblems (11) and (2).
De ini ion 4 Le uτ∈L2(Ω). Then he unc ion uε∈C([τ, +∞), L2(Ω)) is called
a mild solu ion o p oblem (11) i
uε( )=e−A( −τ)uτ+Z
τ
e−A( −s)(b( ) ε(uε( )) + ω( )uε( ))ds,τ≤ < ∞.
We no e ha o any uε∈C([τ, ∞), L2(Ω)) he map ε(uε( , ·)) belongs o
L∞
loc 0,+∞;L2(Ω)⊂L2
loc 0,+∞;L2(Ω).
De ini ion 5 Le uτ∈L2(Ω). Then he unc ion u∈C([τ, +∞), L2(Ω)) is called
a mild solu ion o p oblem (2) i he e exis s hsuch ha h∈L2
loc 0,+∞;L2(Ω),
o any T > τ, h ( , x)∈H0(u( , x)) , o a.a. ( , x), and
u( )=e−A( −τ)uτ+Z
τ
e−A( −s)(b( )h( ) + ω( )u( ))ds,τ≤ < ∞.
Lemma 4 Fo any uτ∈L2(Ω)and g∈L2
loc 0,+∞;L2(Ω) he unc ion u∈
C([τ, +∞), L2(Ω))is a s ong solu ion o p oblem (4) i and only i i is a mild solu-
ion.
P oo The p oo ollows he same lines o [39, Lemma 2], bu we p o ide i in de ail
o he sake o comple eness.
Le u(·)be a s ong solu ion. We ake sequences un
τ∈D(A),gn(·)∈C1[τ, +∞), L2(Ω)
such ha
un
τ→uτin L2(Ω),
gn→gin L2τ, T;L2(Ω)∀T > τ.
Deno e by un(·) he unique classical solu ion o he p oblem
(dun
d =Aun( ) + gn( ), > τ,
un(τ)=un
τ.
Le T > τ and 0 < ε < T −τbe a bi a y. We no e ha wnis he unique s ong
solu ion o p oblem (4) wi h eg( )=gn( )−g( )and w(τ)=un
τ−uτ, so by (7) we
ha e
2d
d (un−u), un−u=d
d 
un( )−u( )
2 o a.a. ∈(τ+ε, T).
16 Tom´as Ca aballo e al.
We now s udy a compa ison p inciple be ween he solu ions in he nonau ono-
mus and au onomous cases.
Le us conside he p oblem





∂u
∂ −∆u ∈bH0(u) + ωu, on Ω×(τ, ∞),
u|∂Ω = 0,
u(τ, x) = uτ(x),
(29)
whe e 0 < b, 0 ≤ω. Le us deno e a s ong solu ion o p oblem (29) by ub,ω (·)
(and we do no ake in o accoun in his no a ion he ini ial ime τas he solu ion
is he same wha e e he alue o τ).
Theo em 3 Fo any ini ial da um uτ≥0 he e exis s a non-nega i e solu ion ub1,ω1(·)
o p oblem (29) wi h u(τ)=uτ, b =b1, ω =ω1such ha
( )≤ub1,ω1( ),∀ ≥τ, (30)
whe e (·)is an a bi a y s ong non-nega i e solu ion o (2) wi h u(τ)=uτ.
On he o he hand, he e exis a non-nega i e solu ion u(·) o (2) wi h u(τ)=uτ
such ha
u( )≥ub0,ω0( ),∀ ≥τ, (31)
whe e ub0,ω0(·)is an a bi a y s ong non-nega i e solu ion o (29) wi h u(τ)=
uτ, b =b0, ω =ω0.
P oo Le F:V→C([τ, τ + 0], L2(Ω)) be de ined by
F(u)( ) = e−A( −τ)uτ+Z
τ
e−A( −s)(b1 ε(u(s)) + ω1u(s))ds, (32)
whe e
V={u∈C([τ, τ+ 0], L2(Ω)) : u(τ) = uτ, u ( )≥0,ku( )−uτk ≤ 1,∀ ∈[τ, τ+ 0]},
and 0>0 sa is ies (17). Le z0∈L2(Ω)be such ha z0(x)= 1 o a.a. x.
Since ε(u(s)) = z0, o any u∈V, and e−A ≥0, we ha e ha F(u)( )≥0.Then,
a guing in he same way as in Theo em 2 we ob ain ha F:V→Vis a con ac ing
map, so Fpossesses a unique ixed poin u∈V, which is a mild solu ion and,
by Lemma 4, coincides wi h he unique s ong solu ion uε o p oblem (11) wi h
b( )=b1, ω ( )=ω1.
Le (·)be a non-nega i e s ong solu ion o (2) such ha (τ)=uτ. Then
he e exis s hsuch ha h∈L∞τ, T;L2(Ω), o any T > τ,h(s, x)∈H0( (s, x))
o a.a. (s, x), and (·)is he unique s ong solu ion o p oblem (19). Mo eo e , (·)
sa is ies (20) and i is he unique ixed poin o he con ac i e map F1:V1→V1
gi en by
F1(u)( ) = e−A( −τ)uτ+Z
τ
e−A( −s)(b(s)h(s) + ω(s)u(s))ds,
whe e
V1={u∈C([τ, τ + 0], L2(Ω)) : u(τ) = uτ,ku( )−uτk ≤ 1,∀ ∈[τ, τ + 0]}.

