Asymp o ic beha iou o nonlocal
eac ion-di usion equa ions∗
Ma ´ıa Anguiano
Dp o. de 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
E-mail: [email p o ec ed]
P.E. Kloeden and T. Lo enz
Fachbe eich Ma hema ik, Johann Wol gang Goe he Uni e si ¨a
D-60054 F ank u am Main, Ge many
E-mail: kloeden / [email p o ec ed]
Abs ac
The exis ence o a global a ac o in L2(Ω) is es ablished o a eac ion-
di usion equa ion on a bounded domain Ω in Rdwi h Di ichle bounda y
condi ions, whe e he eac ion e m con ains an ope a o F:L2(Ω) →
L2(Ω) which is nonlocal and possibly nonlinea . Exis ence o weak solu-
ions is es ablished, bu uniqueness is no equi ed. Compac ness o he
mul i alued low is ob ained ia es ima es ob ained om limi s o Gale kin
app oxima ions. In con as wi h he usual si ua ion, hese limi s apply
o all and no jus o almos all ime ins an s.
1 In oduc ion
A simple popula ion model wi h spa ial dependence is gi en by he eac ion-
di usion equa ion
∂u
∂ = ∆u+u(1 −u) , (1)
on a bounded domain Ω in Rdwi h Di ichle bounda y condi ions. The long
e m dynamics o his model is well unde s ood.
In he abo e model he popula ion a a poin x∈Ω depends only on i s
alue a his poin , apa om he di usi i y e m. Mo e ealis ically, i could
∗Pa ially suppo ed by he Minis e io de Ciencia y Tecnolog´ıa (Spain) and FEDER (Eu-
opean Communi y) g an BFM2002-03068 as well as he DFG g an s KL1203/7, LO273/5)
1
depend on he popula ion size a o he poin s, in pa icula ly a nea by poin s.
Fo example, i could depend on he a e age o e a small neighbou hood
uδ( , x) = ZB(x;δ)
u( , y)dy ZB(x;δ)
dy ,
o some small δ > 0 ins ead o on u( , x) i sel . This leads o a nonlocal PDE
∂u
∂ = ∆u+uδ(1 −uδ) . (2)
Al e na i ely, uδ( , x) could be some o he unc ional o he solu ion u(·, )
e alua ed a he poin x.
The e a e many applica ions o nonlocal e ec s in pa ial di e en ial equa-
ions in he li e a u e, e.g., in combus ion heo y [13] and he Na ie -S okes
equa ions [5]. The nonlocal e m is o en an in eg al ope a o and he equa ions
a e hen called “in eg o-di e en ial” equa ions. Bol zmann equa ions a e a e y
well known class o in eg o-di e en ial, bu a e i s o de unlike hose o in e es
he e. Howe e , he nonlocal e m could be di e en , see, e.g., he e iew a icle
by Ba es [3]. The e is a la ge li e a u e on he exis ence, egula i y and blow–up
o solu ions o nonlocal e olu ion equa ions, see o example [15, 16, 17] and he
pape s ci ed he ein.
Global a ac o s o nonlocal e olu ion equa ions ha e been in es iga ed
ecen ly o he globally modi ied Na ie -S okes equa ions by Ca aballo e al.
[5], o m-Laplacian pa abolic equa ions wi h a nonlocal nonlinea i y by Chen [6]
and by Hilho s e al [8] o a nonlocal Ku amo o–Si ashinsky equa ion. Se e al
aspec s o eac ion-di usion equa ions a e being analyzed o e he las yea s,
pa icula ly, hei asymp o ic beha iou , see o example [14,15] and [17]. In
his pape we conside gene al nonlinea nonlocal e ms in au onomous eac ion-
di usion equa ions, which gene a e s ic mul i alued semi lows. In pa icula ,
we es ablish he exis ence o a global a ac o a e i s p o ing weak solu ions
and he compac ness o a ainabili y se s o he mul i alued semi low. Fo his
we use es ima es ob ained as limi s o Gale kin app oxima ions which hold o
e e y ime ins an and no jus o almos all ime ins an s. The p oblem
is o mula ed in he nex sec ion and dissipa i i y es ima es a e p esen ed in
Sec ion 3 wi h some longe p oo s gi en a he end o he pape in Sec ion 7.
The exis ence o weak solu ions is es ablished in Sec ion 4, while he gene a ion
o a s ic mul i alued semi low and he exis ence o a global a ac o a e shown
in Sec ion 5. Finally an explici example is p esen ed in Sec ion 6.
