Well-posedness and asymptotic behaviour for the Bousinessq system in Rn

XX Cong eso de Ecuaciones Di e enciales y Aplicaciones X Cong eso de Ma em´ a ica Aplicada Se illa, 24-28 sep iemb e 2007 (pp. 1–8) Well-posedness and asymp o ic beha iou o he Boussinesq sys em in Rn. E. J. Villamiza -Roa1, L. C. Fe ei a 2 1Escuela de Ma em´a icas, Uni e sidad indus ial de San ande , A.A. 678, Buca amanga-Colombia. E-mails: [email p o ec ed]. 2Dp o. de Ma em´a ica, Uni e sidade Fede al de Pe nambuco, Reci e-B azil. E-mail: [email p o ec ed]. Palab as cla e: Well-posedness, asymp o ic beha iou , Boussinesq sys em Resumen We analyze he well-posedness o he ini ial alue p oblem o a Con ec ion P o- blem. Mild solu ions a e ob ained in he weak-Lp(Rn) spaces and he exis ence o sel -simila solu ions is showed, while he only small sel -simila solu ion in he Lebes- gue space Lp(Rn) is he null solu ion. The asymp o ic s abili y o solu ions is analyzed and, as a consequence, a c i e ium o sel -simila i y pe sis ence a la ge imes is ob- ained. 1. In oduc ion We conside a iscous incomp essible luid illing he whole space Rn, n ≥2.Due o he Boussinesq app oxima ion (Chand asekha [3]), densi y a ia ions a e neglec ed excep in he g a i a ional e m (buoyancy e m) and hey a e assumed o be p opo ional o empe a u e a ia ions. The ela ionship among he eloci y ield u(x, )∈Rn, he p essu e p(x, )∈Rand he empe a u e θ(x, )∈R,can be desc ibed by he ollowing ini ial alue p oblem ∂u ∂ +u∇u−ν∆u+1 ρ∇p=βθ + 1, x ∈Rn, > 0,(1) ∇ · u= 0, x ∈Rn, > 0,(2) ∂θ ∂ +u∇θ−χ∆θ=h, x ∈Rn, > 0,(3) θ(x, 0) = θ0(x), u(x, 0) = u0(x), x ∈Rn,(4) whe e ep esen s a g a i a ional ec o ield a x,h he e e ence empe a u e and 1 an ex e nal o ce. ρ, ν, β, χ a e posi i e physical cons an s which ep esen , espec i ely, 1 E.J. Villamiza -Roa, L. C. Fe ei a he densi y, he kinema ic iscosi y, he coe icien o olume expansion and he he mal conduc ance. Wi hou loss o gene ali y, we will assume he cons an s ρ, ν, β, χ o be one and he e e ence empe a u e hand he ex e nal o ce 1 o be ze o. The ini ial da a u0 sa is ies he condi ion ∇ · u0= 0 in he dis ibu ional sense. New aspec s o s udies on he Con ec ion P oblem (1)-(4) a e conside ed in his wo k, in ac , we will s udy he sys em (1)-(4) in he whole space Rnin he amewo k o weak−Lpspaces. Fi s ly we p esen esul s o well-posedness in hese spaces and make some conside a ions a ound he well-posedness in he Lebesgue Spaces Lp(see [2]). On he o he hand, we show some esul s abou he exis ence, uniqueness, he asymp o ic s abili y and he sel -simila i y pe sis ence o solu ions o he P oblem (1)-(4) in weak−Lp spaces. Mo eo e , as a consequence o esul s o asymp o ic s abili y, we will show ha he only sel -simila solu ions in Lebesgue spaces Lpis he null solu ion, ein