Ex emal ajec o ies o equa ions wi h discon inuous o cing e m 17
Le us de ine he se b
V={u∈V:u( )≥ ( )}, which is ob iously non-emp y
as ∈b
V.
Fo any u∈b
V, since ε(u(·, s)) = z0and u(s)≥ (s)≥0, we ha e
b1 ε(u(s)) −b(s)h(s) = b1z0−b(s)h(s)≥(b1−b(s)) z0≥0,
ω1u(s)−ω(s) (s)≥0.
Then, by e−A ≥0, we ob ain
F(u)( )− ( )=Z
τ
e−A( −s)(b1 ε(u(s)) −b(s)h(s) + ω1u(s)−ω(s) (s))ds
≥0, o any ∈[τ, τ + 0].
The e o e, F(b
V)⊂b
V. Since Fis a con ac ion in V, i is a con ac ion in b
Vas
well. We deduce ha Fpossesses a unique ixed poin u∈b
V, which is equal o he
solu ion uε o p oblem (11) wi h b( )=b1, ω ( )=ω1. As be o e, by a s anda d
con inua ion a gumen , i ollows ha
uε( )≥ ( ) o any ≥τ.
Passing o he limi in exac ly he same way as in Theo em 2 we ob ain he
exis ence o a solu ion ub1,ω1(·) o p oblem (29) such ha (30) holds.
Le F2:V→C([τ, τ + 0], L2(Ω)) be de ined by
F2(u)( ) = e−A( −τ)uτ+Z
τ
e−A( −s)(b(s) ε(u(s)) + ω(s)u(s))ds. (33)
Le z0∈L2(Ω)be such ha z0(x)= 1 o a.a. x. Since ε(u(s)) = z0, o any
u∈V, and e−A ≥0, we ha e F(u)( )≥0.Then, a guing in he same way as in
Theo em 2 we ob ain ha F2:V→Vis a con ac ing map, so F2possesses a
unique ixed poin u∈V, which is a mild solu ion and, by Lemma 4, coincides
wi h he unique s ong solu ion uε o p oblem (11).
Le ub0,ω0(·)be a non-nega i e s ong solu ion o (29) such ha ub0,ω0(τ)=uτ,
b=b0, ω =ω0. Then he e exis s hsuch ha h∈L∞τ, T;L2(Ω), o any T > τ,
h(s, x)∈H0(ub0,ω0(s, x)) o a.a. (s, x), and ub0,ω0(·)is he unique s ong solu ion
o p oblem (19) wi h b( )=b0, ω ( )=ω0. Mo eo e , ub0,ω0(·)sa is ies (20) and
i is he unique ixed poin o he con ac i e map F3:V1→V1gi en by
F3(u)( ) = e−A( −τ)uτ+Z
τ
e−A( −s)(b0h(s) + ω0u(s))ds.
We de ine he se b
V1={u∈V:u( )≥ub0,ω0( )}, which is non-emp y as
ub0,ω0∈b
V.
As be o e, o any u∈b
V1we ha e
b(s) ε(u(s)) −b0h(s) = b(s)z0−b0h(s)≥(b(s)−b0)z0≥0,
ω(s)u(s)−ω0ub0,ω0(s)≥0,
18 Tom´as Ca aballo e al.
and
F2(u)( )−ub0,ω0( )=Z
τ
e−A( −s)b(s) ε(u(s)) −b0h(s) + ω(s)u(s)−ω0ub0,ω0(s)ds
≥0, o any ∈[τ, τ + 0].
Thus, F2(b
V)⊂b
V. Since F2is a con ac ion in V, i is a con ac ion in b
Vas well,
so F2possesses a unique ixed poin u∗∈b
V, which is equal o he solu ion uε o
p oblem (11). As be o e, by a s anda d con inua ion a gumen , i ollows ha
uε( )≥ub0,ω0( ) o any ≥τ.
Again, passing o he limi we ob ain he exis ence o a solu ion u(·) o p oblem
(29) such ha (31) holds.
3 Cha ac e iza ion o he global a ac o in he au onomous case
In his sec ion we will s udy he au onomous di e en ial inclusion (2) in he scala
case and will deduce om he s ong compa ison p inciple some p ope ies con-
ce ning he s uc u e o he global a ac o .
Hence, we conside he au onomous p oblem