2 Se ing o he p oblem
Le Ω ⊂RNbe a bounded open se , i sa is ies he Poinca ´e inequali y, i.e.,
he e exis s a cons an λΩ>0 such ha
ZΩ
u2(x)dx ≤λ−1
ΩZΩ
(∇u(x))2dx,∀u∈H1
0(Ω).(3)
2
Le (·,·) deno e he scala p oduc in L2(Ω) and k·kL2(Ω) he co esponding
no m in L2(Ω). In addi ion, le h·,·i deno e he duali y p oduc be ween spaces
H1
0(Ω) and H−1(Ω).
Conside he ollowing ini ial bounda y alue p oblem o a nonlocal eac ion–
di usion equa ion wi h ze o Di ichle bounda y condi ion in Ω,





∂u
∂ +A u =F(u) in Ω ×(0, T),
u= 0 on ∂Ω×[0, T],
u(x, 0) = u0(x), o x∈Ω,
(4)
whe e Ais a uni o mly pa abolic ope a o in di e gence o m wi h aij =aji ∈
L∞(Ω), 1 ≤i, j ≤N, o which he e exis cons an s λA, ΛA>0 such ha
λA|η|2≤
N
X
i,j=1
aij(x)ηiηj≤ΛA|η|2
and
Au(x) := −
N
X
i,j=1
∂xjaij(x)∂xiu(x)(5)
o all x∈Ω and η∈RN.
Rema k 1 The ope a o induced by Acan be in e p e ed as
A∈ LH1
0(Ω), H−1(Ω)
and is symme ic wi h
hA , i ≥ λAk∇ k2
L2(Ω) o all ∈H1
0(Ω).
Since H1
0(Ω) is included in L2(Ω) wi h compac injec ion, as a consequence o
he Hilbe -Schmid Theo em he e exis s a nondec easing sequence o posi i e
eal numbe s,
0< λ1≤λ2≤... ≤λk≤......,
wi h limn→∞ λn= +∞and he e exis s an o hono mal basis {wk:k≥1}o
L2(Ω). Mo eo e , {wk:k≥1}is an o hogonal basis o H1
0(Ω) wi h A wk=
λkwk o all k≥1, whe e
(u, )H1
0(Ω)
De .
=hAu, i.
The ope a o F:L2(Ω) →L2(Ω) ul ills he ollowing assump ions:
a) Fis con inuous wi h espec o he L2no m.
3
b) he e exis β∈(0, λAλΩ) and Cβ>0 wi h
(u, F(u)) ≤βkuk2
L2(Ω) +Cβ o e e y u∈L2(Ω),(6)
c) he e is some nondec easing Ψ : [0,∞)→Rsuch ha o all u∈L2(Ω),
kF(u)k2
L2(Ω) ≤ΨkukL2(Ω)(7)
In special cases Fwill also be assumed o sa is y a local Lipschi z condi ion:
d) Fo all R > 0 he e exis s LRsuch ha i , w ∈L2(Ω) wi h k kL2(Ω) ≤R,
kwkL2(Ω) ≤R, hen
kF( )−F(w)kL2(Ω) ≤LRk −wkL2(Ω) . (8)
Rema k 2 In he subsequen s a emen s and p oo s, condi ion (c) can be
easily eplaced by he sligh ly weake assump ion ha he e is a nondec easing
unc ion e
Ψ : [0,∞)→Rsuch ha o e e y u∈H1
0(Ω) wi h Au ∈H1
0(Ω),
|hF(u), A ui| ≤ e
ΨkukL2(Ω).(9)
kF(u)k2
L2(Ω) ≤e
ΨkukH1,2(Ω).(10)
3 Dissipa i i y es ima es
Bo h he exis ence o weak solu ions and o bounded abso bing se s in L2(Ω),
H1
0(Ω), espec i ely, a e based on he ollowing a p io i es ima es. Thei p oo s
do no equi e he uniquenss o weak solu ions o a gi en ini ial alue. Some
o he auxilia y esul s a e o mula ed o Gale kin app oxima ions, and hei
(qui e echnical) p oo s a e pos poned o §7.
P oposi ion 3 I Fsa is ies hypo hesis (b), hen e e y weak solu ion u∈
L20, T;H1
0(Ω)o (4) wi h u0∈L20, T;H−1(Ω) ul ills he es ima es

u( )
2
L2(Ω) ≤M
L+e−L 
u(0)
2
L2(Ω)
λAZ
0k∇uk2
L2(Ω) ds ≤1
2+β
L
u(0)
2
L2(Ω) +Cβ+βM
L
o e e y ∈[0, T]wi h he cons an s M:= 2 Cβ>0and L:= 2 λΩλA−2β > 0.
