EXISTENCE OF INSENSITIZING CONTROLS
FOR A SEMILINEAR HEAT EQUATION WITH
A SUPERLINEAR NONLINEARITY
O. Boda ∗
, M. Gonz´alez-Bu gos†and R. P´e ez-Ga c´ıa†
———
This wo k has been pa ially inanced by D.G.E.S. (Spain), G an PB98–1134.
Abs ac
In his pape we conside a semilinea hea equa ion (in a bounded domain
Ω o IRN) wi h a nonlinea i y ha has a supe linea g ow h a in ini y. We
p o e he exis ence o a con ol, wi h suppo in an open se ω⊂Ω, ha
insensi izes he L2−no m o he obse a ion o he solu ion in ano he open
subse O ⊂ Ω when ω∩ O 6=∅, unde sui able assump ions on he nonlinea
e m (y) and he igh hand side e m ξo he equa ion. The p oo , in ol ing
global Ca leman es ima es and egula izing p ope ies o he hea equa ion,
elies on he sha p s udy o a simila linea ized p oblem and an app op ia e
ixed-poin a gumen . Fo ce ain supe linea nonlinea i ies, we also p o e an
insensi i i y esul o a nega i e na u e. The c ucial poin in his pape is
he echnique o cons uc ion o L –con ols ( la ge enough) s a ing om
insensi izing con ols in L2.
∗Labo a oi e de Ma h´ema iques Appliqu´ees, Uni e si ´e Blaise Pascal (Cle mon -Fe and 2),
63177 Aubi`e e Cedex, F ance, E-mails: Oli ie .Bo[email p o ec ed]cle mon .
†Dp o. de Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa, Ap do. 1160,
41080 Se illa, Spain, E-mails: [email p o ec ed], [email p o ec ed]
1 In oduc ion and main esul s
P oblem o mula ion
Le Ω ⊂IRN,N≥1, be a bounded connec ed open se wi h bounda y ∂Ω∈C2. Le
be a C1 unc ion de ined on IR. Le ωand Obe wo open subse s o Ω ( hough
o be small, in p ac ice). Fo T > 0, we deno e Q= Ω×(0, T) and Σ = ∂Ω×(0, T).
We conside a semilinea hea equa ion wi h pa ially known ini ial condi ion
(∂ y−∆y+ (y) = ξ+ 1ωin Q,
y= 0 on Σ, y(x, 0) = y0(x) + τˆy0(x) in Ω,(1)
whe e ξ∈L2(Q) is a gi en hea sou ce, y0∈L2(Ω) is a gi en ini ial da a (al hough,
by he easons which will be seen la e , we will add ess in his pape he case y0= 0),
ˆy0∈L2(Ω) is unknown wi h kˆy0kL2(Ω) = 1, τis an unknown small eal numbe and
∈L2(Q) is a con ol unc ion o be de e mined. He e 1ωis he cha ac e is ic
unc ion o he con ol se ω.
Le us de ine
φ(y) = 1
2ZZO×(0,T)
|y(x, ;τ, )|2dx d , (2)
y=y(·,·;τ, ) being a solu ion o (1) associa ed o τand . A con ol unc ion is
said o insensi ize φi
∂φ(y(·,·;τ, ))
∂τ ¯¯¯¯τ=0
= 0,∀ˆy0∈L2(Ω) wi h kˆy0kL2(Ω) = 1.(3)
This insensi i i y condi ion means ha we seek a con ol unc ion , ac ing on
ω×(0, T), such ha φis locally insensi i e o small pe u ba ions in he ini ial
condi ion.
In [3] and [15], he exis ence o a con ol sa is ying (3) is p o ed o be equi alen
o he exis ence o a con ol sol ing he ollowing p oblem:
(∂ y−∆y+ (y) = ξ+ 1ωin Q,
y= 0 on Σ, y(x, 0) = y0(x) in Ω,(4)
(−∂ q−∆q+ ′(y)q=y1Oin Q,
q= 0 on Σ, q(x, T) = 0 in Ω,(5)
q(x, 0) = 0 in Ω.(6)
2
Thus he p oblem o seeking a con ol ha insensi izes φboils down o a non-classical
null con ollabili y p oblem. Fi s , i is a null con ollabili y p oblem o backwa d-
o wa d na u e o a cascade sys em o hea equa ions, he i s one o semilinea
ype. In addi ion, he con ol en e s on he second equa ion only indi ec ly h ough
he i s one, while qis he unc ion we wan o lead o ze o a e a ime in e al o
leng h T.
P elimina ies and exis ing esul s
This p oblem, add essed by J.-L. Lions in [15], has been s udied o globally Lipschi z-
con inuous nonlinea i ies and ω∩ O 6=∅( his las hypo hesis is absolu ely essen ial
and o ou knowledge no hing is known when he in e sec ion is emp y). Fi s , in
[3] he au ho s elaxed he no ion o insensi izing con ols, in oducing he so–called
ε–insensi izing con ols: Gi en ε > 0, a con ol is said o ε–insensi ize φi
¯¯¯¯
∂φ(y(·,·;τ, ))
∂τ ¯¯¯¯τ=0¯¯¯¯
≤ε, ∀ˆy0∈L2(Ω) wi h kˆy0kL2(Ω) = 1.
In he abo e-men ioned pape , he exis ence o ε–insensi izing con ols o pa ially
known da a, bo h in he ini ial and bounda y condi ions, was p o ed. This p oblem
is equi alen o an app oxima e con ollabili y p oblem o a sys em o coupled hea
equa ions and i was sol ed by using he echniques in [9].