o cing he need o mo e singula ini ial da a o allow he exis ence o sel -simila solu ions. These sel -simila solu ions co espond, o ins ance, o homogeneous ini ial unc ions o deg ee −1. Finally, om a physical s andpoin , ou analysis can be applied o se e al classes o ex e nal o ces . In ac , we can ake as he g a i a ional ield = (x) = −G∇xφ=Gx |x|3∈L(n 2,∞)(Rn), whe e Gis he g a i a ional cons an , in o de o show he exis ence o global solu ions (u, θ) which a e cons uc ed in di e en unc ional spaces (see Theo em 3.5, Theo em 3.3 and Rema k 4.5). This case can be ega ded as an in e es ing physical case o he B´ena d P oblem. Mo e de ails abou he physical and ma hema ical analysis o sys em (1)-(4) see [2] and he e e ences he ein. 2. Func ion Spaces and De ini ions In his sec ion, we in oduce he unc ional spaces ele an o ou s udy o solu ions ega ding he Cauchy p oblem o sys em (1)-(4). Fo each Lebesgue mensu able unc ion de ined on Rn, he ea angemen ∗is de ined by ∗( ) = ´ın {s > 0 : m({x∈Rn:| (x)|> s})≤ }, > 0. The Lo en z space L(p,q)≡L(p,q)(Rn) is he se o all such ha k k(p,q)=   ³p qR∞ 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 R 0 ∗(s)ds, o > 0.We obse e ha Lp=L(p,p).L(p,∞)a e called he Ma cinkiewicz spaces o weak-Lpspaces. Mo eo e , L(p,q1)⊂Lp⊂L(p,q2)⊂ L(p,∞) o 0 < q1≤p≤q2≤ ∞. See [4]. P oposi ion 2.1 [4] (Gene alized Holde ’s inequali y). Le 1< p1,p2, < ∞. Le ∈ L(p1,q1)and g∈L(p2,q2)whe e 1 p1+1 p2<1, hen he p oduc h= g belongs o L( ,s)whe e 1 =1 p1+1 p2, and s≥1is any numbe such ha 1 q1+1 q2≥1 s. Mo eo e , khk( ,s)≤ 0k k(p1,q1)kgk(p2,q2), 2 Well-posedness and asymp o ic beha iou o he Boussinesq sys em being 0 he conjuga e index o . 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)}.P (o simply P) deno es he p ojec ion ope a o om L on o L σ.The S okes ope a o is deno ed by A (o simply Ao −P∆) and he Laplace ope a o is deno ed by B (o simply Bo −∆). We know ha −A ,−B gene a e uni o mly bounded holomo phic semig oups {e− A } ≥0,{e− B } ≥0o class C0in L σand L , espec i ely. Bo che s and Miyakawa [1] es ablished he ollowing Helmhol z decom- posi ion o he Lo en z spaces. We can ex end P o a bounded ope a o on L( ,d)(Ω), which we deno e by P ,d.Se L( ,d) σ(Ω) = Range(P ,d) and G( ,d)(Ω) = Ke nel(P ,d). Then, L( ,d)(Ω) = L( ,d) σ(Ω) ⊕G( ,d)(Ω).Based on [1], −A, −Bgene a e uni o mly boun- ded analy ic semig oups on L( ,d) σ(Ω) and L( ,d)(Ω), espec i ely. Howe e , no ice ha hese semig oups a e no s ongly con inuous a = 0 i d=∞,since in his case D ,∞(A) and D ,∞(B) a e no dense in L( ,∞) σand L( ,∞), espec i ely. We ecall ha in ou case Ω = Rn, {e− B} ≥0is he hea semig oup gi en as he con olu ion wi h he Gauss-Weie s ass ke - nel: G(x, ) = (4π )−n/2exp(−|x|2/(4 )). Finally, le 1 < p < ∞,1< q < ∞and 1 ≤d≤ ∞.The ollowing no a ion is adop ed o he no m o p oduc in Lo en z spaces L(p,d) σ(Rn)×L(q,d)(Rn) : ° ° °hu θi° ° °(p,q),d =kuk(p,d)+kθk(q,d). I p=qand d=∞,we simply deno e his no m as ° ° °hu θi° ° °(p,∞)=kuk(p,∞)+kθk(p,∞). 