Large time behavior of solutions for a complex-valued quadratic heat equation

Transkript

1 Nonlinear Differ. Equ. Appl. 5, 5 45 c 5 Springer Basel -97/5/55-4 published online March 5, 5 DOI.7/s Nonlinear Differenial Equaions and Applicaions NoDEA Large ime behavior of soluions for a complex-valued quadraic hea equaion Amel Chouichi, Sarah Osmane and Slim Tayachi Absrac. In his paper we sudy he exisence and he asympoic behavior of global soluions for a parabolic sysem relaed o he complexvalued hea equaion wih quadraic nonlineariy: z =Δz + z,>, x R N, wih iniial daa z = u + iv. We show ha if u x c x α and v x c x α, as x wih α, α α N N α, α >, > c is sufficienly small, hen he soluion is global and converges o a self-similar soluion. We also esablish he exisence of four differen self-similar behaviors. These behaviors depend on he values of α and α. In paricular, he real and he imaginary pars of he consruced soluions may have differen behaviors in he L -norm for large ime. Also, he real par may have differen behaviors from hose known for he real-valued quadraic hea equaion. Mahemaics Subjec Classificaion. 35K5, 35K55, 35K65, 35B4. Keywords. Parabolic sysem, Semi-linear parabolic equaions, Global soluions, Large ime behavior, Self-similar soluions, Nonlinear hea equaion.. Inroducion In his paper we consider he complex-valued nonlinear hea equaion: z =Δz + z,. z = z,. where z = z, x is a complex valued funcion, >, x R N, and he iniial condiion z x is a complex valued funcion defined on R N. The aim of his paper is o prove he exisence of global soluions for. and sudy heir asympoic behavior, as, for a class of iniial daa z. In paricular, we consider z = u +iv, where u,v are given precisely in he space L R N CR N bu no in he space L R N. Equaion. has a srong relaion wih he viscous Consanin Lax Majda equaion. See and reference herein. If we wrie z, x = u, x +

2 6 A. Chouichi, S. Osmane and S. Tayachi NoDEA iv, x, where i =, hen he above equaion is rewrien as he parabolic sysem: { u =Δu + u v,.3 v =Δv +uv, u = u,v = v..4 In he case where z is real-valued, ha is v, hen he previous sysem is reduced o he scalar quadraic hea equaion u =Δu + u,.5 where u = u, x is a real valued funcion, >, x R N. In,8, i is proved ha if he iniial value u c x α, as x wih c is a small consan, N>αand α, hen he soluion u of.5 is global and is asympoic, in some Lebesgue spaces, o a self-similar soluion of.5 ifα =, and is asympoic o e Δ c x α ifα>, where e Δ is he hea semigroup. Moreover, in 8 i is proved ha he asympoic behavior holds in he L -norm. Furhermore, he resuling soluion u saisfies, for large, u α,α. The exisence of self-similar soluions of.5 wih iniial daa u x =c x is proved in. See also 5. We refer he reader o he paper 8 for more relaed resuls on he Eq..5. Many resuls have been esablished concerning exisence of self-similar soluions and asympoically self-similar soluions for oher sysems using he mehods used in, 8. In 6, he semi-linear parabolic sysem wih nonlinear erms of he form a j u pj u + b j v qj v, p j,q j >, a j, b j R, j =,, have been sudied. The global exisence and he exisence of asympoically self-similar soluion are shown under some condiions on he parameers. In 7, he semi-linear parabolic sysem wih produc nonlinear erms of he form a j u pj u v qj v, where a j R and p j,q j, j=,. If we rea he Eq.. asin,8, we obain similar resuls as in,8 on he behavior of z. Bu, his will no show he difference on he behavior of he real and he imaginary par of he soluion. On he oher hand, he sysem.3 is no sudied neiher in 6 nor in 7. In fac, he case where he sum and he produc are in he same sysem is no considered. The aim of his paper is o sudy he large ime behavior of global soluions for he sysem.3. The Cauchy problem for he sysem.3 wih an iniial daa u,v L R N CR N,.6 has a unique soluion u, v C,T; L R N, where T = T u,v, denoes he maximal exisence ime of he soluion. Moreover, we have eiher T =, or T< and lim { u., + v., } =..7 T The sysem.3 has been sudied in 3. In paricular, global exisence resuls and large ime behavior are esablished under some appropriae condiions on iniial values. More precisely, he auhors show ha if he iniial daa u,v saisfy for all x R N