The i s esul s on he exis ence and non-exis ence o insensi izing con ols we e
p o ed in [16]. To be p ecise, he au ho showed ha when ≡0 and Ω ω6=∅, he e
exis s y0∈L2(Ω) such ha , o e e y ∈L2(Q), he co esponding solu ion (y, q)
o (4)–(5) sa is ies q(0) 6= 0, ha is o say, he unc ional φcanno be insensi ized
(see Theo em 2 in [16]). On he o he hand, when ω∩ O 6=∅,y0= 0 and
is a C1globally Lipschi z-con inuous unc ion such ha (0) = 0, in [16] i is
p o ed: I ξ∈L2(Q; exp(M/2 )) wi h M>0la ge enough, he e exis s ∈L2(Q)
such ha he solu ion (y, q) o (4)–(5) sa is ies (6) (see Theo em 1 in [16]). He e
L2(Q; exp(M/2 )) s ands o he weigh ed Hilbe space
L2(Q; exp(M/2 )) = {ξ∈L2(Q) : kexp(M/2 )ξkL2(Q)<∞}.
The p oo combines a simila null con ollabili y esul o linea coupled pa abolic
sys ems and an app op ia e ixed-poin a gumen . Mo e p ecisely, he au ho i s
linea izes he sys em and shows i s null con ollabili y wi h con ols uni o mly
bounded in L2(Q) when he po en ials o he linea ized sys em lie in a bounded
se o L∞(Q). Since is a globally Lipschi z-con inuous unc ion, his ac su ices
3
o p o ing ha he ixed-poin mapping maps L2(Q) in o a con ex compac se o
L2(Q).
The main goal o he p esen pape is o analyze he exis ence o con ols ha
insensi ize φ, ha is o say, o s udy he null con ollabili y p ope ies o he coupled
sys em (4)–(5) when has a supe linea g ow h a in ini y. In acco dance wi h
Theo ems 1 and 2 in [16], we will assume om now on ha y0= 0 and ω∩ O 6=∅.
In he s udy o he null con ollabili y o semilinea pa abolic sys ems wi h su-
pe linea nonlinea i ies, addi ional echnical di icul ies a ise. Le us ecall wha
happens in he simple case o one semilinea hea equa ion. Du ing he las yea s,
he con ollabili y p ope ies o
(∂ y−∆y+ (y) = 1ωin Q,
y= 0 on Σ, y(x, 0) = y0(x) in Ω,(7)
wi h a supe linea nonlinea i y (y) has been ho oughly s udied by se e al au-
ho s. As in he sublinea case, he echnique o deal wi h his p oblem combines a
ixed-poin e o mula ion oge he wi h he s udy o he null con ollabili y o linea
p oblems o he o m
(∂ y−∆y+ay = 1ωin Q,
y= 0 on Σ, y(x, 0) = y0(x) in Ω,
whe e a∈L∞(Q). Due o he supe linea g ow h o he nonlinea i y i is necessa y,
o pe o m a ixed poin a gumen , ha he solu ion o he con olled linea p oblem
belongs o L∞(Q). To his aim con ols in L (Q), wi h > (N+ 2)/2, mus be
buil . Mo eo e , in he linea case i is necessa y o analyze how he L –no m o
he con ols depends on kak∞. Le us men ion some pape s on his issue:
1. In [2], V. Ba bu ob ained null L (N)–con ols such ha
k kL (N)(Q)≤C(kak∞)ky0kL2(Ω),
wi h (N)∈·2,2(N+ 2)
N−2¸i N≥3, (N)∈[2,∞) i N= 2, and (N)∈
[2,∞] i N= 1. The p oo o his es ima e is based on a global Ca leman
inequali y o he adjoin p oblem
(−∂ ϕ−∆ϕ+aϕ = 0 in Q,
ϕ= 0 on Σ, ϕ(x, T) = ϕ0(x) in Ω,(8)
4
due o A.V. Fu siko and O.Yu. Imanu ilo (see [11]). In ac , he au ho
p o ed a sha p es ima e o he cons an appea ing on he igh -hand side o
he Ca leman inequali y wi h espec o kak∞. Ne e heless, his echnique
can only be applied o he s udy o he null con ollabili y o he supe linea
hea equa ion (7) when N < 6. Fo o he con ollabili y esul s p o ed in a
simila way, see [1].
2. A second app oach was de eloped by E. Fe n´andez-Ca a and E. Zuazua in [10].
They p o ed he “ e ined” obse abili y inequali y
kϕ(0)k2
L2(Ω) ≤C(T, kak∞)µZZω×(0,T )
|ϕ|dx d ¶2
,(9)
o he solu ions o (8), which implies he exis ence o a null con ol ∈L∞(Q)
sa is ying
k kL∞(Q)≤C(T, kak∞)ky0kL2(Ω).
In his case, he obse abili y inequali y (9) was deduced combining a global
Ca leman es ima e o he adjoin p oblem and he egula izing e ec o he
hea equa ion. The au ho s gi e he explici dependence on Tand kak∞o
he cons an C(T, kak∞) in (9), which is essen ial o p o e hei nonlinea null
con ollabili y esul . This echnique was la e used in [8] o a nonlinea hea
equa ion wi h a supe linea e m (y, ∇y).
The s udy o he null con ollabili y p ope ies o he supe linea coupled sys em
(4)–(5) is mo e in ica e. In his case, as in [1], [2] and [10], null L –con ols (wi h
> (N+2)/2) o he co esponding coupled linea sys em mus also be buil . Again,
L –es ima es o he con ols a e needed. The echnique in oduced by V. Ba bu
could be applied in his case i he condi ion N < 6 is imposed (which does no
seem o be a na u al es ic ion on N). On he o he hand, he app oach p oposed
by E. Fe n´andez-Ca a and E. Zuazua canno be applied since he egula izing e ec
o he associa ed linea adjoin p oblem in ol es wo unc ions ϕand ψ(see (21)–
(22)) while he co esponding “ e ined” obse abili y inequali y should only in ol e
ψ( ecall ha he con ol only appea s in (4)).
In [4], he au ho s in oduced a new echnique o cons uc ion o null L –con ols
o coupled linea pa abolic sys ems. This s a egy made i possible o gene alize
Theo em 1 in [16] o mo e gene al nonlinea i ies. The p oo o his insensi i i y
esul as well as he abo e-men ioned echnique, ske ched in [4], a e de eloped in
he p esen pape . To ou knowledge, his is he i s insensi i i y esul in he
li e a u e o semilinea hea equa ions wi h a supe linea nonlinea i y.