∂u
∂ −∂2u
∂x2∈bH0(u) + ωu, on (0,1) ×(τ, ∞),
u|∂Ω = 0,
u(0, x) = u0(x),
(34)
whe e 0 < b, 0 ≤ω. We assume also h oughou his sec ion ha
0≤ω < π2,
whe e π2is he i s eigen alue o he ope a o −∂2
∂x2in H1
0(0,1).This es ic ion
is necessa y in o de o gua an ee he exis ence o a global a ac o .
Le D(u0) deno e he se o all s ong solu ions o (34) o u0∈L2(Ω) and
τ= 0. We de ine he mul i alued amily o ope a o s G:R+×L2(Ω)→P(L2(Ω)),
whe e P(X) s ands o he se o all non-emp y subse s o he space X, by
G( , u0) = {u( ) : u(·)∈ D(u0)}.
I is well-known [36] ha Gis a s ic mul i alued semi low, i.e., G( +s, u0) =
G( , G(s, u0)), o all , s ≥0, u0∈L2(Ω), possessing a global compac in a ian
a ac o A. This means ha :
–A=G( , A) o all ≥0 (s ic in a iance);
–dis (G( , B),A)→0 as →+∞ o any bounded se B(a ac ing p ope y).
Mo eo e , he a ac o Ais connec ed [37] and bounded in H1
0(Ω) [36]. Also,
Ghas compac alues and he ope a o u07→ G( , u0) is uppe semicon inuous o
any ≥0 (see [36] again).
We ecall he concep o o de -p ese ing mul i alued semi low, which was
in oduced in [11].
Ex emal ajec o ies o equa ions wi h discon inuous o cing e m 19
De ini ion 6 The mul i alued semi low Gis called o de -p ese ing i o any u0≤
0and ≥0 we ha e:
1. The e exis s y∈G( , u0) such ha
y≤y o all y∈G( , 0).
2. The e exis s y∈G( , 0) such ha
y≤y o all y∈G( , u0).
Theo em 2 implies ha he semi low Ggene a ed by he solu ions o (2) is
o de -p ese ing.
We ecall ha xis called a ixed poin (o equilib ium) o he mul i alued
semi low Gi x∈G( , x) o all ≥0.
The se o s ong solu ions R=∪u0∈L2(Ω)D(u0) sa is ies he ollowing p ope -
ies [18]:
(H1)Fo any x∈L2(Ω) he e exis s ϕ∈ R such ha ϕ(0)=x.
(H2)ϕτ(·)=ϕ(·+τ)∈ R o any τ≥0, ϕ(·)∈ R ( ansla ion p ope y).
(H3)Le ϕ1, ϕ2∈ R be such ha ϕ2(0) = ϕ1(s), whe e s > 0. Then he unc ion
ϕ(·),de ined by
ϕ( ) = ϕ1( )i 0 ≤ ≤s,
ϕ2( −s)i s≤ ,
belongs o R(conca ena ion p ope y).
(H4)Fo any sequence ϕn(·)∈ R such ha ϕn(0)→ϕ0in L2(Ω), he e exis s a
subsequence ϕnkand ϕ∈ R such ha
ϕnk( )→ϕ( ),∀ ≥0.
The elemen xis called a ixed poin (o equilib ium) o Ri ϕ( )≡x∈ R.
Since (H1)−(H4)hold, i is well-known ha xis a ixed poin o Gi and only i
i is a ixed poin o R[23, Lemma 7].
Applying Theo em 2 in [11] we ob ain he ollowing esul .
Theo em 4 The e exis wo equilib ia x∗,y∗∈ A such ha :
1. x∗≤z≤y∗ o all z∈ A.
2. I he solu ions co esponding o he ini ial condi ions x∗,y∗a e unique, hen
dis (G( , u0), x∗)→0, as →+∞, o any u0≤x∗,(35)