3. Resul s o well-posedness We de ine he ope a o M:L(p,∞) σ×L(q,∞)→L(p,∞) σ×L(q,∞)by Mhu θi=h−P∆u −∆θi. Wi h he use o he semig oup {e− M } ≥0, he Cauchy p oblem (1)-(4) is con e ed o he in eg al equa ion hu( ) θ( )i=e− M hu0 θ0i−Z 0 e−( −s)M³h(u· ∇u) (u· ∇θ)i−h(θ ) 0i´ds, > 0,(5) in L(p,∞) σ(Rn)×L(q,∞)(Rn).The e m in (5) −Z 0 e−( −s)Mh(u· ∇u)(s) (u· ∇θ)(s)ids will be called he bilinea ec o , and he e m in (5) Z 0 e−( −s)P∆(θ )ds, 3 E.J. Villamiza -Roa, L. C. Fe ei a we will called o coupling e m. Le us emembe he ollowing L(p,d)−L( ,d)es ima es o he semig oup {e− M } ≥0 and gi e he p oo by comple eness. Lemma 3.1 [2] Le 1≤d≤ ∞. Fo all (ϕ, φ)∈L(p,d) σ(Rn)×L(q,d)(Rn),and all > 0, he e exis s a cons an C(p, , s, q)such ha ° ° °∇je− M hϕ φi° ° °( ,s),d ≤C −n 2(γ+j n)° ° °hϕ φi° ° °(p,q),d, whe e γ= 1/p −1/ = 1/q −1/s, wi h 1< p ≤ < ∞and 1< q ≤s < ∞. Nex , le us in oduce sui able ime dependen unc ional spaces in which we will need o s udy he ini ial alue p oblem (1)-(4). De ini ion 3.2 Le n < q < ∞and α= 1 −n/q. We de ine he spaces E≡ {(u, θ) : (u, θ)∈BC((0,∞), L(n,∞) σ×L(n,∞))}, Eq≡ {(u, θ)∈E: α/2(u, θ)∈BC((0,∞), L(q,∞) σ×L(q,∞))}, Fq≡ {(u, θ) : u∈BC((0,∞); L(n,∞) σ), α/2θ∈BC((0,∞); L(q,∞))}, which a e Banach spaces wi h he no ms in E, Eq, Fqde ined, espec i ely, as ° ° °hu θi° ° °E= sup >0° ° °hu θi° ° °(n,∞),° ° °hu θi° ° °Eq =° ° °hu θi° ° °E+ sup >0 α/2° ° °hu θi° ° °(q,∞), ° ° °hu θi° ° °Fq = sup >0 kuk(n,∞)+ sup >0 α/2kθk(q,∞). Theo em 3.3 (i) Le n > 2a posi i e in ege numbe , (u0, θ0)be any pai in L(n,∞) σ× L(n,∞)and small enough wi h espec he ollowing no m k kb= sup >0 β 2k ( )k(b,∞)<∞, β = 2 −n b, b > n 2. Then, he e a e cons an s 0< τ =τ( )<1,δ > 0and ε=ε(δ)>0(ε→0when δ→0) such ha i ° ° °hu0 θ0i° ° °(n,∞)< δ, he ini ial alue p oblem (1)-(4) has a global solu ion (u( , x), θ( , x)) ∈Esa is ying (5), wi h l´ım →0(u( ), φ) = (u0, φ),l´ım →0(θ( ), ϕ) = (θ0, ϕ), o all φ∈L(n0,1) σ(Rn), ϕ ∈L(n0,1) σ(Rn).Mo eo e , i ° ° °hu θi° ° °E<2ε 1−τ, hen he solu ion is unique. (ii) I we assume ha (u0, θ0)∈(L(n,∞) σ×L(p,∞) σ)∩(L(n,∞)×L(p,∞))wi h 1< p0< n, he e a e 0< δp≤δand 0< τp=τp( )≤τsuch ha i ° ° °hu0 θ0i° ° °(p,∞)< δp, hen p e ious solu ion (u, θ) e i ies ha (u, θ)∈BC((0,∞), L(p,∞) σ×L(p,∞)). 