5

Main esul s
The i s ele an esul in his pape is he ollowing one:
Theo em 1.1 Assume ha ω∩ O 6=∅and y0= 0. Le be a C1 unc ion de ined
on IR e i ying ′′ ∈L∞
loc(IR), (0) = 0 and
lim
|s|→∞
′(s)
log(1 + |s|)= 0.(10)
Le ∈µN
2+ 1,∞¶be gi en. Then, o any ξ∈L (Q)such ha
ZZQ
exp µ1
3¶|ξ|2dx d < ∞,(11)
he e exis s a con ol unc ion ∈L (Q)insensi izing he unc ional φgi en by (2).
¤
Condi ion (11) means ha he gi en sou ce e m ξis asked o decay apidly
o ze o close o he ini ial ime = 0. As seen be o e, a simila assump ion, i a
weake one, is equi ed in he case when a globally Lipschi z-con inuous nonlinea i y
is conside ed. Obse e ha hypo hesis (0) = 0 is in acco dance wi h assump ions
on ξ(see [16] o bo h conside a ions).
As usual in he s udy o con ollabili y p oblems o nonlinea equa ions, a con-
ollabili y esul o a linea ized e sion o (4)–(6) will be i s p o ed. We will hen
apply a ixed-poin a gumen o deal wi h he gene al case. The s uc u e o he
p oo is qui e gene al ( o o he con ollabili y esul s p o ed in a simila way, see
[10], [17], ...).
Rema k 1Hypo hesis (10) is ul illed by ce ain supe linea nonlinea i ies such
as
| (s)|=|p1(s)|logα(1 + |p2(s)|) o all |s| ≥ s0>0,
wi h α∈[0,1), whe e p1and p2a e eal a ine unc ions.
Fo nonlinea i ies ∈C1(IR) sa is ying hypo hesis (10), sys em (1) admi s a
global solu ion when he da a ξ, ,y0and ˆy0a e egula enough. Fo ins ance, by
linea iza ion and he la e applica ion o a ixed-poin a gumen , one can p o e ha
o gi en ξand in L (Q) and y0,ˆy0∈W2−2/ , (Ω) ∩W1,
0(Ω), wi h > N/2 + 1,
sys em (1) possesses a unique solu ion in L (0, T;W2, (Ω)). Le us ema k ha
h oughou he pape , his egula i y will be assumed on he da a. ¤
6
Ou second main esul is o a nega i e na u e.
Theo em 1.2 The e exis C1 unc ions e i ying ′′ ∈L∞
loc(IR), (0) = 0 and
| (s)| ∼ |s|logα(1 + |s|)as |s| → ∞,(12)
wi h α > 2, and he e exis sou ce e ms ξ∈L (Q)sa is ying (11), o which i is
no possible o ind con ol unc ions insensi izing he unc ional φgi en by (2).¤
Fo he p oo o his esul , we choose
(s) = Z|s|
0
logα(1 + |σ|)dσ o all s∈IR
and we p o e a localized es ima e in Ω ωo he co esponding solu ion yo (4) ha
shows ha o ce ain sou ce e ms ξ, he con ol canno compensa e he blow up
phenomena occu ing in Ω ω.
In iew o Theo ems 1.1 and 1.2, i would be in e es ing o analyze wha happens
when sa is ies (12), wi h 1 ≤α≤2 (see Subsec ion 6.4 o u he commen s).
The es o his pape will be o ganized as ollows. In he ollowing Sec ion,
we p esen some echnical esul s which will be p o ed in an appendix. Sec ion 3
p o ides an exhaus i e s udy o he linea case. In Sec ion 4, we analyze he non-
linea case and p o e Theo em 1.1. The i h Sec ion is de o ed o he p oo o
Theo em 1.2. In Sec ion 6 we gi e o he insensi i i y esul s and discuss some open
p oblems.
2 Some echnical esul s
In his Sec ion we s a e some echnical esul s which will be used la e . They a e
known esul s on he local egula i y o he solu ions o he linea hea equa ion.
Ne e heless, we include he p oo o hese esul s in an appendix so as o ob ain
he explici dependence o he cons an s on he po en ials, which will be essen ial
in ou analysis.
Fi s , le us p esen he ollowing no a ion, which is used all along his pape .
Fo ∈[1,∞] and a gi en Banach space X,k · kL (X)will deno e he no m in
L (0, T;X). Fo simplici y, he no m in L (Q) will be ep esen ed by k · kL , o
∈[1,∞), and k · k∞will deno e he no m in L∞(Q). Fo ∈[1,∞) and any open
se V ⊂ IRN, we will conside he Banach space
X (0, T;V) = ©u∈L (0, T;W2, (V)) : ∂ u∈L (0, T;L (V))ª,
7
and i s no m, de ined by
kukX (0,T;V)=kukL (W2, (V)) +k∂ ukL (L (V)).
In pa icula , we will conside he space X =X (0, T; Ω) and i s no m, deno ed by
k · kX . The no m in he space L2(0, T;H2(Ω)) ∩C([0, T]; H1(Ω)) will be deno ed
by k · kL2(H2)∩C(H1).
On he o he hand, o α∈(0,1) and u∈C0(Q), we de ine he quan i y
[u]α,α
2= sup
Q
|u(x, )−u(x′, )|
|x−x′|α+ sup
Q
|u(x, )−u(x, ′)|
| − ′|α
2
.
We will conside he space Cα, α
2(Q) = ©u∈C0(Q) : [u]α,α
2<∞ª,which is a Banach
space wi h i s na u al no m |u|α, α
2;Q=kuk∞+ [u]α,α
2.We will also conside he
Banach space de ined by
C1+α, 1+α
2(Q) = (u∈C0(Q) : ∂u
∂xi
∈Cα,α
2(Q)∀i, sup
Q
|u(x, )−u(x, ′)|
| − ′|1+α
2
<∞).
