Pe iodic solu ions in unbounded domains o he Boussinesq
sys em ∗
Elde Jes´us VILLAMIZAR-ROA
Co esponding au ho , Escuela de Ma em´a icas, Uni e sidad Nacional de Colombia,
Sede Medell´ın A.A. 3840, Medell´ın, Colombia, ej illamiza [email p o ec ed]
Ma ´ıa ´
Angeles RODR´
IGUEZ-BELLIDO
Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa,
Ap do. 1160 -41080 Se illa, Spain, [email p o ec ed]
&
Ma ko An onio ROJAS-MEDAR
Dp o. de Ciencias B´asicas, Facul ad de Ciencias, Uni e sidad del B´ıo B´ıo,
Campus Fe nando May, Casilla 447, Chill´an, Chile, ma k[email p o ec ed]
Abs ac
Assuming ha he ex e nal o ces o he sys em a e small enough, he e e ence empe -
a u e being a pe iodic unc ion, we s udy he exis ence, he uniqueness and he egula i y o
ime-pe iodic solu ions o he Boussinesq equa ions in se e al classes o unbounded domains
o Rn.Ou analysis is based on he amewo k o weak-Lpspaces.
Keywo ds: Boussinesq equa ions, S ong Pe iodic Solu ions, Unbounded domains.
MRSubClass: 35B35, 35B40, 35Q35, 76D03.
1 In oduc ion
Le Ωbe as ei he he whole space Rn,n≥3,ei he he hal space Rn
+,n≥3,ei he a bounded
domain in Rn,n≥3,o an ex e io domain in Rn,n≥4,wi h bounda y ∂Ωo class C2+µ(µ>0).
∗The second and hi d au ho s ha e been pa ially suppo ed by M.E.C. (Spain), P ojec MTM2006-07932.
The second au ho has been pa ially suppo ed by by Jun a de Andaluc´ıa, P ojec P06-FQM-02373. The hi d
au ho has been pa ially suppo ed by Fondecy -Chile, No. 1080628.
1
We conside he ollowing nons a iona y Boussinesq equa ions in Ω:
∂u
∂ −ν∆u+(u·∇)u+1
ρ∇p=βθg+Ψ,x∈Ω, ∈R,(1.1)
∇·u=0,x∈Ω, ∈R,(1.2)
∂θ
∂ −χ∆θ+(u·∇)θ= , x∈Ω, ∈R,(1.3)
u=0on ∂Ω,(1.4)
θ= 0 on ∂Ω,(1.5)
whe e g ep esen s he g a i a ional ield a x, is he e e ence empe a u e, Ψis an ex e nal
o ce and ρ,ν,β,χa e posi i e physical cons an s which deno e, espec i ely, he densi y, he
kinema ic iscosi y, he coefficien o olume expansion and he he mal conduc ance. The un-
knowns a e u(x, )∈Rn,p(x, )∈Rand θ(x, )∈R ep esen ing espec i ely, he eloci y ield,
he p essu e and he empe a u e o he luid a poin (x, )∈Ω×R.Boussinesq equa ions de-
sc ibe he e olu ion o he empe a u e and eloci y ield o a iscous incomp essible New onian
luid. Fo an ex ensi e discussion on he physical o igin o he equa ions (1.1)-(1.3), see [4].
We a e in e es ed in he s udy o he ime-pe iodic solu ions o he sys em (1.1)-(1.5) when
he e e ence empe a u e is a pe iodic unc ion wi h he same pe iod. Wi hou loss o gene ali y,
we ha e aken he cons an s ρ,ν,β,χequal o one. To a oid some echnical complexi ies in he
s udy o (1.1)-(1.5), h oughou his pape we assume Ψ=0.Se e al wo ks ha e been made in
he ma hema ical analysis o sys em (1.1)-(1.5); see, o ins ance, [3], [5], [7], [13], [6] and pape s
ci ed he ein. The ime-pe iodic solu ions o he Boussinesq equa ions in bounded domains
was conside ed in [13]. The analysis was made ia he Gale kin’s me hod. Indeed, in [13] i
was conside ed a class o nonlinea e olu ion equa ions in a sepa able Hilbe space gene alizing
se e al models o hyd odynamics. Howe e , he s udy o pe iodic solu ions o sys em (1.1)-(1.5)
has no been in es iga ed in unbounded domains. Hence, he pu pose o he p esen pape is
o p o e he exis ence and uniqueness o s ong pe iodic solu ions o p oblem (1.1)-(1.5) in
he amewo k o Semig oups Theo y on he Lo en z spaces, mo e explici ly, on he heo y o
weak-Lpspaces. We cons uc he ime-pe iodic solu ions using Lp,q −L ,s es ima es o he
semig oups gene a ed by he S okes and Laplace ope a o s. I Ωis an ex e io domain, we need
o assume n≥4 in o de o ob ain he g adien bounds o he semig oups gene a ed by he
S okes and he Laplace ope a o s in L(p,∞)(see Lemma 3.2).
This wo k is mo i a ed by he exis ence esul s o pe iodic solu ions o he Na ie -S okes
equa ions. In unbounded domains, his subjec has been in es iga ed in [11], [14], [15], [18]
and [19]. In pa icula , in [14] was p o ed he exis ence o a unique ime-pe iodic solu ion on
he whole space R3 o small ex e nal o ce. The p oblem in he hal -space R3
+was conside ed
in [15]. In [11], making use o Lp−L es ima es o he semig oup gene a ed by he S okes
ope a o , ime-pe iodic solu ions we e cons uc ed o small ime-pe iodic o ces. The s abili y
2
o hese solu ions was conside ed in [18]. Howe e , he exis ence o s ong ime-pe iodic solu ions
in gene al unbounded domains is s ill an open p oblem. Mo e comple e e e ences, including
esul s o bounded domains, can be ound in [11], [14], [15].
This pape is o ganized as ollows. In Sec ion §2, we gi e some p elimina ies abou Lo en z
spaces and s a e ou main esul s. Sec ion §3 is de o ed o p o e he exis ence and he uniqueness
o s ong pe iodic solu ions.
2 P elimina ies and Resul s
Be o e s a ing ou esul s we in oduce some unc ional spaces. C∞
0,σ(Ω) deno es he se o all
C∞− eal unc ions ϕ=(ϕ1, ..., ϕ
n) wi h compac suppo in Ω,such ha di ϕ=0.The
closu e o C∞
0,σwi h espec o no m L ,1< <∞,is deno ed by L
σ(Ω). Le us ecall he
Helmhol z decomposi ion: L (Ω) = L
σ(Ω)⊕G (Ω),1< <∞,whe e G (Ω) = {∇p∈L (Ω) :
p∈L
loc(Ω)}(c. . [8]). P deno es he p ojec ion ope a o om L (Ω) on o L
σ(Ω).The S okes
ope a o A =−P ∆wi h domain D(A )={u∈H2, (Ω) : u|∂Ω=0}∩L
σ.I iswellknown
ha −A gene a es a uni o mly bounded analy ic semig oup {e− A } ≥0o class C0in L
σ(c. .