dis (G( , u0), y∗)→0, as →+∞, o any u0≥y∗.(36)
We obse e ha in [11] he ollowing addi ional assump ion conce ning he
o de ela ion ’≤’ was assumed: o any bounded se B he e exis s a, d such ha
a≤y≤d o all y∈B,
which means ha Bis con ained in an in e al [a, d]. Though his assump ion is
no ue in he space L2(Ω), in he p oo o Theo em 2 in [11] i is only necessa y
o use his p ope y o he global a ac o A, and no o an a bi a y bounded
se B. Since Ais bounded in H1
0(Ω), which is con inuously embedded in C([0,1]),
20 Tom´as Ca aballo e al.
he global a ac o is in ac con ained in an in e al [a, d], so Theo em 2 in [11] is
applicable.
The ixed poin s o Gwe e desc ibed explici ly in [4]. The e exis s an in ini e bu
coun able numbe o ixed poin s, deno ed by 0= 0, +
1(x), −
1(x), ..., +
n(x), −
n(x), ...,
whe e ±
j(x) possess exac ly j+ 1 ze oes in [0,1] and +
j(x) = − −
j(x). The exac
exp ession o he poin +
1is gi en by
+
1(x) = b0
ω0cos (√ω0x)+b0(1−cos (√ω0))
ω0sin (√ω0)sin (√ω0x)−b0
ω0
,
which is he unique solu ion o he bounda y- alue p oblem
u00 +ω0u=−b0, u (0)=u(1)= 0.
F om he analysis in [4] we in e ha
−
1(x)≤ ±
j(x)≤ +
1(x) o all j≥1,
and hen x∗= −
1,y∗= +
1so Theo em 4 implies ha
−
1≤z≤ +
1 o all z∈ A.
Also, since he poin s ±
1a e s able [4, Theo em 6.3], he solu ions co esponding
o he ini ial condi ions ±
1a e unique. Hence, he con e gences (35), (36) hold
ue.
Finally, we ema k ha in [4, Theo em 6.3] he s uc u e o he global a ac o
was s udied.
We ecall ha a map φ:R→L2(Ω) is a comple e ajec o y i
φ(·+h)|[0,∞)∈ R, o all h∈R.
The global a ac o Aconsis s o all bounded comple e ajec o ies and i consis s
i ac o he ixed poin s and all comple e bounded ajec o ies ψ(·)connec ing
wo ixed poin s, ha is,
ψ( )→z1as →+∞,(37)
ψ( )→z2as → −∞,
whe e zja e ixed poin s. Pa ial esul s ela ed o how he ixed poin s a e con-
nec ed we e gi en in [4, Theo em 6.3]: i zmeans ha he e is a connec ion
om o z, hen:
1. 0 ±
j,∀j≥1;
2. +
j ±
j−1, −
j ±
j−1,∀j≥2;
3. +
j ±
1, −
j e±
1,∀j≥2;
4. I ±
k ±
j(k, j 6= 0), hen
±
kn ±
jn,∀n≥1;
5. I 1 ≤k≤j, hen ±
k ±
j, ±
k 0a e o bidden.
Now we ha e comple ed his desc ip ion by showing ha all hese bounded
comple e ajec o ies lie inside he in e al [ −
1, +
1].
Ex emal ajec o ies o equa ions wi h discon inuous o cing e m 21
4 Cha ac e iza ion o he pullback a ac o in he nonau onomous case
In his sec ion we will ea he nonau onomous di e en ial inclusion (2) in he
scala case, ha is, we conside he nonau onomous p oblem