The ollowing holds:
P oposi ion 2.1 Le a∈L∞(Q)and F∈L2(Q)be gi en. Le us conside a
solu ion y∈L2(0, T;H2(Ω)) ∩C([0, T]; H1(Ω)) o
(∂ y−∆y+ay =Fin Q,
y= 0 on Σ, y(x, 0) = 0 in Ω.(13)
a) Le V ⊂ Ω( esp. B ⊂⊂ Ω) be an open se . Le us suppose ha F∈
L (0, T;L (V)) ( esp. F∈L (0, T;L (Ω B)), wi h ∈(2,∞). Then, o
any open se V′⊂⊂ V ( esp. B ⊂⊂ B′⊂⊂ Ω) one has
y∈X (0, T;V′)( esp. y∈X (0, T; Ω B′)).
Mo eo e , he e exis posi i e cons an s C=C(Ω, T, N, , V,V′)( esp. C=
C(Ω, T, N, , B,B′)) and K=K(N)such ha
kykX (0,T;V′)≤C(1 + kak∞)K£kFkL (L (V)) +kykL2(H2)∩C(H1)¤.(14)
( esp. kykX (0,T ;Ω B′)≤C(1 + kak∞)KhkFkL (L (Ω B)) +kykL2(H2)∩C(H1)i).
8
b) Assume, in addi ion, ha F∈L (0, T;W1, (V)), wi h as abo e, and ∇a∈
Lγ(Q)N, wi h
γ=






max µ , N
2+ 1¶i 6=N
2+ 1,
N
2+ 1 + εi =N
2+ 1,
(15)
and εbeing an a bi a ily small posi i e numbe . Then, o any open se
V′⊂⊂ V, one has
y∈L (0, T;W3, (V′)), ∂ y∈L (0, T;W1, (V′))
and, o a new posi i e cons an C=C(Ω, T, N, , V,V′), he ollowing es i-
ma e holds
kykL (W3, (V′)) +k∂ ykL (W1, (V′)) ≤CH£kFkL (W1, (V)) +kykL2(H2)∩C(H1)¤,
whe e
H=H(N, kak∞,k∇akLγ) = (1 + kak∞)K+1(1 + k∇akLγ),
K=K(N)being as in (14).¤
We will also use he ollowing esul , which is immedia ely ob ained ew i ing
Lemma 3.3, p. 80, in [14] wi h ou no a ion.
Lemma 2.2 Le Ω⊂IRN,N≥1, be an open se wi h ∂Ω∈C2. The ollowing
con inuous embeddings hold:
i. I < N
2+ 1, hen X ֒→Lp(Q), whe e 1
p=1
−2
N+ 2.
ii. I =N
2+ 1, hen X ֒→Lq(Q) o all q < ∞.
iii. I N
2+ 1 < < N + 2, hen X ֒→Cβ, β
2(Q), wi h β= 2 −N+ 2
.
i . I =N+ 2, hen X ֒→Cl, l
2(Q) o all l∈(0,1).
. I > N + 2, hen X ֒→C1+α, 1+α
2(Q), whe e α= 1 −N+ 2
.¤
9
Le us ema k ha in he con ex o he hea equa ion, he p e ious echnique p o-
ides a new me hod o cons uc ion o egula con ols s a ing om con ols in
L2(Q). This will make i possible o gi e a new p oo o known null con ollabili y
esul s o nonlinea hea equa ions. A local esul on he null con ollabili y o
he classical hea equa ion and a local esul on insensi izing con ols o a semi-
linea hea equa ion, bo h wi h nonlinea Fou ie bounda y condi ions, can also be
ob ained by using a simila cons uc ion (see [7] and [6]).
4 The nonlinea case: p oo o Theo em 1.1
In his Sec ion, we will apply an app op ia e ixed-poin a gumen o ea he
nonlinea case. In a i s s ep, will be assumed o be a C2 unc ion. The gene al
case will be s udied in Subsec ion 4.2.
4.1 The case when is a C2 unc ion
Le ∈C2(IR) be a unc ion e i ying (0) = 0 and (10). Le ξ∈L (Q) sa is y
hypo hesis (11), wi h > N/2 + 1.
Le us de ine
g(s) = 


(s)
si s6= 0,
′(0) i s= 0.
Then g, ′∈C0(IR) and (s) = g(s)s o all s∈IR. Since (0) = 0, hypo hesis
(10) on ′implies a simila one on g, ha is,
lim
|s|→∞
g(s)
log(1 + |s|)= 0.
Thus, o each ε > 0, he e exis s a posi i e cons an Cε(which only depends on ε
and on he unc ion ) such ha
|g(s)|+| ′(s)| ≤ Cε+εlog(1 + |s|) o all s∈IR.(37)
Le us ecall ha , o > N
2+ 1, we de ined
Z =C0(Q)∩L (0, T;W1, (Ω)).
16

Fo any z∈B(0; R)⊂Z ,R > 0 o be de e mined la e , we conside he linea
con ollabili y p oblem
(∂ y−∆y+g(z)y=ξ+ 1ωin Q,
y= 0 on Σ, y(x, 0) = 0 in Ω,(38)
(−∂ q−∆q+ ′(z)q=y1Oin Q,
q= 0 on Σ, q(x, T) = 0 in Ω,(39)
q(x, 0) = 0 in Ω. (40)
Le us obse e ha (38)–(39) a e o he o m (16)–(17) wi h po en ials
(a=az=g(z)∈L∞(Q),
b=bz= ′(z)∈L∞(Q)∩L (0, T;W1, (Ω)) (γ= in his case).
In iew o Co olla y 3.4, o any z∈B(0; R)⊂Z he e exis s a con ol z∈
L (Q) such ha he co esponding solu ion (yz, qz) o (38)–(39) lies in Z ×Z and
sa is ies (40). Mo eo e , es ima es
kyzkZ ≤C1(Ω, ω, O, T, z)µkξkL + exp µC
2Hz¶°
°
°
°
exp µCMz
2 ¶ξ°
°
°
°L2¶(41)
and
k zkL ≤C2(Ω, ω, O, T, z)µkξkL + exp µC
2Hz¶°
°
°
°
exp µCMz
2 ¶ξ°
°
°
°L2¶(42)
hold, whe e
C1(Ω, ω, O, T, z) = exp [C(1 + kg(z)k∞+k ′(z)k∞)] ,
C2(Ω, ω, O, T, z) = exp [C(1 + kg(z)k∞+k ′(z)k∞)] (1 + k ′′(z)∇zkL ),
Mz= 1 + kg(z)k2/3
∞+k ′(z)k2/3
∞+kg(z)− ′(z)k1/2
∞,
Hz= 1 + kg(z)k∞+k ′(z)k∞+kg(z)k2/3
∞+k ′(z)k2/3
∞+kg(z)− ′(z)k1/2
∞,
and C=C(Ω, ω, O, T)>0.