[9]).
We deno e by Bq he Laplace ope a o in Lq(Ω),1<q<∞,wi h homogeneous Di ichle
bounda y condi ions: Bq=−∆wi h domain D(Bq)=W2,q(Ω)�W1,q
0(Ω).We ecall ha he
ope a o −Bqgene a es a uni o mly bounded analy ic semig oup {e− Bq} ≥0in Lq(Ω) o class
C0.
Now we in oduce some p elimina ies abou he Lo en z spaces. Fo de ails see [1]. Le
1<p≤∞and 1 ≤q≤∞.A Lebesgue measu able unc ion de ined on a domain Ω⊂Rn
belongs o Lo en z space L(p,q)(Ω) i he quan i y
� �(p,q)=

�q
p�∞
0� 1
p ∗∗( )�qd
�1
q,i 1<p<∞,1≤q<∞
sup >0 1
p ∗∗( ),i 1<p≤∞,q=∞
,
is ini e, whe e
∗∗( )=1
�
0
∗(s)ds, ∗( )=in {s>0:m{x∈Ω:| (x)|>s}≤ }, >0,
wi h mdeno ing he Lebesgue measu e on Rn.The spaces L(p,q)wi h he no m � �(p,q)a e Ba-
nach spaces. No e ha Lp(Ω) = L(p,p)(Ω). When q=∞,L
(p,∞)(Ω) a e called he Ma cinkiewicz
spaces o weak-Lpspaces. Mo eo e , L(p,q1)(Ω)⊂Lp(Ω)⊂L(p,q2)(Ω)⊂L(p,∞)(Ω) o 0 <
q1≤p≤q2≤∞. We ecall ha he space C∞
0(Ω) is no dense in L(p,∞)(Ω).Bo che s
and Miyakawa [2] es ablished he ollowing Helmhol z decomposi ion o he Lo en z spaces ex-
ending he ope a o P o a bounded ope a o on L( ,d)(Ω),which we deno e by P ,d.Se ing
L( ,d)
σ(Ω) = Range(P ,d) and G( ,d)(Ω) = Ke nel(P ,d), hen L( ,d)(Ω) = L( ,d)
σ(Ω)⊕G( ,d)(Ω),
3
wi h L( ,d)
σ(Ω) = {u∈L( ,d)(Ω) : ∇·u=0,u·n|∂Ω=0}and G( ,d)(Ω) = {∇ ∈L( ,d)(Ω) :
∈L( ,d)
loc (¯
Ω)}.Fo simplici y, we shall abb e ia e he p ojec ion ope a o and he S okes and
Laplace ope a o s on Lo en z spaces by P,A,B, espec i ely. In iew o [2], he ope a o s
−A,−Bgene a e bounded analy ic semig oups on L(p,q)
σ(Ω) and L(p,q)(Ω), espec i ely. How-
e e , we ecall ha i q=∞, his semig oups a e no s ongly con inuous a =0.
Applying he ope a o Pon he equa ions (1.1)-(1.2), om (1.1)-(1.5) we ob ain he ollowing
p oblem o pa abolic ype:
u +Au +P[(u·∇)u]=P(θg), ∈R(2.6)
θ +Bθ+(u·∇)θ= , ∈R.(2.7)
The sys em (2.6)-(2.7), wi h pe iodic in ime condi ions, has associa ed he ollowing sys em o
in eg al equa ions
u( )=−�
−∞
e−( −s)AP[(u·∇)u]ds +�
−∞
e−( −s)AP(θg)ds (2.8)
θ( )=−�
−∞
e−( −s)B(u·∇)θds+�
−∞
e−( −s)B ds. (2.9)
Th oughou his pape we assume he ollowing assump ions on he ex e nal o ce and he
ield g:
Assump ion 1.
(CASE 1). I Ωis ei he he whole space Rn,a bounded domain in Rn,wi h bounda y o
class C2+µ(µ>0),o he hal space Rn
+,n≥3,we conside , ˜ ,q, ˜q e i ying 2 < ,˜ <n, n
2<
q, ˜q<n, 1
−1
˜ <min{2
n−1
q,2
n−1
˜q}.
(CASE 2). I Ωis an ex e io domain in Rn,n≥4,wi h bounda y o class C2+µ(µ>0),we
conside , ˜ ,q, ˜qsuch ha 2n
(n−1) ≤ , ˜ <n, n
2<q,˜q<n, 1
−1
˜ <min{2
n−1
q,2
n−1
˜q}.
Fo each , ˜ and q, ˜qwe assume ha sa is ies
∈BC(R;L(˜p,∞)(Ω)∩L(˜
l,∞)(Ω)),(2.10)
o 1 <˜p,˜
l<∞wi h 1
˜ +2
n<1
˜p,1
˜q<1
˜
l<1
˜q+1
np o ided n≥4 in bo h CASES (1,2). (No e
ha as n<2˜q,˜ <n, hen he inequali y 1/˜q<1/˜
l<1/˜q+1/n implies ha 1/˜
l<2/n +1/˜ ).
I n=3,in he CASE 1, we assume ha sa is ies
� ∈BC(R,L
(˜
l,∞)(Ω)) such ha
(s)=Bδ
˜p,∞h(s) o some h∈BC(R,D(Bδ
˜p,∞)),(2.11)
4
o 1 <˜p < min{˜ , ˜q},and δ>0 sa is ying 3
2˜p+δ>1+max{1+ 3
2˜ ,1
2+3
2˜q}and 1/˜q<1/˜
l<
1/˜q+1/3, whe e Bδ
˜p,∞deno es he powe δo he ope a o Bon L˜p,∞.Wi h espec o he ield
gwe make he ollowing assump ions:
g∈L(a,∞)(Ω)∩L(b,∞)(Ω),
whe e aand ba e such ha : 1
a>2
n+1
−1
˜ ,1
b<1
n+1
q−1
˜ ,(b>1,a>1).
Rema k 2.1 Condi ion (2.11) can be eplaced by (s)=∇·G(s),G(s)=(G1,...,G
n)∈
BC(R;L(˜p,∞)(Ω)) wi h ∇G( )∈BC(R;L(˜p,∞)(Ω))n o 1<˜p<∞wi h 1/˜ +1/3<1/˜p. This
implies ha (s)=∆h(s) o some h∈BC(R;D(B˜p,∞)).