4 Well-posedness and asymp o ic beha iou o he Boussinesq sys em Theo em 3.4 (Regula iza ion) Unde he assump ions o Theo em 3.3, le n < q < ∞, such ha 1 b+1 q>1 n. I k ( )kb= sup >0 β 2k ( )k(b,∞)is small enough, he e a e cons an s 0< τq( )<1and 0< δq≤δsuch ha i ° ° °hu0 θ0i° ° °(n,∞)< δq, hen he solu ion (u, θ)o Theo em 3.3 belongs o Eq. In he case n > 2, he assump ion k kb<∞can be changed by he ollowing one: sup >0k ( )k(n 2,∞)<∞. Indeed we will p o e he ollowing heo em: Theo em 3.5 Le (u0, θ0)∈L(n,∞) σ×L(n,∞)whe e n > 2and assume ha belongs o BC((0,∞), L(n 2,∞)). I n < q < ∞and sup >0k k(n 2,∞)is su icien ly small, hen he e a e cons an s 0< τ =τ( )<1,δ > 0and ε=ε(δ)>0(ε→0when δ→0) such ha i ° ° °hu0 θ0i° ° °(n,∞)< δ, hen he ini ial alue p oblem o (1)-(4) has a global solu ion (u( , x), θ(x, )) ∈Fqsa is ying (5) oge he wi h l´ım →0(u( ), φ) = (u0, φ),l´ım →0(θ( ), ϕ) = (θ0, ϕ), o all φ∈L(n0,1) σ(Rn), ϕ ∈L(n0,1) σ(Rn).Mo eo e , i ° ° °hu θi° ° °Fq ≤2ε 1−τ, hen he solu ion is unique in he space Fq. Fu he mo e, i we assume ha (u0, θ0)∈(L(n,∞) σ∩L(p,∞))×(L(n,∞) σ∩L(p,∞)), wi h q0< p0<n 2, he e a e 0< δp≤δand 0< τp=τp( )≤τsuch ha i ° ° °hu0 θ0i° ° °(n,∞)< δp, hen p e ious solu ion (u, θ)sa is ies (u, θ)∈BC((0,∞), L(p,∞) σ×L(p,∞)). 3.1. Ske ch o P oo s o he well-posedness Theo ems The p oo s o he well-possedness Theo ems ollows basically om he nex lemma in a gene ic Banach space and lemmas 3.7, 3.8, 3.9 ( see [2]). Lemma 3.6 Le Xbe a Banach space wi h no m k · kX,T:X→Xa linea con inuous map wi h no m τ < 1and B:X×X→Xa con inuous bilinea map, ha is, he e exis s a cons an K > 0such ha o all x1and x2in XkB(x1, x2)kX≤Kkx1kXkx2kX. Then, o 0< ε < (1−τ)2 4Kand o any ec o y∈X,y6= 0, such ha kykX< ε, he e exis s a solu ion x∈X o he equa ion x=y+B(x, x) + T(x)such ha kxkX≤2ε 1−τ. The solu ion xis unique in he closed ball B2ε 1−τ:= B(0,2ε 1−τ)⊂X. Mo eo e , he solu ion depends con inuously on yin he ollowing sense: I k˜ykX≤ε,˜x= ˜y+B(˜x, ˜x) + T(˜x), and k˜xkX≤2ε 1−τ, hen kx−˜xkX≤1−τ (1−τ)2−4Kε ky−˜ykX. Lemma 3.7 I (u0, θ0)∈L(n,∞) σ×L(n,∞).Then e− M hu0 θ0i∈E, wi h ° ° °e− M hu0 θ0i° ° °E≤ C° ° °hu0 θ0i° ° °(n,∞)and e− M hu0 θ0i*hu0 θ0iwhen →0+,whe e he limi is aken in he 5 E.J. Villamiza -Roa, L. C. Fe ei a weak-s a opology o he L(n,∞) σ×L(n,∞).Mo eo e ° ° °e− M hu0 θ0i° ° °Eq ≤C° ° °hu0 θ0i° ° °(n,∞),and i (u0, θ0)∈(L(p,∞) σ×L(p,∞)) hen ° ° °e− M hu0 θ0i° ° °(p,∞)≤C° ° °hu0 θ0i° ° °(p,∞). Lemma 3.8 Le n, b be as in he Theo em 3.3 and T(θ) = R 0e( −s)P∆(θ )(s)ds. Then kT(θ)k(n,∞)≤Ck kbsup >0 kθk(n,∞),kT(θ)k(p,∞)≤Ck kbsup >0 kθk(p,∞). Mo eo e , i n, b sa is y he assump ions o Theo em 3.4, hen kT(θ)kEq≤Ck kbsup >0 α 2kθk(q,∞). Lemma 3.9 I 1< p < q < ∞ hen o all φ∈L(p,1)(Rn)hold: s1 2(n p−n q+1)k∇e−sM φk(q,1) ≤Ckφk(p,1), s1 2(n p−n q)ke−sM φk(q,1) ≤Ckφk(p,1), Z∞ 0 s1 2(n p−n q)−1 2k∇e−sM φk(q,1)ds ≤Ckφk(p,1),Z∞ 0 s1 2(n p−n q)−1ke−sM φk(q,1)ds ≤Ckφk(p,1). 