Due o hypo hesis (11) on ξ, one has
ZZQ
exp µCMz
¶|ξ|2dx d =ZZQ
exp µCMz
−1
3¶exp µ1
3¶|ξ|2dx d
≤exp ¡C M3/2
z¢ZZQ
exp µ1
3¶|ξ|2dx d ,
(43)
17
C=C(Ω, ω, O, T) being a new posi i e cons an .
Then, om inequali ies (41), (42) and (43), and using he con exi y o he eal
unc ion s7→ s3/2, i can be es ima ed
||yz||Z ≤C1(Ω, ω, O, T, z)µkξkL +°
°
°
°
exp µ1
2 3¶ξ°
°
°
°L2¶,(44)
and
|| z||L ≤C2(Ω, ω, O, T, z)µkξkL +°
°
°
°
exp µ1
2 3¶ξ°
°
°
°L2¶
≤˜
C(Ω, ω, O, T, R)µkξkL +°
°
°
°
exp µ1
2 3¶ξ°
°
°
°L2¶,
(45)
whe e ˜
C(Ω, ω, O, T, R) is a posi i e cons an independen o z.
Le us de ine
A:z∈B(0; R)⊂Z 7−→ A(z)⊂L (Q),
wi h
A(z) = { ∈L (Q) : (y, q) sa is ies (38)–(40), e i ying (45)},
and le Λ be he se - alued mapping de ined on Z as ollows:
Λ : z∈B(0; R)⊂Z 7−→ Λ(z)⊂Z ,
wi h
Λ(z) = {y∈Z : (y, q) sol es (38)–(39) wi h ∈ A(z), y sa is ying (44)}.
Le us p o e ha Λ ul ills he assump ions o Kaku ani’s ixed-poin Theo em.
In he i s place, one can check ha Λ(z) is a non-emp y closed con ex subse o
Z o ixed z∈Z , due o he linea i y o sys ems (38) and (39).
In ac , in iew o Theo em 9.1 in [14] and (45), Λ(z) is uni o mly bounded in X ,
he space in oduced in Sec ion 2. Recall ha o > N/2 + 1, X is con inuously
embedded in he H¨olde space Cβ, β
2(Q), wi h β= 2 −(N+ 2)/ (see Lemma 2.2).
Then, he e exis s a compac se K⊂Z ,Konly depending on R, such ha
Λ(z)⊂K∀z∈B(0; R).(46)
Le us now p o e ha Λ is an uppe hemicon inuous mul i alued mapping, ha
is o say, o any bounded linea o m µ∈Z′
, he eal- alued unc ion
z∈B(0; R)⊂Z 7−→ sup
y∈Λ(z)
hµ, yi
18
is uppe semicon inuous. Co espondingly, le us see ha
Bλ,µ =(z∈B(0; R) : sup
y∈Λ(z)
hµ, yi ≥ λ)
is a closed subse o Z o any λ∈IR, µ∈Z′
(see [8] o a simila p oo ). To his
end, we conside a sequence {zn}n≥1⊂Bλ,µ such ha
zn→zin Z .
Ou aim is o p o e ha z∈Bλ,µ. Since all he Λ(zn) a e compac se s, by he
de ini ion o Bλ,µ, o any n≥1 he e exis s yn∈Λ(zn) such ha
hµ, yni= sup
y∈Λ(zn)
hµ, yi ≥ λ. (47)
Recalling now he de ini ion o A(zn) and Λ(zn), le n∈ A(zn), qn∈Z be such
ha (yn, qn) sol es (38)–(40) wi h con ol nand po en ials g(zn), ′(zn). F om (44)
and (45), ynand nsa is y
||yn||Z ≤C1(Ω, ω, O, T, zn)µkξkL +°
°
°
°
exp µ1
2 3¶ξ°
°
°
°L2¶,
and
|| n||L ≤C2(Ω, ω, O, T, zn)µkξkL +°
°
°
°
exp µ1
2 3¶ξ°
°
°
°L2¶.
Thus, {yn}( esp. { n}) is uni o mly bounded in Z ( esp. in L (Q)). In pa icula ,
(46) gi es us {yn} ⊂ K. Hence he e exis subsequences, s ill deno ed by {yn}and
{ n}, such ha
yn→ys ongly in Z ,
and
n→ weakly in L (Q).
Since gand ′′ a e con inuous unc ions, one also has
g(zn)→g(z) in C0(Q),
′(zn)→ ′(z) in C0(Q),
and
′′(zn)∇zn→ ′′(z)∇zin L (Q)N.
19
Passing o he limi , one deduces ha yand he associa ed unc ion qsol e (38)–(40)
wi h con ol unc ion (and po en ials g(z), ′(z)). Mo eo e , ¯yand ¯ sa is y (44)
and (45), ha is, ∈ A(z) and y∈Λ(z). Then, aking limi s in (47), i holds ha
sup
y∈Λ(z)
hµ, yi ≥ hµ, yi ≥ λ,
whence i is deduced ha z∈Bλ,µ and hence, Λ is uppe hemicon inuous.