Ou main esul s a e s a ed as ollows:
Theo em 2.2 Le be a pe iodic unc ion wi h pe iod τ>0(i.e, o all ∈R, ( )= ( +τ))
sa is ying Assump ion 1. Then, i he quan i ies
sup
s∈R
� (s)�(˜p,∞)+sup
s∈R
� (s)�(˜
l,∞),n≥4,in he CASES 1 and 2,
sup
s∈R
�h(s)�(˜p,∞)+sup
s∈R
� (s)�(˜
l,∞),n=3,in he CASE 1,
�g�(b,∞)+�g�(a,∞),in he CASES 1 and 2,
a e small enough, hen he e exis s a pe iodic solu ion (u,θ)o (2.8),(2.9), wi h he same pe iod
τo he ex e nal o ce, in he class u∈BC(R;L( ,∞)
σ(Ω)),θ∈BC(R;L(˜ ,∞)(Ω)),wi h ∇u∈
BC(R;L(q,∞)(Ω))n,∇θ∈BC(R;L(˜q,∞)(Ω))n.Mo eo e , wi hin his class, i
sup
s∈R
�u(s)�( ,∞)+sup
s∈R
�∇u(s)�(q,∞),sup
s∈R
�θ(s)�(˜ ,∞)+sup
s∈R
�∇θ(s)�(˜q,∞)
a e small enough, hen he solu ion is unique.
Theo em 2.3 Unde he assump ions o Theo em 2.2, i is H¨olde con inuous wi h alues in
L( ∗,∞)(Ω)and g∈L( ∗,∞)(Ω), hen, o all n<
∗<q
∗=nq/(n−q), he pe iodic solu ion
gi en by Theo em 2.2 sa is ies
1. u∈BC(R;L(n,∞)
σ(Ω)) ∩C1(R;L ∗
σ(Ω)),θ∈BC(R;L(n,∞)(Ω)) ∩C1(R;L ∗(Ω)).
2. Anu∈C(R;L ∗
σ(Ω)),Bnθ∈C(R;L ∗(Ω)).
3. Fo all ∈R,(2.6) and (2.7) a e sa is ied in L ∗
σ(Ω)and L ∗(Ω), espec i ely.
5

3 Exis ence, Uniqueness and Regula i y o Pe iodic Solu ions
In his sec ion we p o e Theo em 2.2 and Theo em 2.3. Le us i s ecall he H¨olde ’s inequali y
and some L( ,∞)−L(p,∞)es ima es o he semig oups {e− A} ≥0,{e− B} ≥0.
P oposi ion 3.1 (Gene alized H¨olde ’s inequali y [16]) Le 1<p
1,p2, <∞, ∈
L(p1,q1)(Ω)and g∈L(p2,q2)(Ω)whe e 1
p1+1
p2<1, hen g belongs o L( ,s)(Ω)whe e 1
=1
p1+1
p2,
and s≥1is any numbe such ha 1
q1+1
q2≥1
s. Mo eo e ,
� g�( ,s)≤C( )� �(p1,q1)�g�(p2,q2).(3.12)
Lemma 3.2 ([2], [19]).
1. Le Ωbe ei he he whole space Rn,aboundeddomaininR3wi h bounda y ∂Ωo class
C2+µ(µ>0),o he hal space Rn
+,n≥3.Then
�∇je− Aa�( ,1) ≤c −n/2(1/p−1/ )−j/2�a�(p,1),1<p≤ <∞,
o all a∈L(p,1)
σ(Ω),j =0,1and all >0,whe e c=c(n, p, ).
2. Le Ωbe an ex e io domain in Rn,n≥4wi h bounda y ∂Ω,o class C2+µ(µ>0). Then
�e− Aa�( ,1) ≤c −n/2(1/p−1/ )�a�(p,1),1<p≤ <∞,
�∇e− Aa�( ,1) ≤c −n/2(1/p−1/ )−1/2�a�(p,1),1<p≤ ≤n,
o all a∈L(p,1)
σ(Ω)and all >0,whe e c=c(n, p, ).
Rema k 3.3 Es ima es in Lemma 3.2 o e− Aaand ∇e− Aain he no m L( ,∞)wi h espec o
he da a in L(p,∞)a e ue, because hey a e ob ained by duali y. Simila es ima es hold o he
semig oup {e− B} ≥0.The es ima es abo e hold in he pa icula case o Lpspaces (c. . [11]).
Lemma 3.4 ([2]). Le Ωbe as he CASE 1 and CASE 2 and suppose ha 1<q<n,1≤d≤∞
and q∗=nq/(n−q).I φ∈L(p,∞)(Ω) o some p<∞and ∇φ∈L(q,d)(Ω)n, hen φ∈L(q∗,d)(Ω)
and he es ima e �φ�(q∗,d)≤C�∇φ�(q,d)holds wi h C>0independen o φ.
We deno e by X he space o scala unc ions {u∈BC(R;L(˜ ,∞)):∇u∈BC(R;L(˜q,∞))n}
wi h he no m �·�
Xde ined as
�u�X≡sup
s∈R
�u(s)�(˜ ,∞)+sup
s∈R
�∇u(s)�(˜q,∞).
6
We also de ined by Y he space o ec o unc ions {u∈BC(R;L( ,∞)
σ(Ω)) : ∇u∈BC(R;L(q,∞)(Ω))n}
wi h he no m �·�
Yde ined as
�u�Y≡sup
s∈R
�u(s)�( ,∞)+sup
s∈R
�∇u(s)�(q,∞).
Xand Ya e Banach spaces. We de ine he ollowing ope a o s F1and Gon Y×Yand Y×X,
espec i ely, by
F1(u, )( )=−�
−∞
e−( −s)AP[(u·∇) ](s)ds, (3.13)
G(u,θ)( )=−�
−∞
e−( −s)B(u·∇)θ(s)ds. (3.14)
3.1 P oo o Theo em 2.2
We cons uc a pe iodic solu ion o P oblem (2.8)-(2.9) acco ding o he ollowing scheme:
um+1( )=F(um,θ
m)( ),θ
m+1( )=θ0( )+G(um,θ
m)( ),(3.15)
whe e
u0( )=�
−∞
e−( −s)AP(θ0g)ds, θ0( )=�
−∞
e−( −s)B ds, (3.16)
F(um,θ
m)( )=F1(um,um)( )+
�
−∞
e−( −s)A{P(gθm)}ds, (3.17)
G(um,θ
m)( )=−�
−∞
e−( −s)B(um·∇)θm(s)ds. (3.18)
Rema k 3.5 In (1.1), when Ψis no ze o, in he scheme abo e we conside
u0( )=�
−∞
e−( −s)AP(Ψ)(s)ds and um+1 =u0( )+F(um,θ
m)( ).
Le us i s ob ain some es ima es o app oxima ions abo e. We shall need he ollowing
lemmas.
Lemma 3.6 Le , ˜ ,q and ˜qbe as Theo em 2.2. Then, we ha e
sup
s∈R
�F1(u, )�( ,∞)≤c1sups∈R�u(s)�( ,∞)�sup
s∈R
� (s)�( ,∞)+sup
s∈R
�∇ (s)�(q,∞)�,
sup
s∈R
�∇F1(u, )�(q,∞)≤c1sup
s∈R
�∇ (s)�(q,∞)�sup
s∈R
�u(s)�( ,∞)+sup
s∈R
�∇u(s)�(q,∞)�,
sup
s∈R
�G(u,θ)�(˜ ,∞)≤c2sup
s∈R
�u(s)�( ,∞)�sup
s∈R
�θ(s)�(˜ ,∞)+sup
s∈R
�∇θ(s)�(˜q,∞)�,
sup
s∈R
�∇G(u,θ)�(˜q,∞)≤c2sup
s∈R
�∇θ(s)�(˜q,∞)�sup
s∈R
�u(s)�( ,∞)+sup
s∈R
�∇u(s)�(q,∞)�,
o e e y u, ∈Y,θ∈X, whe e c1=c1(n, , q),c
2=c2(n, , q, ˜ , ˜q).