∂u
∂ −∂2u
∂x2∈b( )H0(u) + ω( )u, on (0,1) ×(τ, ∞),
u|∂Ω = 0,
u(τ, x) = u0(x),
(38)
whe e b( ), ω ( ), H0(u) a e as in Sec ion 2. Addi ionally, we assume in he sequel
ha
ω1< π2.
We s a by de ining a mul i alued p ocess associa ed wi h he s ong solu ions
o p oblem (38) and showing ha i possesses a pullback a ac o . A e ha we
will gi e a cha ac e iza ion o his a ac o in a simila manne as in Theo em 4
o he au onomous case. In pa icula , we will show he exis ence o a bounded
comple e non-degene a e ajec o y a −∞ which is unique in he class o non-
degene a e bounded comple e ajec o ies in he whole line.
Le Dτ(uτ) be he se o all s ong solu ions o (38) wi h ini ial condi ion uτa
ime τand le Rτ=∪x∈L2(Ω)Dτ(x). In he same way as in he au onomous case,
one can p o e ha he se s Rτsa is y he ollowing p ope ies:
(K1)Fo any τ∈Rand x∈L2(Ω) he e exis s ϕ∈ Rτsuch ha ϕ(τ)=x.
(K2)ϕs=ϕ|[τ+s,∞)∈ Rτ+s o any s≥0, ϕ∈ Rτ( ansla ion p ope y).
(K3)Le ϕ, ψ ∈ R be such ha ϕ∈ Rτ,ψ∈ R and ϕ(s) = ψ(s) o some s≥ ≥τ.
Then he unc ion θde ined by
θ( ) := ϕ( ), ∈[τ, s],
ψ( ), ∈[s, ∞),
belongs o Rτ(conca ena ion p ope y).
(K4)Fo any sequence ϕn∈ Rτsuch ha ϕn(τ)→ϕ0in L2(Ω), he e exis s a
subsequence ϕnkand ϕ∈ Rτsuch ha
ϕnk( )→ϕ( ),∀ ≥τ.
We de ine he mul i alued amily o ope a o s U:R2
≥×L2(Ω)→P(L2(Ω)),
whe e R2
≥={( , s)∈R2: ≥s}, by
U( , s, x) = {u( ) : u(·)∈ Ds(x)}.
I easily ollows om (K1)−(K3) ha Uis a s ic mul i alued p ocess, ha is,
U( , , ·)=Id is he iden i y map and U( , s, x)=U( , τ, U (τ, s, x)) o all s≤τ≤ ,
x∈L2(Ω).Mo eo e , (K4)implies ha he g aph o he map x7→ U( , s, x) is
closed o all ( , x)∈Rd.
We ecall ha he amiliy o se s {K( )} ∈Ris called pullback a ac ing o U
i i a ac s e e y bounded se Bin he pullback sense, ha is,
dis (U( , s, B), K( )) →0,as s→ −∞.(39)
Lemma 5 The p ocess Uhas a pullback a ac ing amily o compac se s {K( )} ∈R.