The p oo is gi en in §7.
4
P oposi ion 4 Suppose ha condi ions (a)–(c)hold o Fand ha T < ∞.
Then he e exis s posi i e cons an s C0,C1and C2depending only on β,Cβ,
ΛA,λAand λΩsuch ha e e y weak solu ion u∈L20, T ;H1
0(Ω)o (4) wi h
u0∈L20, T;H−1(Ω)and ku0kL2(Ω) ≤ρsa is ies he a p io i es ima es
ku(s)k2
L2(Ω) ≤C1+e−C0sρ2

∇u( )
2
L2(Ω) ≤C1·max{1,1
}· 1+e−C2 ρ2+ ΨC1·(1 + e−C2 ρ2)
o e e y 0≤s < ≤T.
The p oo , gi en below, uses analogous es ima es o he Gale kin app oxima-
ions in he ollowing lemma, which is p o ed in he §7.
Lemma 5 Le {wk:k≥1}be an o hogonal basis o H1
0(Ω) as in Rema k 1.
Fo each n∈N, suppose ha un( ) =
n
X
k=1
unk( )·wkis a solu ion o



d
d (un( ), wk) + hAun( ), wki= (F(un( )), wk)
(un(0), wk) = (u0, wk), k = 1 . . . n.
(11)
I F:L2(Ω) →L2(Ω) sa is ies he hypo heses (b)and (c), hen he e exis
posi i e cons an s C1,C2and C3depending only on β,Cβ,ΛA,λAand λΩ
such ha whene e ku0kL2(Ω) ≤ρ, he ollowing es ima es holds o e e y s0,s,
∈(0, T]wi h 0< s0≤s≤

∇un( )
2
L2(Ω) ≤C1·max 1,1
·1+e−C2 ρ2+ ΨC1(1 + e−C2 ρ2)
Z
sku0
n(ξ)kL2dξ ≤C3·√ −s·cons (s0, T, ρ).
ku0
n( )kH−1(Ω) ≤cons (β, Cβ,ΛA, λA, λΩ, ρ).
P oo o P oposi ion 4. The inclusions u∈L20, T;H1
0(Ω)and u0∈
L20, T;H−1(Ω)always imply ha u∈C0[0, T ]; L2(Ω), see [7, §5.9].
Le {wk:k≥1}be an o hogonal basis o H1
0(Ω) as in Rema k 1. Then o
each n∈N,
un: [0, T]→H1
0(Ω), 7−→
n
X
k=1 u( ), wkwk
is induced by he o hogonal p ojec ion o u( ) on span{w1. . . wn}in L2(Ω).
I sol es he pe u bed nonlocal Gale kin p oblem
(d
d (un( ), wk) + hAun( ), wki= (G( ), wk)
(un(0), wk) = (u0, wk), k = 1 . . . n,
(12)
wi h he map G: [0, T]→L2(Ω) de ined by G( ) := F(u( )) o each ∈[0, T],
which depends only on ime in combina ion wi h he weak solu ion u(·) (bu no
5

on unexplici ly). In pa icula , G ul ills he condi ions (a)–(c) (uni o mly wi h
espec o ime). Hence, Lemma 5 p o ides a p io i es ima es o each Gale kin
app oxima ion un(·) depending essen ially only on he L2no m o he ini ial
alue u0.
Since {wk, k ≥1}is an o hono mal basis o L2(Ω), he sequence (un( ))n∈N
con e ges o u( ) in L2(Ω) a each ime ∈[0, T]. The L2bound o he H1
0
no m implies a weakly con e gen subsequence o (un)n∈Nin L20, T;H1
0(Ω)
and, i s weak limi is u∈L20, T ;H1
0(Ω)(again). Fo each s0∈(0, T), we
e en ha e a L∞bound o he H1
0no ms o (un)n∈Nin [s0, T] and so Lemma 7
below gua an ees he same es ima es holds o 
∇u( )
2
L2(Ω) a e e y ime ∈
[s0, T], no jus o Lebesgue-almos all such .
Fixing s0∈(0, T) a bi a ily, he Gale kin app oxima ions o any weak solu ion
u(·) wi h ku(0)kL2(Ω) ≤ρa e equi–con inuous w. . . L2(Ω) in [s0, T ] due o he
second es ima e in Lemma 5. Now he poin wise con e gence o he Gale kin
app oxima ions o u(·) implies he ollowing s a emen di ec ly:
Lemma 6 Suppose ha he condi ions (a)–(c)hold o Fand ha T < ∞.