7
P oo . The p oo is an applica ion o Lemma 3.2. In ac ,
G(u,θ)( )=−� −1
−∞
e−( −s)B(u·∇)θ(s)ds −�
−1
e−( −s)B(u·∇)θ(s)ds =G1( )+G2( ).
Then o all ψ∈C∞
0and o all ∈R,we ha e
|(G1( ),ψ)|≤� −1
−∞
�∇e−( −s)Bψ�(( ˜ /( +˜ ))�,1)�θu�( ˜ /( +˜ ),∞)ds
≤csup
s∈R
�θ(s)�(˜ ,∞)sup
s∈R
�u(s)�( ,∞)� −1
−∞
( −s)−n/2 −1/2�ψ�(˜ �,1).
Hence, by duali y, o all ∈R,�G1( )�(˜ ,∞)≤csup
s∈R
�θ(s)�(˜ ,∞)sup
s∈R
�u(s)�( ,∞).
�G2( )�(˜ ,∞)≤�
−1
( −s)−n/2(1/ +1/˜q−1/˜ )�u(s)�( ,∞)�∇θ(s)�(˜q,∞)ds
≤csup
s∈R
�u(s)�( ,∞)sup
s∈R
�∇θ(s)�(˜q,∞).
Now, using Lemma 3.2 and Lemma 3.4 ( o d=∞), we ge
�∇G(u,θ)�(˜q,∞)≤� −1
−∞
�∇e−( −s)B(u·∇)θ(s)�(˜q,∞)ds
+�
−1
�∇e−( −s)B[(u·∇)θ](s)�(˜q,∞)ds
≤csup
s∈R
�u(s)�( ,∞)sup
s∈R
�∇θ(s)�(˜q,∞)� −1
−∞
( −s)−n/2 −1/2ds
+csup
s∈R
�u(s)�(q∗,∞)sup
s∈R
�∇θ(s)�(˜q,∞)�
−1
( −s)−n/2qds
≤c�sups∈R�u(s)�( ,∞)+sup
s∈R�∇u(s)�(q,∞)�sups∈R�∇θ(s)�(˜q,∞),
o all ∈Rand c=c(n, , q).This comple e he p oo o he las wo es ima es o lemma. The
i s wo es ima es a e ob ained simila ly.
Lemma 3.7 Le θ0be de ined as in (3.16). Then θ0∈X.
P oo . I sa is ies (2.10), hen using Lemma 3.2 we ob ain
�θ0( )�(˜ ,∞)≤csup
s∈R
� (s)�(˜p,∞)� −1
−∞
( −s)−n/2(1/˜p−1/˜ )ds
+csup
s∈R
� (s)�(˜
l,∞)�
−1
( −s)−n/2(1/˜
l−1/˜ )ds.
This is alid o all ∈R.The cons an c=c(n, ˜ , ˜p, ˜
l).F om (2.10), ha is, 1/˜ +2/n < 1/˜pand
1/˜
l<2/n +1/˜ , we conclude ha each in eg al abo e is ini e and consequen ly, �θ0( )�(˜ ,∞)≤
8
c1sup
s∈R
� (s)�(˜p,∞)+c2sup
s∈R
� (s)�(˜
l,∞),whe e c1=c(n, ˜p, ˜ ) and c2=c(n, ˜
l, ˜ ). A simila analysis
p o es ha
�∇θ0( )�(˜q,∞)≤�c1sup
s∈R
� (s)�(˜p,∞)� −1
−∞
( −s)−n/2(1/˜p−1/˜q)−1/2ds
+�c2sup
s∈R
� (s)�(˜
l,∞)�
−1
( −s)−n/2(1/˜
l−1/˜q)−1/2ds,
o all ∈Rand �c1=c(n, �p, �q) and �c2=c(n,�l, �q). As 1/˜p>1/˜ +2/n > 1/n +1/˜qand
1/˜
l<1/˜q+1/n, he wo in eg als abo e con e ge.
Now, i n=3, he p e ious analysis is w ong because i will be necessa y 3/2(1/˜p−1/˜ )>1,
wi h ˜p>1 and his is no possible. Consequen ly, we assume a new condi ion; in ac , i
sa is ies (2.11), using he ollowing es ima e (which is a consequence o he analy ic p ope ies
o he semig oup):
�Bδe− Ba�(˜p,∞)≤C
−δ�a�(˜p,∞),∀a∈L(˜p,∞), >0,c=c(˜p, δ),δ≥0,
and using Lemma 3.2, we ob ain
�θ0( )�(˜ ,∞)≤� −1
−∞
�e−( −s)BBδh(s)�(˜ ,∞)ds +�
−1
�e−( −s)B (s)�(˜ ,∞)ds
≤c� −1
−∞
( −s)−3/2(1/˜p−1/˜ )�Bδe−( −s)B/2h(s)�(˜p,∞)ds +c�
−1
( −s)−3/2(1/˜
l−1/˜ )� (s)�(˜
l,∞)ds
≤c� −1
−∞
( −s)−3/2(1/˜p−1/˜ )−δ�h(s)�(˜p,∞)ds +sup
s∈R
� (s)�(˜
l,∞)�
−1
( −s)−3/2(1/˜
l−1/˜ )ds
≤c�sups∈R�h(s)�(˜p,∞)+sup
s∈R� (s)�(˜
l,∞)�,
o all ∈Rwi h c=c(n, ˜ , ˜p, ˜
l,δ).A simila es ima e can be ob ained o �∇θ0�(˜q,∞),(n= 3).
This p o es he lemma.
Now we will es ima e he e ms F(um,θ
m) and G(um,θ
m).We s a wi h he ollowing
lemma
Lemma 3.8 The e ms �F(um,θ
m)�Y,�G(um,θ
m)�Xgi en by (3.17),(3.18) sa is y
�F(um,θ
m)�Y≤2c1�um�2
Y+c3�θm�X,(3.19)
�G(um,θ
m)�X≤2c2�um�Y�θm�X,(3.20)
whe e c1,c
2a e as in Lemma 3.6 and c3depends on gbu is independen o m.
P oo . We will p o e ha
����
−∞
e−( −s)AP(gθm)(s)ds���Y≤c3�θm�X.(3.21)
9
By duali y, o 0< <
0+T, we ge
����
0
e−( −s)B( m·∇wm)(s)ds���(n/α,∞)≤C1,2Km,2Km,1( − 0)−(1−α)/2.