22 Tom´as Ca aballo e al.
P oo Fo any s ong solu ion mul iplying equali y (3) by uwe ha e
1
2
d
d kuk2+


∂u
∂x( )



2
=Z1
0
( , x)u( , x)dx
≤b( )Z1
0|u( , x)|dx +ω( )ku( )k2
≤b1ku( )k+ω1ku( )k2
≤b2
1
4ε0+(ε0+ω1)ku( )k2,
whe e ε0is chosen such ha ε0+ω1< π2. Then, as π2is he i s eigen alue o
he ope a o −∂2u
∂x2in H1
0(Ω), we ob ain
d
d kuk2+δku( )k2≤d
d kuk2+δ
π2


∂u
∂x( )



2
≤b2
1
2ε0=C1,
whe e δ= 2(π2−ω1−ε0)>0. By G onwall’s lemma we ge
ku( )k2≤e−δ( −s)ku(s)k2+C1
δ o all ≥s. (40)
Also, in eg a ing o e ( −α, ), whe e 0 < α ≤1,we ha e
Z
−α


∂u
∂x



2
d ≤π2C1
δ+π2
δku( −α)k2.(41)
Fu he , we mul iply (3) by du
d and use Co olla y 1 o ob ain ha



du
d 



2
+1
2
d
d 


∂u
∂x



2
≤b( )Z1
0∂u
∂x( , x)dx +ω( )Z1
0
u( , x)∂u
∂x( , x)dx (42)
≤b1


∂u
∂x ( )


+ω1ku( )k


∂u
∂x ( )



≤b2
1
2+ω2
1
2ku( )k2+


∂u
∂x ( )



2
.
Fo s≤ −α≤ ≤ we in eg a e o e he in e al ( , ). Hence, by (40) and (41)
we ha e



∂u
∂x( )



2
≤


∂u
∂x( )



2
+b2
1+ω2
1Z
ku(τ)k2dτ + 2 Z



∂u
∂x (τ)



2
dτ
≤


∂u
∂x( )



2
+b2
1+ (ω2
1+2π2
δ)e−δ( −α−s)ku(s)k2+C1
δ(ω2
1+ 2π2+2π2
δ).
In eg a ing now wi h espec o he a iable o e he in e al ( −α, )and using
again (40) and (41) we ge
α


∂u
∂x( )