Fo e e y ρ > 0, he subse o C0[0, T ]; L2(Ω)consis ing o all weak solu ions
u∈L20, T;H1
0(Ω)o (4) wi h u0∈L20, T;H−1(Ω)and ku(0)kL2(Ω) ≤ρ
is equi-con inuous in he subin e al [s0, T] o e e y s0∈(0, T).
Lemma 7 Le X, Y be Banach spaces such ha Xis e lexi e, and he in-
clusion X⊂Yis con inuous. Assume ha (un)n∈Nis a bounded sequence in
L∞( 0, T;X)such ha un* u weakly in Lp( 0, T;X) o some p∈[1,+∞)
and u∈C0([ 0, T]; Y).
Then, o e e y ∈[ 0, T ],u( )belongs o Xand sa is ies
ku( )kX≤sup
n≥1kunkL∞( 0,T ;X).
P oo o Lemma 7. We deno e
C:= sup
n≥1kunkL∞( 0,T ;X).
As (un)n∈Nis a bounded sequence in L∞( 0, T ;X), he e exis a subsequence
(uµ) and ∈L∞( 0, T;X) such ha uµ
∗
* in L∞( 0, T;X), i.e.,
ZT
0hw∗( ), uµ( )id −→ ZT
0hw∗( ), ( )id ∀w∗∈L1( 0, T ;X0),
whe e by h·,·i we deno e he duali y p oduc be ween X0and X.
In pa icula , we ha e his con e gence o all w∗∈Lp0( 0, T ;X0). Then, uµ*
weakly in Lp( 0, T;X), and as we also ha e un* u weakly in Lp( 0, T ;X), hen
=u.
6
Then, uµ
∗
* u in L∞( 0, T;X) and by he ∗−weak lowe semicon inui y o he
no m, we ob ain
kukL∞( 0,T ;X)≤lim in
µ→∞ kuµkL∞( 0,T ;X)≤C. (13)
Now ix ∈[ 0, T]. By (13) he e exis s a sequence ( n)n∈Nin [ 0, T ] such ha
n→ and u( n)∈Xwi h ku( n)kX≤C o all n∈N.
As Xis e lexi e, he e exis a subsequence ( µ) and x∈Xsuch ha u( µ)* x
weakly in X. The inclusion X⊂Yis assumed o be con inuous and so,
u( µ)* x weakly in Y. (14)
Due o u∈C0([ 0, T]; Y), we ha e in addi ion ha u( n)→u( ) in Y. This
implies u( ) = x∈X.
Finally he weak lowe semi-con inui y o he no m implies o e e y ∈[ 0, T ]
ku( )kX=kxkX≤lim in
µ→∞ ku( µ)kX≤C
4 Exis ence o weak solu ions
P oposi ion 8 Suppose ha hypo heses (a)–(c)hold o F:L2(Ω) →L2(Ω).
Then, o e e y u0∈L2(Ω), he e exis s a weak solu ion u∈L20, T;H1
0(Ω)
o p oblem (4). Mo eo e , ubelongs o C0[0, T]; L2(Ω)and, u|(0,T ]is locally
bounded in H1
0(Ω).
Fu he mo e, i Fis locally Lipschi z as in hypo hesis (d), hen he weak solu-
ion is unique.
P oo . The p oo is based on a sequence o Gale kin app oxima ions and he
a p io i es ima es in §3. Le {wk, k ≥1}be an o hogonal basis o H1
0(Ω) as in
Rema k 1 and o each n∈N, conside a solu ion un( ) =
n
X
k=1
unk( )wko (11).
Fix s0∈(0, T) a bi a ily.
By P oposi ion 3, he sequence (un)n∈Nis bounded in L20, T ;H1
0(Ω)Due o
Lemma 5, (u0
n)n∈Nis bounded bo h in L2s0, T ;L2(Ω)and L2(0, T ;H−1(Ω)).
Hence Alaoglu’s Theo em p o ides a subsequence (again deno ed by) (un)n∈N
and unc ions u∈L20, T;H1
0(Ω), ∈L2s0, T;L2(Ω),w∈L20, T;H−1(Ω)
wi h 








un−→ uweakly in L20, T;H1
0(Ω)
u0
n−→ weakly in L2s0, T;L2(Ω)
u0
n−→ wweakly in L20, T;H−1(Ω).