4. Sel -Simila i y Assuming ha ( , x) = λ2 (λ2 , λx) is smoo h and ha (u( , x), θ( , x)) is a smoo h solu ion o he con ec ion p oblem (1)-(4), i is s aigh o wa d o check ha (u, θ)λ( , x) = λ(u(λ2 , λx), θ(λ2 , λx)) is also a solu ion o he Sys em (1)-(4). In ac , we can look o pa icula solu ions o he Sys em (1)-(4) sa is ying (u( , x), θ( , x)) = (u( , x), θ( , x))λ( , x),(6) o any > 0, x∈Rnand λ > 0.These solu ions a e called sel -simila solu ions o he sys em and i is clea ha aking →0+, o mally in (6), (u(0, x), θ(0, x)) should be a homogeneous unc ion o deg ee −1. This ema k gi es he hin ha a sui able space o ind sel -simila solu ions should be one con aining homogeneous unc ions wi h ha exponen . The space L(n,∞)is he only weak-Lpspace such ha |x|−1∈L(p,∞). Mo eo e , in case ha such a sel -simila solu ion exis s, i s no m is in a ian by he scaling ans o ma ion, (u( , x), θ( , x)) →(u( , x), θ( , x))λ=λ(u(λ2 , λx), θ(λ2 , λx)). Mo eo e , homogenous unc ions o any o de do no belong o any s ong Lpspace. All o hese ac s ein o ce he idea ha weak-Lpspaces wi h he igh homogenei y a e he mos ele an spaces o inding global non- i ial sel -simila solu ions o he Con ec ion P oblem. 4.1. Decay Es ima es in weak −Lpand Lp Theo em 4.1 Le (u0, θ0)as in he Theo em 3.4 and ≥pis ini e and sa is ies 1 p+1 q− 1 <1 nand 1 b+1 p−1 <2 n. Then he solu ion o he Theo em 3.4 sa is ies (n 2p−n 2 )u∈BC((0,∞); L( ,∞) σ)n, (n 2p−n 2 )θ∈BC((0,∞); L( ,∞)). Mo eo e , his heo em is ue by elaxing he assump ions o n≥2, and e en subs i u ing weak-Lpspaces by hei s onge coun e pa s. 6 Well-posedness and asymp o ic beha iou o he Boussinesq sys em 4.1.1. P oo o Theo em 4.1. Le (u0, θ0)∈L(p,∞) σ×L(p,∞). As a di ec consequence o Lemma 3.1 we ha e ha sup >0 −p 2γ° ° °e− M hu0 θ0i° ° °( ,∞)≤C° ° °hu0 θ0i° ° °(p,∞).Now, by he second pa o Theo em 3.3, we al eady know ha i he ini ial da a (u0, θ0)∈(L(p,∞) σ∩L(n,∞) σ)×(L(p,∞)∩L(n,∞)), hen he solu ion (u( ), θ( )) sa is ies sup >0(ku( )k(p,∞)+kθ( )k(p,∞))<∞.Thus, in o de o conclude he p oo o he Theo em 4.1, we need a lemma whe e we es ima e he no m sup >0 −p 2γk·k ,∞o bilinea ec o e m and he linea ope a o e m T(θ), using he no m k · kEq+ sup >0k·kp,∞o he solu ion. Fo his, we p o e he ollowing lemmas. Lemma 4.2 Le pand bas in he Theo em 4.1 and ≥p, hen sup >0 (n 2p−n 2 )kT(θ)( )k( ,∞)≤Ck kbsup >0 kθk(p,∞). Lemma 4.3 Le pas in he Theo em 4.1 and ≥p, hen sup >0 ρ ° ° °Z 0 e−( −s)Mh(u1· ∇u2) (u2· ∇θ1)i° ° °( ,∞)≤Csup >0° ° °hu1 θ1i° ° °(p,∞)sup >0 α 2ku2k(q,∞), whe e ρ= ( n 2p−n 2 ). 4.2. Sel -Simila Solu ion in he spaces L(n,∞). The aim o his subsec ion is o desc ibe he p incipal esul s ela i e o he exis ence and he uniqueness o sel -simila i y solu ions in he L(n,∞)-spaces. Theo em 4.4 Le (u0, θ0)∈L(n,∞) σ×L(n,∞). Assume ha u0, θ0a e homogeneous unc- ions o deg ee −1, ha is, u0(λx) = λ−1u0(x), θ0(λx) = λ−1θ0(x) o all x∈Rn,x6= 0 and all λ > 0and as in Theo em 3.3 and Theo em 3.5, sa is ies he scale ela ion ( , x) = λ2 (λ2 , λx).Then, i ° ° °hu0 θ0i° ° °(n,∞)< ε he solu ion (u, θ)gi en by Theo em 3.3 and Theo em 3.5 is sel -simila , i.e., u( , x) = λu(λ2 , λx), θ( , x) = λθ(λ2 , λx), o all x∈Rn,x6= 0 and all λ > 0. Mo eo e , in case o Theo em 3.3, i he ini ial da a is smalle ° ° °hu0 θ0i° ° °(n,∞)< εq, he p e ious unique sel -simila solu ion becomes egula ized. Rema k 4.5 [B´ena d P oblem] No e ha , in he case o Theo em 3.5, we can ake as he g a i a ional ield = (x) = −G∇x(1 |x|) = Gx |x|3∈L(n 2,∞)(Rn),whe e Gis he g a i a ional cons an . This case can be ega ded as he B´ena d p oblem (see [3]) which co esponds o he in e es ing physical case. This conside a ion is also ue o he modi ied Theo em 3.3, whe e we assume ∈BC([0,∞); L(n 2,∞)(Rn)),ins ead o sup >0 β 2k ( )k(b,∞)<∞, and we sea ch solu ion in he space Eq. 7 E.J. Villamiza -Roa, L. C. Fe ei a 5. S abili y in L(n,∞). We analyze he la ge ime beha io o solu ions o Sec ion 3. In sho , we will show ha pe u ba ions o he ini ial da a a e negligible o la ge imes. Theo em 5.1 Assume ha (u, θ)and ( , φ)a e solu ions o (1)-(4) as in he Theo em 3.3 co esponding o he ini ial condi ions (u0, θ0)and ( 0, φ0)∈L(n,∞) σ×L(n,∞), espec i ely. Suppose ha l´ım →∞ ° °e ∆(θ0−φ0)° °(n,∞)= l´ım →∞ ° °e P∆(u0− 0)° °(n,∞)= 0, hen l´ım →∞ ku( )− ( )k(n,∞)= 0,l´ım →∞ kθ( )−φ( )k(n,∞)= 0. Mo eo e , assume (u, θ)and ( , φ)a e solu ions o (1)-(4) gi en by Theo em 3.4 co es- ponding o ini ial condi ions (u0, θ0)and ( 0, φ0)∈L(n,∞) σ×L(n,∞)sa is ying ha l´ım →∞ α 2° °e ∆(θ0−φ0)° °(q,∞)= l´ım →∞ ° °e P∆(u0− 0)° °(q,∞)= 0, hen l´ım →∞ α 2ku( )− ( )k(q,∞)= 0,l´ım →∞ α 2kθ( )−φ( )k(q,∞)= 0. Theo em 5.2 Assume ha (u, θ)and ( , φ)a e solu ions o (1)-(4) as in he Theo em 3.5 co esponding o he ini ial condi ions (u0, θ0)and ( 0, φ0)∈L(n,∞) σ×L(n,∞), espec i ely. Suppose ha l´ım →∞ α 2° °e ∆(θ0−φ0)° °(q,∞)= 0 and ha l´ım →∞ ° °e P∆(u0− 0)° °(n,∞)= 0, hen l´ım →∞ ku( )− ( )k(n,∞)= 0,l´ım →∞ α 2kθ( )−φ( )k(q,∞)= 0. Co olla y 5.3 Le (u0, θ0)∈Ln σ×Ln(Lebesgue space) be as in he Theo em 3.4. Then he co esponding solu ion sa is ies l´ım →∞ ku( )kLn= 0,l´ım →∞ kθ( )kLn= 0.As a consequence, he unique sel -simila solu ion in Ln(Rn)×Ln(Rn)is he null solu ion. Ag adecimien os E J Villamiza -Roa was suppo ed by COLCIENCIAS, Colombia, P oyec o COLCIENCIAS- BID III e apa and UIS and L C F Fe ei a was suppo ed by CAPES, B azil. Re e encias [1] W Bo che s, and T Miyakawa, 1995 On s abili y o ex e io Na ie -S okes lows. Ac a Ma h. 147, 311-382. [2] L C F Fe ei a, E J Villamiza -Roa, 2006 Well-posedness and asymp o ic beha iou o he con ec ion p oblem in Rn,Nonlinea i y 19, 2169-2191. [3] S Chand asekha , 1981 Hid odinamic and Hyd omagne ic S abili y. Do e , New Yo k. [4] R O’Neil, 1963 Con olu ion ope a o s and L(p, q)spaces. Duke Ma h. J. 30, 129-142. 8