Finally, le us see ha he e exis s R > 0 such ha
Λ¡B(0; R)¢⊂B(0; R).(48)
Le R > 0 be, o be de e mined. Fo any z∈B(0; R)⊂Z , om (44) and (37) i is
obse ed ha each y∈Λ(z) sa is ies
||y||Z ≤exp [C(1 + Cε+εlog(1 + ||z||∞))] µkξkL +°
°
°
°
exp µ1
2 3¶ξ°
°
°
°L2¶
≤exp [C(1 + Cε)] (1 + R)Cε µkξkL +°
°
°
°
exp µ1
2 3¶ξ°
°
°
°L2¶,
wi h C=C(Ω, ω, O, T)>0. Thus, choosing ε= 1/2C, we ge
||y||Z ≤C(1 + R)1/2µkξkL +°
°
°
°
exp µ1
2 3¶ξ°
°
°
°L2¶,
om which we in e he exis ence o R > 0 la ge enough such ha (48) is sa is ied.
The Kaku ani Fixed-poin Theo em hus applies, which ends he p oo when
is a C2 unc ion.
Rema k 2Le us no ice ha wo di e en nonlinea i ies o which he posi i e
cons an Cεin (37) coincide, lead o he same R > 0. This ac will be used in he
ollowing Subsec ion when s udying he gene al case. ¤
4.2 The gene al case
I is now assumed ha is a C1 unc ion sa is ying ′′ ∈L∞
loc(IR), (0) = 0 and (10)
and le ξbe as abo e.
We conside a unc ion ρ∈ D(IR) such ha
ρ≥0 in IR,supp ρ⊂[−1,1] and ZIR
ρ(s)ds = 1.
20
Fo any n≥1, le us se
ρn(s) = nρ(ns) o all s∈IR,
Fn=ρn∗ , n(·) = Fn(·)−Fn(0)
and
gn(s) = 


n(s)
si s6= 0,
′
n(0) i s= 0.
Due o he p ope ies o ρnand he con olu ion, as well as he hypo hesis on ,
one can p o e ha nand gnha e he ollowing p ope ies:
(i)gnand ′′
na e con inuous unc ions and n(0) = 0 o all n≥1.
(ii) n→ in C1(K) o all compac se K⊂IR.
(iii)gn→guni o mly on compac se s o IR.
(i ) Fo any gi en M > 0 he e exis s a posi i e cons an C(M) such ha
sup
|s|≤M
(|gn(s)|+| ′
n(s)|+| ′′
n(s)|)≤C(M)
o all n≥1.
( ) I also holds ha :
lim
|s|→∞
| ′
n(s)|+|gn(s)|
log(1 + |s|)= 0 uni o mly in n,
ha is, o any ε > 0 he e exis s Mε>0 such ha
|gn(s)|+| ′
n(s)| ≤ εlog(1 + |s|) o all |s| ≥ Mεand n≥1.
In pa icula , he las wo p ope ies imply ha , o any ε > 0, he e exis s
Cε>0, only depending on εand no on he unc ion n, such ha
|gn(s)|+| ′
n(s)| ≤ Cε+εlog(1 + |s|) o all s∈IR and n≥1.(49)
As i was p o ed in he p e ious Subsec ion, o any n≥1 he e exis s a con ol
unc ion n∈L (Q), wi h supp n⊂ω×[0, T], such ha he ollowing cascade o
sys ems
(∂ yn−∆yn+ n(yn) = ξ+ n1ωin Q,
yn= 0 on Σ, yn(x, 0) = 0 in Ω,(50)
21

(−∂ qn−∆qn+ ′
n(yn)qn=yn1Oin Q,
qn= 0 on Σ, qn(x, T) = 0 in Ω,(51)
admi s a solu ion (yn, qn)∈Z ×Z sa is ying
qn(x, 0) = 0 in Ω.(52)
Mo eo e , es ima es
||yn||Z ≤ C1(Ω, ω, O, T, yn)µkξkL +°
°
°
°
exp µ1
2 3¶ξ°
°
°
°L2¶,(53)
and
|| n||L ≤ C2(Ω, ω, O, T, yn)µkξkL +°
°
°
°
exp µ1
2 3¶ξ°
°
°
°L2¶,(54)
hold, whe e
C1(Ω, ω, O, T, yn) = exp [C(1 + kgn(yn)k∞+k ′
n(yn)k∞)] ,
C2(Ω, ω, O, T, yn) = exp [C(1 + kgn(yn)k∞+k ′
n(yn)k∞)] (1 + k ′′
n(yn)∇ynkL ),
and C=C(Ω, ω, O, T)>0.
Le us ecall ha ynis, o any n≥1, a ixed poin o a se - alued mapping
Λnde ined om Z on o i sel . In iew o es ima es (53) and (54), and aking in o
accoun (49) and Rema k 2, one deduces, a guing as in he p e ious Subsec ion,
ha he e exis s R > 0 la ge enough, and independen o n, such ha
Λn¡B(0; R)¢⊂B(0; R).
In addi ion, {yn}( esp. { n}) is uni o mly bounded in Z ( esp. in L (Q)). Indeed,
easoning as in Subsec ion 4.1, {yn}is uni o mly bounded in he space X and
hence, being g a e han N/2 + 1, he e exis s a compac se Kin Z such ha
{yn} ⊂ K.
Thus, up o a subsequence, one has
yn→ys ongly in Z
and
n→ weakly in L (Q),
wi h ∈L (Q) and y∈K⊂Z . Taking now in o accoun p ope ies (ii) and (iii),
one also has
gn(yn)→g(y) and ′
n(yn)→ ′(y) in C0(Q).
22
Hence, passing o he limi in (50)–(52), one in e s ha yand he co esponding
qsol e (38)–(40), ha is, we ha e ound a con ol in L (Q) insensi izing φ. This
ends he p oo o Theo em 1.1. ¤
5 P oo o Theo em 1.2
This Sec ion is de o ed o p o ing he insensi i i y esul o a nega i e na u e s a ed
in Theo em 1.2. To do so, we will show ha o ce ain unc ions as in he
s a emen and ce ain sou ce e ms ξ∈L (Q) anishing o ∈(0, 0), wi h 0∈
(0, T), wha e e he con ol is, he co esponding solu ion y o (4) blows up be o e
he ime =Tand hence, he unc ional φcanno be insensi ized. We will ollow
he p oo o Theo em 1.1 in [10], whe e he lack o null con ollabili y o a semilinea
hea equa ion is p o ed.