Now, using Lemma 3.4 we ob ain
�
0
�e−( −s)B (s)�(n/α,∞)≤c�
0
�∇e−( −s)B (s)�(q,∞)
≤c� �BC(R;L(l,∞))( − 0)−(1−α)/2+3/2−n/2l,
o all 0< <
0+Twi h c=c(n, q, l).Since 1/l < 1/q +1/n, we ha e (1 −α)/2<3/2−n/2l
and hence he abo e es ima e yields
( − 0)(1−α)/2����
0
e−( −s)B (s)���(n/α,∞)≤c� �BC(R,L(l,∞))T(1−α)/2.
Consequen ly,



sup
0< < 0+T
( − 0)(1−α)/2�wm+1( )�(n/α,∞)
≤K0,2+c� �BC(R;L(l,∞))T(1−α)/2+C1,2Km,1Km,2.
Then, we can ake Km+1,1,K
m+1,2being espec i ely,
Km+1,1=K0,1+C1,1K2
m,1+C2,1Km,2,(3.45)
Km+1,2=K0,2+c� �BC(R;L(l,∞))T(1−α)/2+C1,2Km,1Km,2.(3.46)
Se ing Km= max(Km,1,K
m,2),m=1,2,..., om (3.43), (3.44), (3.45) and (3.46) we ha e
Km+1 ≤K0+�
CK2
m+C2,1Km,
K0=c1T(1−α)/2max{�a�(n/α,∞),�b�(n/α,∞)+� �BC(R;L(l,∞))}(3.47)
and �
C= max{C1,1,C
1,2}.I we conside
C2,1<1,K
0<(1 −C2,1)2
4�
C,(3.48)
we ha e ha
Km<(1 −C2,1)−�(1 −C2,1)2−4�
CK0
2�
C≡k< 1
2�
C,∀m=0,1,2,... (3.49)
Assuming (3.48) and wo king as Subsec ion 3.2, due o he uni o m es ima e wi h espec
o m, we can conclude he exis ence o a couple ( ,w) such ha (3.40) holds and sa is ying
(3.41) −(3.42). Thus, we inish he p oo o Lemma 3.12.
16

Lemma 3.13 I K0de ined by (3.47) is small enough, hen he limi ( ,w)gi en by Lemma
3.12 sa is ies he ollowing es ima e
( − 0)1/2∇( (·),w(·)) ∈BC(( 0,
0+T); L(n,∞)(Ω)×L(n,∞)(Ω))n,(3.50)
wi h
lim
m→∞ sup
0< < 0+T
( − 0)1/2�∇ m( )−∇ ( )�(n,∞)=0,
lim
m→∞ sup
0< < 0+T
( − 0)1/2�∇wm( )−∇w( )�(n,∞)=0.
P oo . The p oo is done by induc ion. In ac , we will p o e ha
sup
0< < 0+T
( − 0)1/2�∇ m( )�(n,∞)≤Jm,1,(3.51)
sup
0< < 0+T
( − 0)1/2�∇wm( )�(n,∞)≤Jm,2,(3.52)
o some cons an s Jm,1,J
m,2which a e independen o 0,m=0,1,...
No e ha by Lemma 3.2
�∇ 0( )�(n,∞)≤C( − 0)−1/2�a�(n,∞),�∇w0( )�(n,∞)≤C( − 0)−1/2�b�(n,∞),
whe e C=C(n) is independen o 0.Hence we can ake J0,1and J0,2, being espec i ely,
C�a�(n,∞),C�b�(n,∞).
Supposed inequali ies (3.51)-(3.52) a e ue. Then
���∇�
0
e−( −s)AP[( m·∇) m]���(n,∞)≤�
0
( −s)−n/2(α/n)−1/2� m�(n/α,∞)�∇ m�(n,∞)
≤cKm,1Jm,1�
0
( −s)−α/2−1/2(s− 0)α/2−1ds
≤cKm,1Jm,1( − 0)−1/2B((1 −α)/2,α/2) ≤C3,1kJm,1( − 0)−1/2,
o all 0< <
0+T, whe e C3,1=C3,1(n, q) is independen o 0.Mo eo e
���∇�
0
e−( −s)AP(gwm)ds���(n,∞)≤�
0
( −s)−n/2(α/n)−1/2�g�(n,∞)�wm�(n/α,∞)ds
≤c�g�(n,∞)Km,2�
0
( −s)−(α+1)/2(s− 0)−(1−α)/2ds
≤cB((1 −α)/2,(1 + α)/2)k�g�(n,∞)≤C4,1k�g�(n,∞).
The e o e,
sup
0< < 0+T
( − 0)1/2�∇ m+1�(n,∞)≤J0,1+C3,1kJm,1+C4,1k�g�(n,∞)T1/2.
Now, o any , 0< <
0+T,
���∇�
0
e−( −s)B( m·∇wm)ds���(n,∞)≤�
0
( −s)−n/2(α/n)−1/2� m�(n/α,∞)�∇wm�(n,∞)
≤cKm,1Jm,2�
0
( −s)−α/2−1/2(s− 0)α/2−1ds ≤C2,2kJ
m,2( − 0)−1/2,
17
whe e C2,2is independen o 0.As
���∇�
0
e−( −s)B (s)ds���(n,∞)≤c( − 0)−1/2� �BC(R;L(n,∞)),
we conclude ha
sup
0< < 0+T
( − 0)1/2�wm+1( )�(n,∞)≤J0,2+C2,2kJm,2+c� �BC(R;L(n,∞)).
Then we can ake Jm+1,1and Jm+1,2being espec i ely,
Jm+1,1=J0,1+C3,1kJm,1+C4,1k�g�(n,∞)T1/2,
Jm+1,2=J0,2+C2,2kJm,2+c� �BC(R;L(n,∞)) .
Le Jm= max{Jm,1,J
m,2},m=1,2,... and
J0= Max{J0,1+C4,1k�g�(n,∞)T1/2,J
0,2+c� �BC(R;L(n,∞))},
hen
Jm+1 ≤J0+k�
CJm,
whe e �
C= max{C3,1,C
2,2}.Consequen ly, i
k<1/�
C(3.53)
we ha e a uni o m es ima e o he sequence {Jm}gi en by Jm≤J0
1−
�
Ck ≡J, m =0,1,...
Assuming (3.53), we can see ha he limi s ,w sa is y (3.50) and he p oo o Lemma 3.13
is inished.
Lemma 3.14 The limi ( ,w)gi en by Lemma 3.12 and Lemma 3.13 e i ies
( − 0)1/4( (·),w(·)∈BC(( 0,
0+T); L2n
σ(Ω)×L2n(Ω)),
wi h
lim
m→∞ sup
0< < 0+T
( − 0)1/4� m( )− ( )�2n=0,
lim
m→∞ sup
0< < 0+T
( − 0)1/4�wm( )−w( )�2n=0.