7
In pa icula , u0= =wholds Lebesgue-almos e e ywhe e, which a simple
check o he dis ibu ional de i a i e p ope y e eals. This implies ha u∈
L2(0, T;H−1(Ω)) ∩C0[0, T]; L2(Ω). S anda d a gumen s conclude u(0) = u0
om un(0) −→ u0in L2(Ω) o n→ ∞ (see e.g. [7, §7.1]).
The sequence (un)n∈Nis equi–con inuous in C0[s0, T ]; L2(Ω)by Lemma 6.
Mo eo e , he se un( ) ∈[s0, T], n ∈Nis ela i ely compac in L2(Ω)
as a consequence o Lemma 5 and he Sobole Embedding Theo em. Hence
he A zel`a–Ascoli Theo em p o ides a u he subsequence (again deno ed by)
(un)n∈Nwi h
un( )−→ u( ) in L2(Ω) uni o mly o ∈[s0, T].
Hypo hesis (a) on he con inui y o Fimplies ha F(un( )) → F(u( )) in L2(Ω)
o e e y ∈[s0, T].
Taking he limi as n→∞gi es ha u∈L2s0, T;H1
0(Ω)is a weak solu ion
o he pa ial di e en ial equa ion in p oblem (4) wi h u0∈L2s0, T;H−1(Ω).
Finally, u|[s0,T ]: [s0, T ]→H1
0(Ω) is bounded due o P oposi ion 4.
I Fsa is ies he local Lipschi z condi ion (d) in addi ion he uniqueness o
he weak solu ion ollows by a s anda d a gumen .
5 The mul i alued semi low o weak solu ions
De ine ΦF: [0,∞)×L2(Ω) → PL2(Ω), whe e PL2(Ω)consis s o all
nonemp y subse s o L2(Ω), by
ΦF( , u0) := u( )∈L2(Ω) ∃u(·)∈L20, ;H1
0(Ω):
u0∈L20, ;H−1(Ω), u(0) = u0and
uis weak solu ion o (4) in Ω ×(0, ).
(15)
I is clea ha his mul i alued map ΦF o ms a s ic mul i alued semi low on
L2(Ω) as in he he ollowing de ini ion o Kapus yan e al. [9, De ini ion 2.1].
De ini ion 9 Le (X, d)be a me ic space and, P(X)consis s o all i s
nonemp y subse s.
A map Φ : [0,∞)×X→ P(X)is called s ic mul i alued semi low (m–semi-
low) on Xi i sa is ies he ollowing condi ions:
(A) Φ(0, x) = {x} o all x∈X
(B) Φ( +s, x) = Φ( , Φ(s, x)) o all s, ≥0, x ∈X.
Rema k 10 Kapus yan e al. [9, De ini ion 2.1] de ine, in ac , a mo e gen-
e al m–semi low Φ : [0,∞)×X→ P(X), which sa is ies condi ion (A), bu
8
ins ead o he equali y condi ion (B), sa is ies he ollowing weake inclusion
condi ion
(C) Φ( +s, x)⊂Φ( , Φ(s, x)) o all s, ≥0, x ∈X.
The eason is ha he coun e pa o he se alued mapping ΦFabo e o many
sys ems, in pa icula he 3-dimensional Na ie –S okes equa ions, is es ic ed
o weak solu ions ha sa is y an ene gy inequali y. Howe e , such ene gy in-
equali ies only hold (o can only be p o ed o hold) o almos all ime ins an s.
Hence a conca en a ion o weak solu ions sa is ying he ene gy inequali y on ad-
jacen ime in e als may no sa is y he ene gy inequali y on he conca ena ed
ime in e al, which means ha he s ic p ope y (B) need no hold. see also
Mo illas & Vale o [12]. This si ua ion does no a ise o he mapping ΦFde ined
abo e o he p oblem (4) unde conside a ion in his pape .
E e y s ic mul i alued semi low Φ : [0,∞)×X→ P(X) (in he sense
o De ini ion 9) induces a map ˜
Φ : [0,∞)× P(X)→ P(X) in he ollowing
canonical way:
˜
Φ( , M)De .
=[
x∈M
Φ( , x) o all nonemp y M⊂X.