Le us conside he ollowing unc ion
(s) = Z|s|
0
logα(1 + σ)dσ o all s∈IR,
wi h α > 2. I is easy o check ha is a con ex unc ion, (s)s < 0 o all s < 0
and
| (s)|∼|s|logα(1 + |s|) as |s| → ∞.
Le ρ∈ D(Ω) be such ha
ρ≥0 in Ω, ρ ≡0 in ω, and ZΩ
ρ(x)dx = 1.
Fo a ixed 0∈(0, T), we se
ξ(x, ) = (0 i ∈[0, 0],
−(M+k) i ∈( 0, T],
whe e Mis a posi i e cons an which will be chosen la e and kis gi en by
k=1
2ZΩ
ρ ∗µ2|∆ρ|
ρ¶dx.
He e ∗is he con ex conjuga e o he con ex unc ion ( he unc ion ρcan be
aken in such a way ha ρ ∗(2|∆ρ|/ρ)∈L1(Ω), see [10]).
23
Le ybe a solu ion o (4), associa ed o a con ol and ξ, de ined in he maximal
in e al [0, T∗). Mul iplying he equa ion in (4) by ρand in eg a ing in Ω, we ge
d
d ZΩ
ρy dx =ZΩ
ρ∆y dx −ZΩ
ρ (y)dx +ZΩ
ρξ dx.
We also se
z( ) = −ZΩ
ρ(x)y(x, )dx, ∀ ∈[ 0, T∗).
Using he p ope ies o (see [10] o he de ails), we ob ain





z′( )≥M+1
2 (z( )), ∈[ 0, T∗),
z( 0) = z0=−ZΩ
ρ(x)y(x, 0)dx.
De ining
G(z0;s) = Zs
z0
2
(σ) + 2Mdσ, ∀s≥z0,
we can p o e ha
T∗≤ 0+ sup
∈[ 0,T∗)
G(z0;z( )) ≤ 0+Z∞
z0
2
(σ) + 2Mdσ.
Thus, o M > 0 la ge enough, he solu ion yblows up in L1(Ω) be o e T. This
ends he p oo .
6 Fu he commen s, esul s and open p oblems
6.1 On he cons uc ion o egula con ols o pa abolic
null con ollabili y p oblems
The echnique o cons uc ion o egula con ols (s a ing om L2–con ols) in o-
duced in he p esen pape can be applied o he s udy o he null con ollabili y o
(7) no only when jus depends on he s a e ybu also when = (y, ∇y). In ac ,
con ols in Cα, α
2(Q), wi h α∈(0,1], can be ob ained o po en ials egula enough.
Le us desc ibe his s a egy in he linea case. We conside he null con ollabili y
p oblem
(∂ y−∆y+B· ∇y+ay = 1ωin Q,
y= 0 on Σ, y(x, 0) = y0in Ω,(55)
24
y(x, T) = 0 in Ω,
whe e y0∈L2(Ω), a∈L∞(Q) and B∈L∞(Q)N. The p e ious null con ollabili y
p oblem is equi alen o





∂ q−∆q+B· ∇q+aq =−η′( )Y+ 1ωin Q,
q= 0 on Σ, q(x, 0) = 0 in Ω,
q(x, T) = 0 in Ω,
(56)
whe e q=y−η( )Y,η∈C∞([0, T]) sa is ies
η≡1 in [0, T/3], η ≡0 in [2T/3, T],
and Ysol es (55) wi h = 0. Suppose ha he e exis s a con ol ˆ ∈L2(Q) sol ing
(56), wi h supp ˆ ⊂B0×[0, T], and le ˆqbe he associa ed s a e (he e B0⊂⊂ ωis
a non-emp y open se ). Then q= (1 −θ(x))ˆq oge he wi h
=θ(x)η′( )Y+ 2∇θ· ∇ˆq+ (∆θ)ˆq−(B· ∇θ)ˆq
sol e he null con ollabili y p oblem (56), whe e θ∈ D(ω) e i ies θ≡1 in B0.
Following Subsec ion 3.2, one can p o e ha ∈L∞(Q) and
k k∞≤C(T, Ω, ω, kak∞,kBk∞)kˆ kL2.
The explici dependence o he cons an Con kak∞and kBk∞can also be ob ained.
Mo eo e , i B∈Cα, α
2(Q), he con ol is p o ed o lie in Cα, α
2(Q) and e i ies a
Cα,α
2–es ima e simila o he p e ious one.
This s a egy has been used in [6] o cons uc h¨olde ian con ols o a null
con ollabili y p oblem o he hea equa ion wi h nonlinea bounda y Fou ie con-
di ions (see also [7]).
6.2 O he insensi i i y esul s
The p oo o Theo em 1.1 can be adap ed o p o e o he insensi i i y esul s o
sys em (1).
1. Theo em 1.1 is s ill ue unde a sligh ly mo e gene al condi ion on . To be
p ecise, he e exis s l1(Ω, ω, O, T)>0 such ha , i hypo hesis (10) is eplaced
by
lim sup
|s|→∞
| ′(s)|
log(1 + |s|)≤l1,
a con ol unc ion insensi izing he unc ional gi en by (2) can be ound o
any gi en sou ce e m ξas in Theo em 1.1.
25
Then, ecalling ha w0|V0=y|V0, one in e s om (69) and (68) ha
y∈L (0, T;W2,p0(V0))
and he ollowing es ima e is sa is ied
kykL (W2,p0(V0)) ≤C(1 + kak∞)¡kFkL (L (V)) +kykL2(H2)∩C(H1)¢.
I ≤2 + 4
N, ha is, i p0= , he i s poin is al eady p o ed. Le us now
suppose ha > 2 + 4
N(i.e. 2 < p0=2N
N −4< ). We will now use he ollowing
Lemma, which will be p o ed a he end o his Appendix.