P oo . As he p e ious lemmas, he p oo is done by induc ion. In ac , we will p o e ha he e
exis some cons an s Nm,1,N
m,2,which a e independen o 0,such ha
� m( )�2n≤Nm,1( − 0)−1/4,�wm( )�2n≤Nm,2( − 0)−1/4.(3.54)
18
Since L(p0,∞)∩L(p1,∞)⊂Lpand � �p≤C(p0,p
1,λ)� �1−λ
(p0,∞)� �λ
(p1,∞)p o ided ha p0�=
p1,0<λ<1 and 1/p =(1−λ)/p0+λ/p1,we ha e
� 0( )�2n≤C( − 0)−1/4�a�(n,∞),�w0( )�2n≤C( − 0)−1/4�b�(n,∞),
whe e C=C(n) is independen o 0.Hence, we de ine N0,1and N0,2as C�a�(n,∞)and C�b�(n,∞),
espec i ely.
Assuming ue (3.54) o a gi en m, we can p o e ha (3.54) holds o he case m+1.In ac ,
no e ha o all φ∈C∞
0,σ,ϕ∈C∞
0,we ge
����−�
0
e−( −s)AP[( m·∇) m](s)ds, φ����≤�
0
� m⊗ m�n�∇e−( −s)Aφ�n�ds
≤C�
0
� m�2
2n( −s)−3/4�φ�(2n)�ds ≤c( − 0)−1/4B(1/4,1/2) N2
m,1�φ�(2n)�
����−�
0
e−( −s)B( m·∇wm)(s)ds, ϕ����≤�
0
�wm· m�n�∇e−( −s)Bϕ�n�ds
≤C�
0
� m�2n�wm�2n( −s)−3/4�ϕ�(2n)�ds ≤c( − 0)−1/4B(1/4,1/2) Nm,1Nm,2�ϕ�(2n)�.
Hence by duali y
����
0
e−( −s)AP( m·∇ m)(s)ds���2n≤C1,1N2
m,1( − 0)−1/4,
����
0
e−( −s)B( m·∇wm)(s)ds���2n≤C1,2Nm,1Nm,2( − 0)−1/4.
We also no e ha
����
0
e−( −s)AP(gwm)(s)ds���2n≤�
0
( −s)−1/2�g�(n,∞)�wm(s)�(2n)ds
≤c�g�(n,∞)B(1/2,3/4) ( − 0)1/4≤C2,1Nm,2( − 0)1/4
and
����
0
e−( −s)B (s)ds���2n≤�
0
( −s)−1/4� (s)�(n,∞)ds ≤c� �BC(R;L(n,∞))( − 0)3/4.
The inequali ies abo e imply ha
sup
0< < 0+T
( − 0)1/4� m+1�2n≤N0,1+C1,1N2
m,1+C2,1Nm,2T1/2
sup
0< < 0+T
( − 0)1/4�wm+1�2n≤N0,2+C1,2Nm,1Nm,2+c� �BC(R;L(n,∞))T.
As be o e, se ing Nm= max(Nm,1,N
m,2),m=1,2,...and N0= max(N0,2+c� �BC(R;L(n,∞))T,N0,1),
we ob ain Nm+1 ≤N0+�
CN2
m+�
C2,1Nm,wi h �
C2,1=C2,1T1/2whe e �
C= max(C1,1,C
1,2).I
we conside
C2,1<1,N
0<(1 −C2,1)2
4�
C,(3.55)
19
we ha e ha he sequence {Nm}m=∞
m=0 is bounded wi h
Nm≤(1 −C2,1)−�(1 −C2,1)2−4N0�
C
2�
C,m=0,1,...
Assuming (3.55) and wo king as Lemma 3.12 and Lemma 3.13, we conclude he p oo .
Lemma 3.15 The limi ( ,w)gi en by Lemma 3.12 and Lemma 3.13 e i ies
( ,w)∈BC(( 0,
0+T); L(n,∞)
σ(Ω)∩L(q∗,∞)
σ(Ω)×L(n,∞)(Ω)∩L(q∗,∞)(Ω)),(3.56)
wi h
sup
0< < 0+T
� m( )�(n/s,∞)≤M1,s,m,s=α, s =1,(3.57)
sup
0< < 0+T
�wm( )�(n/s,∞)≤M2,s,m,s=α, s =1,(3.58)
and M1,s,m,M
2,s,m independen o 0.
P oo . Compu a ions simila o Lemma 3.12, Lemma 3.13, yield o
M1,α,0=C�a�(n/α,∞),M
1,1,0=C�a�(n,∞),
M2,α,0=C�b�(n/α,∞),M
2,1,0=C�b�(n,∞),
whe e C=C(n, s) is independen o 0.Suppose by induc ion ha (3.57), (3.58) a e ue. No e
ha
����−�
0
e−( −s)AP( m·∇ m)(s)ds, φ����≤����
0
( m⊗ m(s),∇e−( −s)Aφ)ds���
≤�
0
� m�(n/α,∞)� m�(n/s,∞)�∇e−( −s)Aφ�(n/(n−α−s),1) ds
≤CKm,1M1,s,m �
0
( −s)−α/2−1/2(s− 0)−(1−α)/2·�φ�(n/(n−s),1) ds
≤CkM1,s,mB((1 −α)/2,(1 + α)/2)�φ�(n/(n−s),1),
o all φ∈C∞
0,σand all 0< <
0+Tand C=C(n, q, s) independen o 0. Consequen ly, by
duali y, o s=1,α,we ha e
sup
0< < 0+T����
0
e−( −s)AP( m·∇ m)(s)ds���(n/s,∞)≤C5,1kM
1,s,m,(3.59)
whe e C5,1independen o 0.No e ha
����
0
e−( −s)AP(gwm)(s)ds���(n/s,∞)≤c�g�(b,∞)�
0
( −s)−n/2b�wm(s)�(n/s,∞)ds
≤c�g�(b,∞)( − 0)1−n/2b≤C6,1M2,s,m ( − 0)1−n/2b,
20
����
0
e−( −s)B (s)ds���(n/α,∞)≤c� �BC(R;L(l,∞))Ta,
wi h a=α
2−n
2l+1>0.Mo eo e , o all ϕ∈C∞
0and all 0< <
0+T,
����−�
0
e−( −s)B( m·∇wm)(s)ds, ϕ����≤����
0
(wm· m(s),∇e−( −s)B)ϕ)ds���
≤�
0
� m�(n/α,∞)�wm�(n/s,∞)�∇e−( −s)Bϕ�(n/(n−α−s),1) ds
≤cK
m,1M2,s,m ��
0( −s)−α/2−1/2(s− 0)−(1−α)/2ds��ϕ�(n/(n−s),1)
≤ckM
2,s,m B((1 −α)/2,(1 + α)/2) �ϕ�(n/(n−s),1).