Ob iously, i sa is ies he co esponding condi ions
(A∗)˜
Φ(0, M) = M o all M∈ P(X)
(B∗)˜
Φ( +s, M) = ˜
Φ( , ˜
Φ(s, M)) o all s, ≥0, M ∈ P(X)
o in he gene al nons ic case
(C∗)˜
Φ( +s, M)⊂˜
Φ( , ˜
Φ(s, M)) o all s, ≥0, M ∈ P(X)
Hence one usually does no dis inguish be ween ˜
Φ and Φ.
5.1 Global a ac o s o mul i alued semi lows:
Gene al esul s
De ini ion 11 Aglobal a ac o o a s ic mul i alued semi low Φon a me ic
space (X, d)is a nonemp y subse Ao Xwhich is Φ-in a ian , i.e. Φ( , A) =
A o all ≥0, and a ac s e e y bounded subse Bo X, i.e.
dis X(Φ( , B), A)→0as → ∞,
whe e dis Xis he Hausdo semi–dis ance on P(X). Fo a gene al non-s ic
mul i alued semi low he a ac o Ais only equi ed o be Φ-nega i ly in a ian ,
i.e. A⊂Φ( , A) o all ≥0 [11, De ini ion 6].
The exis ence o a global a ac o is usually concluded om he compac ness
o asymp o ic compac ness o he mul i alued mapping Φ( , ·) o > 0:
9
we obse e ha i we conside m≥1, we ha e
ZΩ|u|mdx ≤|Ω|
|B(0; δ)|m/2kukm
L2(Ω) . (18)
We also ob ain
ZΩ
(F(u))2dx ≤C|Ω|
|B(0; δ)|kuk2
L2(Ω)
+C|Ω|
|B(0; δ)|2kuk4
L2(Ω)
+2C|Ω|
|B(0; δ)|3/2kuk3
L2(Ω) ,
and hen we ha e (7) wi h ΦkukL2(Ω)=C|Ω|
|B(0;δ)|kuk2
L2(Ω) +C|Ω|
|B(0;δ)|2kuk4
L2(Ω) +
2C|Ω|
|B(0;δ)|3/2kuk3
L2(Ω).
We obse e ha i u∈L2(Ω), hen F(u)∈L∞(Ω) ⊂L2(Ω) .
Now, we p o e ha Fis con inuous. We suppose ha un−→ us ongly
in L2(Ω), hen we ha e o p o e ha F(un)−→ F(u) s ongly in L2(Ω). I
is su icien o p o e ha he e exis s {uµ}⊂{un}such ha F(uµ)−→ F(u)
s ongly in L2(Ω).
As un−→ us ongly in L2(Ω), hen (see [4]) he e exis s {uµ}⊂{un}such
ha
uµ(x)−→ u(x) a.e. in Ω.
Then, as is con inuous we ha e
(uµ(x)) −→ (u(x)) a.e. in Ω. (19)
We obse e ha a guing as be o e we ob ain
|uµ(x)−u(x)|=1
|B(x;δ)|ZB(x;δ)∩Ω
(uµ(y)−u(y)) dy
≤1
|B(0; δ)|1/2kuµ−ukL2(Ω) −→ 0,
and hen
uµ(x)−→ u(x) a.e. in Ω. (20)
F om (19) and (20) we ha e
F(uµ)(x)−→ F(u)(x) a.e. in Ω. (21)
16

On he o he hand, we obse e ha kuµkL2(Ω) is bounded because Ω is a
bounded domain and uµ−→ us ongly in L2(Ω). Then, we ha e
|F(uµ(x))| ≤ C|uµ(x)||1−uµ(x)|(22)
≤C1
|B(0; δ)|1/2kuµkL2(Ω) 1 + 1
|B(0; δ)|1/2kuµkL2(Ω)!
≤e
C,
whe e e
Cis a posi i e cons an .
Then, om (21), (22) and by he Domina ed Con e gence Theo em, we
oba in
F(uµ)−→ F(u) s ongly in L2(Ω) ,
hen Fis con inuous.
F om Theo em 8 we ha e ha he e exis s a weak solu ion u∈L20, T ;H1
0(Ω)
o p oblem (16) wi h u0∈L20, T;H−1(Ω).Acco ding o Theo em 15, he
p oblem (16) de ines a mul i alued semi low in L2(Ω), which possesses a com-
pac global a ac o A. Also, Ais he minimal closed a ac ing se .