2
≤π2C1
δ+π2
δku( −α)k2
+b2
1+ (ω2
1+2π2
δ)(e−δ( −α−s)ku(s)k2) + C1
δ(ω2
1+ 2π2+2π2
δ)
≤C2+C3e−δ( −α−s)ku(s)k2,(43)
Ex emal ajec o ies o equa ions wi h discon inuous o cing e m 23
whe e C2, C3>0 a e some cons an s.
We de ine he amily K( ) by
K( ) = {y∈H1
0(Ω) : kyk2
H1
0≤2C2
α}.
The compac embedding H1
0(Ω)⊂L2(Ω) implies ha K( ) a e ela i ely compac
in L2(Ω). Also, as K( ) is weakly closed in H1
0(Ω), i is closed in L2(Ω). Thus,
K( ) a e compac in L2(Ω). Finally, we ob ain eadily om (43) ha o any
bounded se Band any ∈R he e exis s T(B, ) such ha U( , s, B)⊂K( ) o
all s≤T(B, ). Thus, (39) ollows.
We ecall he de ini ion o pullback a ac o .
De ini ion 7 The amily o compac se s {A( )} ∈Ris called a pullback a ac o
i :
1. I is pullback a ac ing.
2. A( )⊆U( , s, A(s)), o all ≥s(nega i e semi-in a iance);
3. {A( )} ∈Ris minimal in he sense ha i {K( )} ∈Ris a pullback a ac ing
amily o closed se s, hen A( )⊂K( ) o all ∈R.
The pullback a ac o is s ic ly in a ian i A( ) = U( , s, A(s)), o any ≥s.
Theo em 5 The p ocess Upossesses a s ic ly in a ian pullback a ac o {A( )} ∈R.
Mo eo e , ∪ ∈RA( )is bounded in H1
0(Ω)and ∪ ∈RA( )is compac in L2(Ω).
P oo Since he e exis s a pullback a ac ing amily o compac se s {K( )} ∈R
and he map x7→ U( , τ, x) has closed g aph o all ≥τ(by (K4)), he e exis s
a compac pullback a ac o {A( )} ∈R, which sa is ies A( )⊂K( ) o all ∈R
(see Theo em 1 in [22]). Mo eo e , since A( )⊂K( ) and he de ini ion o K( )
we deduce ha ∪ ∈RA( ) is bounded in H1
0(Ω) and ∪ ∈RA( ) is compac in L2(Ω).
Using his and he ac ha Uis a s ic p ocess we also ob ain ha he pullback
a ac o is s ic ly in a ian (see Lemma 2.5 in [12] o P oposi ion 4.3 in [19]).
A map γ:R→L2(Ω) is called a comple e ajec o y i
ϕ=γ|[τ,+∞)∈ Rτ, o all τ∈R.(44)
I is ob ious ha any comple e ajec o y sa is ies
γ( )∈U( , s, γ (s)) o all s≤ . (45)
The comple e ajec o y γis said o be bounded i ∪ ∈Rγ( )is a bounded se .
By he pullback a ac ing p ope y and (45) i is easy o see ha i γ(·)is a
bounded comple e ajec o y, hen γ( )⊂ A( ) o any ∈R, whe e {A( )} ∈Ris
he pullback a ac o .
We ha e he ollowing cha ac e iza ion o he pullback a ac o .
Lemma 6 A( )={γ( ):γis a bounded comple e ajec o y}.
24 Tom´as Ca aballo e al.
P oo Since (K1)−(K4)a e sa is ied and ∪ ∈RA( ) is bounded, he esul ollows
ei he om [12, Co olla y 2.10] o [12, Co olla y 2.12].
We will also p o e ha he se s A( ) a e compac in H1
0(Ω).
Lemma 7 Le un
τ→uτin L2(Ω)and le un∈ Dτ(un
τ). Then he e exis s a subse-
quence {unk}and u∈ Dτ(uτ)such ha
unk→uin C([τ+ , T], H1
0(Ω)) o all 0< < T −τ,T > τ. (46)
P oo We know by (K4) ha he e exis s u∈ Dτ(uτ) such ha , up o a subse-
quence, un( )→u( )in L2(Ω) o any ≥τ. We need o p o e ha (46) holds o
his solu ion u.
We ix an in e al [τ+ , T]. Taking in (43) s=τ, −α=τ, we ob ain a
cons an D1=D1( ) such ha

un( )
H1
0(Ω)≤D1,∀ ∈[τ+ , T].(47)
By in eg a ing (42) o e (τ+ , T )and using (40) and (47) we ha e a cons an
D2=D2(τ, , T ) sa is ying
ZT
τ+ 


dun
d 



2
ds ≤D2.(48)
Hence, Ascoli-A zel´a heo em implies, passing o a subsequence, ha
un→uin C([τ+ , T], L2(Ω)).
Also, om (48) and equali y (3) we ge a cons an D3=D3(τ, , T ) such ha
ZT
τ+ 


