Periodic solutions in unbounded domains for the Boussinesq system

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