7 P oo s o he dissipa i i y es ima es
P oo o P oposi ion 3. The uni o m ellip ici y condi ion o he linea
ope a o Aand assump ion (b) on he nonlocal ope a o Fused in he ene gy
equali y, 1
2
d
d kuk2
L2(Ω) +ZΩ
u(A u)dx =ZΩ
uF(u)dx (23)
gi e 1
2
d
d kuk2
L2(Ω) +λAk∇uk2
L2(Ω) ≤βkuk2
L2(Ω) +Cβ.(24)
Hence, by he Poinca ´e inequali y (3),
1
2
d
d kuk2
L2(Ω) +λΩλAkuk2
L2(Ω) ≤βkuk2
L2(Ω) +Cβ.
i.e.,
d
d kuk2
L2(Ω) ≤M−Lkuk2
L2(Ω) , (25)
whe e M:= 2 Cβ>0 and L:= 2 λΩλA−2β > 0.
Mul iplying by eL , and in eg a ing be ween 0 and , i ollows
eL ku( )k2
L2(Ω) ≤ ku(0)k2
L2(Ω) +MZ
0
eLs ds
≤ ku(0)k2
L2(Ω) +MZ
−∞
eLs ds,
17
i.e., o e e y ∈[0, T],
ku( )k2
L2(Ω) ≤M
L+e−L ku(0)k2
L2(Ω) . (26)
Finally, in eg a ing equa ion (24) wi h espec o ime, we ob ain
λAZ
0k∇uk2
L2(Ω) ds
≤1
2ku0k2
L2(Ω) −ku( )k2
L2(Ω)+βZ
0kuk2
L2(Ω) ds +Cβ
≤ ku0k2
L2(Ω) 1
2+βZ
0
e−L s ds+Cβ+βM
L .
P oo o Lemma 5. Le {wk:k≥1}deno e an o hogonal basis o H1
0(Ω)
as men ioned in Rema k 1. Fo each n≥1, le
un( ) =
n
X
k=1
unk( )wk,
be a Gale kin app oxima ion o p oblem (4) sa is ying he ini e dimensional
sys em o o dina y di e en ial equa ions in he sense ha
(d
d (un( ), wk) + hAun( ), wki= (F(un( )), wk)
(un(0), wk)=(u0, wk), k = 1, . . . , n.
(27)
We obse e ha
Aun( ) =
n
X
k=1
unk( )Awk=
n
X
k=1
unk( )λkwk∈span {w1, . . . , wn}.
Then, mul iplying equa ion (27) by unk( )λk, summing om k= 1 o n, and
using p ope y (c) o F, gi es
hu0
n( ), Aun( )i+kAun( )k2
L2(Ω) = (F(un( )), Aun( ))
≤1
2kF(un( ))k2
L2(Ω) +kAun( )k2
L2(Ω)
≤Ψ(un( )) + 1
2kAun( )k2
L2(Ω) (28)
18
o Lebesgue-almos all ∈[0, T ]. On he o he hand,
hu0
n( ), Aun( )i=−ZΩ
N
X
i,j=1
∂xjaij(x)∂xiunu0
ndx (29)
=
N
X
i,j=1 ZΩ
aij(x)∂xiun(∂ ∂xjun)dx
=d
d 1
2·A[un, un](30)
o he symme ic bilinea o m
A[u, ] := ZΩ
N
X
i,j=1
aij(x)∂xiu ∂xj dx,u, ∈H1
0(Ω).
since i is assumed ha aij =aji, wi h i, j = 1, .., N, and hese coe icien s do
no depend on .
In eg a ing om sand , hen gi es
1
2·A[un( ), un( )] ≤1
2·A[un(s), un(s)] + Z
s
Ψkun(ξ)kL2(Ω)dξ. (31)
Hence, i ollows om he inequali ies o coe ci i y and con inui y
λAk∇un(s)k2
L2(Ω) ≤ A[un(s), un(s)] ≤ΛAk∇un(s)k2
L2(Ω)
ha
λA
2·k∇un( )k2
L2(Ω) ≤ΛA
2·k∇un(s)k2
L2(Ω) +Z
s
Ψkun(ξ)kL2(Ω)dξ. (32)
In eg a ing now he a iable sbe ween 0:= max{0, −1}and , gi es
λA
2·min{ , 1}·k∇un( )k2
L2(Ω) ≤ΛA
2Z
max{0, −1}k∇un(s)k2
L2(Ω) ds +
Z
max{0, −1}
Ψkun(s)kL2(Ω)ds.
On he o he hand, in eg a ing (24) wi h espec o ime in [ 0, ], i ollows ha
kun( )k2
L2(Ω) −kun( 0)k2
L2(Ω) + 2 λAZ
0k∇un(s)k2
L2(Ω) ds
≤2βZ
0kun(s)k2
L2(Ω) ds + 2 Cβ.