∂2un
∂x2



2
ds ≤D3.
These inequali ies and he Compac ness Theo em [26, p.58] imply ha
un→uweakly s a in L∞(τ+ , T;H1
0(Ω)),
un→uweakly in L2(τ+ , T;H2(Ω)),
dun
d →du
d weakly in L2(τ+ , T;L2(Ω)),
un→us ongly in L2(τ+ , T;H1
0(Ω)),
un( )→u( ) in H1
0(Ω) o a.a. ∈(τ+ , T).
In iew o (9) we ob ain also ha un, u ∈C([τ+ , T ], H1
0(Ω)).
Now, making use o (42) and (47) we deduce he exis ence o D4=D4( )
sa is ying

un( )
2
H1
0(Ω)≤
un(s)
2
H1
0(Ω)+D4( −s), o τ+ ≤s≤ ≤T,
and he same inequali y is ue o he limi unc ion u. Hence, he unc ions
Jn( ) = kun( )k2
H1
0(Ω)+D4 ,J( ) = ku( )k2
H1
0(Ω)+D4 a e con inuous and non-
inc easing in he in e al [τ+ , T ]. Mo eo e , Jn( )→J( ) o a.a. ∈(τ+ , T ).
Ex emal ajec o ies o equa ions wi h discon inuous o cing e m 25
Le n∈[τ+ , T] and n→ 0∈(τ+ , T]. We choose m∈(τ+ , 0) such
ha m→ 0and Jn( m)→J( m) o each m. I is impo an o obse e ha
when we ix m he elemen s na e g ea e han m o nbig enough. By he abo e
p ope ies we ha e
Jn( n)−J( 0) = Jn( n)−Jn( m) + Jn( m)−J( m) + J( m)−J( 0)
≤Jn( m)−J( m) + J( m)−J( 0)≤ε,
i n≥N(ε, m(ε)), whe e ε > 0. We in e ha lim sup ku( n)k2
H1
0(Ω)≤ ku( 0)k2
H1
0(Ω).
Since u( n)→u( 0) weakly in H1
0(Ω), hen lim in ku( n)k2
H1
0(Ω)≥ ku( 0)k2
H1
0(Ω),
so lim ku( n)k2
H1
0(Ω)=ku( 0)k2
H1
0(Ω)and hus
un( n)→u( 0)in H1
0(Ω).
As his a gumen is alid also in he in e al [τ+
2, T], his con e gence holds o
n→τ+ as well.
Using a s anda d diagonal p ocedu e we ob ain ha (46) is ue.
Co olla y 6 The se s A( )a e compac in H1
0(Ω).
P oo Le yn∈ A( ), ∈R. Since A( ) is compac in L2(Ω), up o a subsequence
yn→yin L2(Ω). The in a iance o A( ) implies he exis ence o solu ions un(·)∈
R −1such ha un( ) = ynand un( −1) ∈ A( −1). Again, passing o a subsequence
un( −1)→uin L2(Ω). Hence, by Lemma 7 we ob ain he exis ence o u(·)∈
D −1(u) such ha yn=un( )→u( )in H1
0(Ω). This p o es ha he se s A( ) a e
ela i ely compac in H1
0(Ω). As hey a e closed in L2(Ω), so hey a e in H1
0(Ω).
Thus, A( ) a e compac in H1
0(Ω).
Fu he we a e going o gi e a deepe cha ac e iza ion o he pullback a ac-
o by showing ha any bounded comple e ajec o y is con ained in an in e al
de ined by wo special bounded comple e ajec o ies.
Le w+
bi,ωi,i= 0,1, deno e he posi i e ixed poin +
1o p oblem (29) o he
pa ame e s b=bi,ω=ωi.
Theo em 6 The e exis s a bounded comple e ajec o y ξMsuch ha any comple e
bounded ajec o y γsa is ies
−ξM( )≤γ( )≤ξM( ) o all ∈R.(49)
Mo eo e ,
w+
b0,ω0≤ξM( )≤w+
b1,ω1,(50)
−ξM( )≤y≤ξM( ) o all y∈ A( ),(51)
−ξM( )≤lim in
s→−∞ u( )≤lim sup
s→−∞
u( )≤ξM( ),(52)
uni o mly o u∈ Ds(uτ),uτ∈B, whe e Bis bounded.