Thus, o s=1,α,
sup
0< < 0+T����
0
e−( −s)B( m·∇wm)(s)ds���(n/s,∞)≤C4,2kM
2,s,m,(3.60)
whe e C4,2is independen o 0.Hence, om (3.59)-(3.60) we can ake
M1,s,m+1 =M1,s,0+C5,1kM
1,s,m +C6,1M2,s,m,(3.61)
M2,s,m+1 =M2,s,0+c� �BC(R;L(l,∞))Ta+C4,2kM
2,s,m.(3.62)
Se ing
Ms,m = max{M1,s,m,M
2,s,m},
Ms,0= max{M1,s,0,M
2,s,0+c� �BC(R;L(l,∞))Ta},
˘
C= max{C5,1,C
4,2},
om (3.61),(3.62) we ob ain Ms,m+1 ≤Ms,0+k˘
CM
s,m +C6,1Ms,m, o m=0,1,...,s =1,α.
Then, i
k˘
C+C6,1<1,(3.63)
we ha e Ms,m ≤Ms,0
1−k˘
C−C6,1,m=0,1,...,s=1,α,which yields o (3.56) wi h (3.57)-(3.58).
Now, we con inue he p oo o P oposi ion 3.11. We can see ha unde condi ions (3.48),
(3.53) and (3.63), he limi ( ,w) belongs o he class equi ed in P oposi ion 3.11. Mo eo e ,
hen ollowing con e gences hold in L(n,∞)
σ(Ω), L(n,∞)
σ(Ω) and L(n,∞)(Ω), espec i ely
�
0
e−( −s)AP[( m·∇) m](s)ds −→ �
0
e−( −s)AP[( ·∇) ](s)ds,
�
0
e−( −s)AP(wmg)(s)ds −→ �
0
e−( −s)AP(wg)(s)ds,
�
0
e−( −s)BP[( m·∇)wm](s)ds −→ �
0
e−( −s)AP[( ·∇)w](s)ds, (3.64)
21

uni o mly in ∈( 0,
0+ ) as m→∞.In ac , no e ha by Lemma 3.2, Lemma 3.12 and
Lemma 3.13, we ha e
����
0
e−( −s)AP[( m·∇) m](s)ds −�
0
e−( −s)AP[( ·∇) ](s)ds���(n,∞)
≤�
0
( −s)−α/2� m(s)− (s)�(n/α,∞)�∇ m(s)�(n,∞)ds
+�
0
( −s)−α/2� (s)�(n/α,∞)�∇( m(s)− (s))�(n,∞)ds
≤cB(1 −α/2,1/2) Jsup
0<s< 0+T
(s− 0)(1−α)/2� m(s)− (s)�(n/α,∞)
+B(1 −α/2,(α+ 1)/2) ksup
0<s< 0+T
(s− 0)1/2�∇ m(s)−∇ (s)�(n,∞)),
which con e ges o 0. On he o he hand
����
0
e−( −s)AP(wmg)(s)ds −�
0
e−( −s)AP(wg)(s)ds���(n,∞)
≤�
0
�e−( −s)AP((wm−w)g)(s)�(n,∞)ds
≤�
0
( −s)−n/2b+(1−α)/2�g�(b,∞)�wm(s)−w(s)�(n/α,∞)
≤cT1−n/2bsup
0<s< 0+T
(s− 0)(1−α)/2�wm(s)−w(s)�(n/α,∞)−→ 0.
Analogously we ob ain (3.64). Now we will p o e he weak con inui y on he ini ial dada. Fi s ly,
we no e ha o any φ∈L(n�,1)
σ(Ω) and ϕ∈L(n�,1)(Ω) we ha e
|(e−( − 0)Aa−a,φ)|=|(a,e
−( − 0)Aφ−φ)|
≤�a�(n,∞)�e−( − 0)Aφ−φ�(n�,1) →0, → +
0.
|(e−( − 0)Bb−b, ϕ)|=|(b, e−( − 0)Aϕ−ϕ)|
≤�b�(n,∞)�e−( − 0)Bϕ−ϕ�(n�,1) →0, → +
0.
As � �(q∗,∞),�w�(q∗,∞)≤cand ( − 0)1/4� �2n,( − 0)1/4�w�2n≤c, we ha e
lim
→ 0��
0
e−( −s)AP(wg)(s)ds −�
0
e−( −s)AP[( ·∇) ](s)ds, φ�=0,
lim
→ 0��
0
e−( −s)B (s)ds −�
0
e−( −s)B( ·∇w)(s)ds, ϕ�=0.
Indeed, no e ha when → +
0,
�
0�e−( −s)B[( ·∇)w],ϕ
�≤�
0
� �(q∗,∞)�w�2n�∇e−( −s)Bϕ�(1−1/q∗−1/2n,1)
≤c�
0
(s− 0)−1/4( −s)−n/2q+1/4�ϕ�(n�,1) ds
≤c( − 0)1−n/2qB(3/4,−n/2q+5/4) �ϕ�(n�,1) →0,
22
�
0�e−( −s)AP[( ·∇) ],φ�≤�
0
� �(q∗,∞)� �2n�∇e−( −s)Aφ�(1−1/q∗−1/2n,1)
≤c�
0
(s− 0)−1/4( −s)−n/2q+1/4�φ�(n�,1) ds
≤c( − 0)1−n/2qB(3/4,−n/2q+5/4) �φ�(n�,1) →0,
�
0�e−( −s)AP(wg),φ�ds ≤�
0
�e−( −s)AP(wg)�(n,∞)�φ�(n�,1) ds
≤c�w�q∗�g�(n,∞)�φ�(n�,1) �
0
(s− 0)−n/2q∗ds
≤c( − 0)1−n/2q∗�w�q∗�g�(n,∞)�φ�(n�,1) →0,
and �
0�e−( −s)B (s),ϕ
�ds ≤�
0
�e−( −s)B (s)�(n,∞)�ϕ�(n�,1) ds
≤c�
0
� �(n,∞)ds �ϕ�(n�,1) ≤c( − 0)→0.
Collec ing all he p e ious con e gences and le ing m→∞in (3.36)-(3.37), we see ha ( ,w)
is a solu ion o (3.34)-(3.35). Finally, we will es ima e he ime-in e al To exis ence in e ms
o he p esc ibed da a. As kis de e mined by (3.49), he e exis s a cons an ˜
kindependen o
0such ha i K0≤˜
k, hen condi ions (3.48),(3.53), (3.63) a e sa is ied. Now, om (3.47) we
see ha Tmay be chosen as (3.33).
Rema k 3.16 The solu ion ( ,w)o in eg al equa ions (3.34)-(3.35) sa is ies ( ,w)∈BC( 0,
0+
T;Lp
σ(Ω)×Lp(Ω)), o all p∈(n, q∗),wi h � �p≤C� �1−λ
(n,∞)�u�λ
(q∗,∞),�w�p≤C�w�1−λ
(n,∞)�w�λ
(q∗,∞),
whe e λis such ha 1/p =(1−λ)/n +λ/q∗.