19
Then, using P oposi ion 3,
ΛAZ
0k∇un(s)k2
L2(Ω) ds ≤ΛA
2β+ 1
2λA
sup
s∈[ 0, ]kun(s)k2
L2(Ω) +CβΛA
λA
≤C2·1 + e−L ku0k2
L2(Ω)(33)
o all ≥0 wi h a cons an C2=C2(β, ΛA, λA, λΩ, Cβ)>1. Thus, whene e
ku0kL2(Ω) ≤ρ,
k∇un( )k2
L2(Ω) ≤2C2
λAmin{ , 1}1 + e−L ρ2+ ΨM
L+e−L ρ2.(34)
Mo eo e , his uppe bound holds o e e y ∈(0, T] — in con as o he
immedia e consequence o inequali y (28) and he coe ci i y o A.
The nex s ep is o es ablish an es ima e o ku0
nkL∞(0,T ;H−1(Ω)). This ol-
lows essen ially he a gumen s o he s anda d ene gy es ima e, aking he
p eceding a p io i bounds in o conside a ion.
Fo e e y ∈H1
0(Ω), he e is a unique ep esen a ion = 1+ 2wi h 1∈
span{w1, . . . , wn}and ( 2, wk) = 0 o k= 1, . . .,nsince {wk, k ≥1}is an
o hono mal basis o L2(Ω).
Fo Lebesgue-almos e e y ∈[0, T ] i ollows ha
u0
n( ), 1+Aun( ), 1=F(un( )), 1
hu0
n( ), i= (u0
n( ), ) = u0
n( ), 1
=F(un( )), 1−Aun( ), 1
≤(kF(un( ))kL2+ ΛAk∇un( )kL2)k 1kH1
0(Ω).
Then, inequali y (7) in assump ion (c) o Fimplies ha
hu0
n( ), i ≤ pΨ (kun( )kL2)+ΛAk∇un( )kL2k 1kH1
0(Ω).
Hence, in iew o he es ima es (26) and (34),
ku0
n( )kH−1(Ω) ≤pΨ (kun( )kL2)+ΛAk∇un( )kL2
≤cons β, Cβ,ΛA, λA, λΩ,ku0kL2(Ω).
Finally, mul iplying (27) by u0
nk( ) and summing om k= 1 o n, i ollows
20
om inequali y (10) in assump ion (c) o F ha
hu0
n( ), u0
n( )i+hAun( ), u0
n( )i= (F(un( )), u0
n( ))
≤ kF(un( ))k2
L2(Ω) +1
4ku0
n( )k2
L2(Ω)
≤Ψkun( )kL2(Ω)+1
4ku0
n( )k2
L2(Ω).
Using (30), his can be e o mula ed as
ku0
n( )k2
L2(Ω) +d
d 1
2A[un, un]≤Ψkun( )kL2(Ω)+1
4ku0
n( )k2
L2(Ω)
i.e., o Lebesgue-almos e e y ∈[0, T ],
ku0
n( )k2
L2(Ω) +d
d A[un, un]≤2 Ψkun( )kL2(Ω).
Now ix s0∈(0, T[ a bi a ily and in eg a e be ween s0and T o ob ain
ZT
s0ku0
n(ξ)k2
L2(Ω) dξ +A[un(T), un(T)] ≤A[un(s0), un(s0)]
+ 2 ZT
s0
Ψkun(ξ)kL2(Ω)dξ.
The gene al inequali ies o coe ci i y and con inui y
0≤λAk∇un( )k2
L2(Ω) ≤ A[un( ), un( )] ≤ΛAk∇un( )k2
L2(Ω) ,
imply ha
ZT
s0ku0
n(ξ)k2
L2(Ω) dξ ≤ΛAk∇un(s0)k2
L2(Ω) + 2 ZT
s0
Ψkun(ξ)kL2(Ω)dξ.
Hence, om he es ima es (26), (34) and he mono onici y o Ψ one concludes
ha
ZT
s0ku0
n(ξ)k2
L2(Ω) dξ ≤cons (β, ΛA, λA, λΩ, Cβ, s0, T, ku0kL2(Ω)).
Finally, H¨olde ’s inequali y gua an ees he claimed inequali y o e e y s, ∈
[s0, T] wi h s< , i.e.,
Z
sku0
n(ξ)kL2(Ω) dξ ≤cons β, ΛA, λA, λΩ, Cβ, s0, T, ku0kL2(Ω)·√ −s .
21

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23