Lemma A.1 Le a∈L∞(Q)and F∈L2(Q)∩L (0, T;L (V)) be, wi h V ⊂ Ωan
a bi a y open se and ∈(2,∞). Le y∈L2(0, T;H2(Ω))∩C([0, T]; H1(Ω)) sa is y
(13). Le ω0and ω1be wo open subse s o Ωsuch ha ω1⊂⊂ ω0⊂ V. Le us
assume ha y∈L (0, T;W2, 0(ω0)), wi h 0∈[2, ). Then,
y∈L (0, T;W2, 1(ω1)) and ∂ y∈L (0, T;L 1(ω1)),
wi h
1=


min µ , N 0
N− 0¶i 0< N,
i 0≥N.
(70)
Fu he mo e, he ollowing es ima e holds:
kykL (W2, 1(ω1)) +k∂ ykL (L 1(ω1)) ≤C(1 + kak∞)¡kFkL (L (V)) +kykL (W2, 0(ω0))¢,
whe e Cis a posi i e cons an depending on Ω,T,N, ,ω0and ω1.¤
We apply his Lemma o i= 1,...,I, eplacing ω0,ω1, 0and 1, espec i ely,
by Vi−1,Vi,pi−1and pi, wi h
1
pi
=1
p0
−i
N o i= 1,...,I−1 and pI= .
This yields y∈L (0, T;W2,pi(Vi)), ∂ y∈L (0, T;Lpi(Vi)), 1 ≤i≤I, and he co -
esponding es ima es. In o de o de e mine I, obse e ha we may go on applying
Lemma A.1 while pi−1< N and pi< , ha is o say, while i < N( −2) −4
2 .
32

Thus, in Is eps, wi h
I=·N( −2) −4
2 ¸+ 1
([σ] being he in ege pa o he eal numbe σ), we ha e y∈X (0, T;V′) oge he
wi h es ima e (14), wi h K=N
2+ 2, which is a uni o m bound o I+ 1.
b) Suppose now ha F∈L (0, T;W1, (V)), as abo e, and ∇a∈Lγ(Q)N, wi h
γgi en by (15). Le us conside a new open se ˜
Vsuch ha
V′⊂⊂ ˜
V ⊂⊂ V.
In iew o poin a), y∈X (0, T;˜
V) and
kykX (0,T;˜
V)≤C(1 + kak∞)K£kFkL (L (V)) +kykL2(H2)∩C(H1)¤.(71)
To end he p oo , le us see ha , in addi ion,
∂iy∈X (0, T;V′),1≤i≤N,
and ha he ollowing es ima e is sa is ied
k∂iykX (0,T;V′)≤C(1 + kak∞)K+1(1 + k∂iakLγ)£kFkL (W1, (V)) +kykL2(H2)∩C(H1)¤,
whe e ∂iydeno es he de i a i e o ywi h espec o xi, 1 ≤i≤N. To do so, le
us se
wi=ζ1∂iy o a ixed i∈ {1,...,N},
wi h ζ1∈ D(˜
V) a unc ion such ha ζ1≡1 in V′. Then, wisol es (67) wi h G=Gi
gi en by
Gi=ζ1∂iF−ζ1a∂iy−ζ1y∂ia−2∇ζ1· ∇ (∂iy)−(∆ζ1)∂iy. (72)
Le us see ha Gi∈L (Q). We will s udy in de ail he e m ζ1y∂ia. Unde
assump ions on y,aand F, i is di ec o see ha he o he e ms in (72) lie in
L (Q).
Le us obse e ha ζ1y∈X . In iew o Lemma 2.2, since ∇a∈Lγ(Q)N, wi h γ
gi en by (15), and ecalling ha he H¨olde space Cl, l
2(Q) is con inuously embedded
in C0(Q), o all ∈(2,∞) one may in e ha he e m in o conside a ion, ζ1y∂ia,
lies in L (Q) and one has
kζ1y∂iakL ≤Ckζ1ykX k∂iakLγ.
33
Hence, coming back o (72), Gi∈L (Q) and one can es ima e
kGikL ≤Chk∂iFkL (L (V)) + (kak∞+k∂iakLγ+ 1) kykX (0,T ;˜
V)i.(73)
The egula izing e ec o he hea equa ion (see [12] and [13]) yields wi∈X , wi h
kwikX ≤CkGikL .(74)
Then, om (74), (73) and (71), one deduces
kwikX ≤C(1 + kak∞)K+1 (1 + k∂iakLγ)£kFkL (W1, (V)) +kykL2(H2)∩C(H1)¤,
wi h C=C(Ω, T, N, , V,V′) and K=K(N) he same as abo e.
Finally, jus aking in o accoun ha wi≡∂iyin V′, poin b) is p o ed. ¤
Rema k 3Following he p e ious p oo , one easily obse es ha he same esul
can be ob ained when eplacing hypo hesis y∈L2(0, T;H2(Ω)) ∩C([0, T]; H1(Ω))
by
y∈L2(0, T;H1
loc(Ω)) ∩L∞(0, T;L2
loc(Ω)).
This ac jus a ec s he numbe Io s eps equi ed o p o e poin a). ¤
We end his Appendix by gi ing he
P oo o Lemma A.1: Le us conside a unc ion ζ∈ D(ω0) such ha ζ≡1 in
ω1and le us se u=ζy. Then, usol es (67), wi h
G=ζF −[ζay + 2∇ζ· ∇y+ (∆ζ)y].
The egula i y o yand usual Sobole embeddings gi e G∈L (0, T;L 1(Ω)),
whe e 1is gi en by (70), and he es ima e
kGkL (L 1(Ω)) ≤C(1 + kak)∞£kFkL (L (V)) +kykL (W2, 0(ω0))¤(75)
(he e Cdepends on he open se s ω0and ω1).
Then, again due o he egula izing p ope ies o he hea equa ion, one deduces
ha
u∈L (0, T;W2, 1(Ω)), ∂ u∈L (0, T;L 1(Ω))
and
kukL (W2, 1(Ω)) +k∂ ukL (L 1(Ω)) ≤CkGkL (L 1(Ω)).
Finally, aking in o accoun ha u|ω1=y|ω1and inequali y (75), he esul
ollows. ¤
34
Acknowledgemen s. The au ho s hank he e e ees o hei in e es ing com-
men s and sugges ions.
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