No e ha being aand belemen s o L(n,∞)
σ(Ω)∩L(q∗,∞)
σ(Ω) and L(n,∞)(Ω)∩L(q∗,∞)(Ω),
espec i ely, we ha e ha aand bbelong o space L ∗
σ(Ω) and L ∗(Ω), espec i ely. Consequen ly,
he no ms �e−( − 0)Aa� ∗≤C�a� ∗and �e−( − 0)Bb� ∗≤C�b� ∗a e ini e. Mo eo e , i is no
difficul o see ha
�����
0
e−( −s)AP[( ·∇) ],φ����≤C��
0
� �2
∗( −s)−n
2(1
( ∗)�−1
( ∗/2)�)−1
2��φ�( ∗)�
≤C� �2
∗��
0
( −s)−n
2 ∗−1
2ds��φ�( ∗)�≤C�φ�( ∗)�,
o all φ∈C∞
0,σ.By duali y, we ha e ha �
0e−( −s)AP[( ·∇) ](s)ds ∈L ∗
σ(Ω).
Analogously, we can see ha
�
0
e−( −s)B( ·∇w)(s)ds +�
0
e−( −s)AP(wg)(s)ds ∈L ∗(Ω),�
0
e−( −s)B (s)ds ∈L ∗(Ω).
23
The e o e we conclude ha equali ies (3.34)-(3.35) a e sa is ied in L ∗
σ(Ω) and L ∗(Ω), espec-
i ely. The p oo o P oposi ion 3.11 is inished. �
P oo o Theo em 3.10. By he hypo hesis o Theo em 3.10, we can apply P oposi ion
3.11 and hence, he e exis T∈(0,1] and unc ions and wsa is ying:













∈BCw([ 0,
0+T); L(n,∞)
σ(Ω)) ∩BC([ 0,
0+T); L(q∗,∞)
σ(Ω)),
w∈BCw([ 0,
0+T); L(n,∞)(Ω)) ∩BC([ 0,
0+T); L(q∗,∞)(Ω)),
( − 0)1/2∇ ∈BC(( 0,
0+T); L(n,∞)(Ω))n,
( − 0)1/2∇w∈BC(( 0,
0+T); L(n,∞)(Ω))n,
in such a way ha o all 0∈R he in eg al sys em (3.34)-(3.35) is sa is ied in he L ∗-no m.
Fi s ly, we s udy he uniqueness o ha solu ion. Le ( 1,w
1) ano he solu ion o (3.34)-
(3.35), in he class gi en by P oposi ion 3.11, wi h he same ini ial condi ion. Le (V,W)=
( − 1,w−w1). No e ha (V( ),W( )) ∈Ln
σ(Ω)×Ln(Ω) o all 0 < <T,wi h
sup
0< <T
�V( )�n<∞,sup
0< <T
�W( )�n<∞.(3.65)
In ac , aking n<q<min{2n, ∗}we ha e o all φ∈C∞
0,σand all 0< <T,
|(V( ),φ)|≤����
0
( − 1⊗ 1,∇e−( −s)Aφ)ds���+����
0
(e−( −s)AP(Wg),φ)ds���
≤C(n, q)
��sup
0<s<T
sn/2(1/n−1/q)� �q�2
+�sup
0<s<T
sn/2(1/n−1/q)� 1�q�2��φ�n�
+C(n, q)��sup
0<s<T
sn/2(1/n−1/q)�w(s)�q+sup
0<s<T
sn/2(1/n−1/q)�w1(s)�q�
�sup
0<s<T
s1/2�g�(n,∞)���φ�n�.
Analogously, o all ϕ∈C∞
0and all 0< <T
|(W( ),ϕ)|≤C(n, q)�sup
0<s<T
sn/2(1/n−1/q)� (s)�qsup
0<s<T
sn/2(1/n−1/q)�w(s)�q
+sup
0<s<T
sn/2(1/n−1/q)� 1(s)�qsup
0<s<T
sn/2(1/n−1/q)�w1(s)�q��ϕ�n�.
24
By duali y we can conclude (3.65). We de ine KV( )≡sup
0< <T
�V( )�(n,∞)and KW( )≡
sup
0< <T
�W( )�(n,∞).Le psa is ying 1/p =1−1/n −1/ ∗.Then
|(V( ),φ)|≤����
0
(V⊗ (s)− 1,∇e−( −s)Aφ)ds���+����
0
(e−( −s)AP(Wg)(s),φ)ds���
≤�
0
(� (s)� ∗+� 1(s)� ∗)�V(s)�n�∇e−( −s)Lφ�pds
+�
0
( −s)−1/2�W(s)�n�g�(n,∞)�φ�n�ds
≤CK
V( )�sup
0<s< 0+
(s− 0)n/2(1/n−1/ ∗)� (s)� ∗
+sup
0<s< 0+
(s− 0)n/2(1/n−1/ ∗)� 1(s)� ∗�
��
0
( −s)−n/2(1/n�−1/p)−1/2(s− 0)−n/2(1/n−1/ ∗)ds��φ�n�
+KWsup
0<s< 0+
(s− 0)1/2�g�(n,∞)��
0
( −s)−1/2(s− 0)−1/2ds��φ�n�,
o all φ∈C∞
0,σand o all 0< <T.Then, using duali y we ob ain
�V( )�n≤KV( )�sup
0<s< 0+
(s− 0)n/2(1/n−1/ ∗)� (s)� ∗+
+sup
0<s< 0+
(s− 0)n/2(1/n−1/ ∗)� 1(s)�∗
�+C2KW( )sup
0<s< 0+
(s− 0)1/2�g�(n,∞).(3.66)
Analogously,
|(W( ),ϕ)|≤CK
V( )sup
0<s< 0+
(s− 0)n/2(1/n−1/ ∗)�w1(s)� ∗�ϕ�n�
+CK
W( )sup
0<s< 0+
(s− 0)n/2(1/n−1/ ∗)� (s)� ∗�ϕ�n�,
o all ϕ∈C∞
0and all 0< <T.By duali y,
�W( )�n≤C3KV( )sup
0<s< 0+
(s− 0)n/2(1/n−1/ ∗)�w1(s)� ∗
+C4KW( )sup
0<s< 0+
(s− 0)n/2(1/n−1/ ∗)� (s)� ∗.(3.67)
Le K( ) = max{KV( ),K
W( )}.Then we ha e ha
max {�V( )�n,�W( )�n}≤C∗K( )�sup
0<s< 0+
(s− 0)n/2(1/n−1/ ∗)� (s)� ∗
+sup
0<s< 0+
(s− 0)n/2(1/n−1/ ∗)� 1(s)� ∗+sup
0<s< 0+
(s− 0)n/2(1/n−1/ ∗)�w(s)� ∗
+sup
0<s< 0+
(s− 0)n/2(1/n−1/ ∗)�w1(s)� ∗�+C∗∗ K( )sup
0<s< 0+
(s− 0)1/2�g�(n,∞).
We de ine he cons an �
C=C∗+C∗∗.F om hypo hesis o his heo em, he e exis s 0<
1≤T,
such ha
sup
0<s< 0+
(s− 0)n/2(1/n−1/ ∗)(�w(s)� ∗+�w1(s)� ∗+� (s)� ∗+� 1(s)� ∗)
+sup
0<s< 0+
(s− 0)1/2�g�(n,∞)<1
2�
C, o 0< <
1.
25