3 Vol. 5 The complex valued nonlinear hea equaion 7 u x Av x <,.8 wih A R is a consan, hen, he soluion of.3 saisfying.6 exiss globally in ime. See 3, Theorem., p.. Moreover, in 3 he auhors sudy he large ime behavior for he case where he iniial daa are posiive and asympoically consans. More precisely, hey impose he following condiions on iniial daa: u,v C R N, u N,u N, v L, L >,.9 lim u x =N, lim v x =N,N >N >,. x x and hey prove ha he soluions converges o he saionary soluions of.3. See 3, Theorem.4, p. 4. In he presen paper, a differen class of iniial values are considered. In paricular, we do no impose a resricion on he sign of he iniial daa. Also, we do no consider he condiions.8 and.9. In our case. is saisfied for N = N =. We have obained he following heorem. Theorem.. Global exisence Le N be an ineger, α and α be posiive real numbers saisfying α and α α. Suppose ha N N >, α α >. Consider u,v L R N CR N be such ha u c η x α, v c η x α,. where η is an L cu-off funcion η near he origin and having a compac por. Then, here exiss ε> such ha if max c, c <ε, he soluion of.3 wih iniial daa u,v exiss globally in ime. The case of α = α is proved in,8. In fac, alhough in,8 only he real valued nonlinear hea equaion is considered, bu he case α = α can be proved using exacly he same calculaions of, 8. Tha is by working direcly on he Eq... Our resul is new for α α. For general formulaion see Theorem.6 below. The mehod which we have employed is differen from 3. Our ideas are based on he echnique used in,6 8. The exisence of self-similar soluions for he sysem.3, ha is., has been proved in. I is proved here ha if u = c x, v = c x, c sufficienly small, hen here exiss a self-similar soluion of he sysem.3. Similar calculaions, give he exisence of a self-similar soluion of he sysem.3 wih iniial daa c x,c x, wih c,c are wo small consans. In order o sudy he differen asympoic behaviors of global soluions for he sysem.3, we consruc self-similar soluions for he following asympoic sysem: { w =Δw + w,. w =Δw +w w, where w, x andw, x are real valued funcions, >, x R N.Wehus have he following heorem.

4 8 A. Chouichi, S. Osmane and S. Tayachi NoDEA Figure. Asympoic behavior. Assume he condiion on α,α : α, α α, α,α <N/. If α α > wih α >, hen, we have LB for z case i. If α,α are on he line passing hrough he poins,.5 and, wih α >, hen, we have LNLB for Rez, LB for Imz case ii. If α,α are on he line passing hrough he poins, and, wih α >, hen, we have LB for Rez, NLB for Imz case iii. Finally, if α,α =,, we have NLB for z case iv. Where LB linear behavior, LNLB linear and nonlocal behavior, NLB nonlinear behavior Theorem.. Self-similar soluions Le N be an ineger, α be posiive real number saisfying α N >, >. Consider u = c x, v = c x α. Then, here exiss ε> such ha if max c, c <ε, here exiss a global self-similar soluion for he sysem w =e Δ c x + α e sδ w s ds,.3 w =e Δ c x α + e sδ w sw s ds..4 See Theorem 3.3 below for he general version of he previous resul. In his paper we also discuss all possible asympoic self-similar behaviors for he sysem.3 which are described in he heorem below and illusraed by he Fig.. Theorem.3. Differen asympoic behavior Le N be an ineger, α and α be wo real numbers saisfying

5 Vol. 5 The complex valued nonlinear hea equaion 9 α, α N N α, > and α α >. Le U =u, v be he global soluion of.3 wih iniial value ηu,v consruced by Theorem., u = c x α,v = c x α such ha max c, c <ε.hereη and ε are as in Theorem.. Then, for ε sufficienly small, here exiss a real number δ >, such ha for all δ saisfying <δ<δ, here exiss a posiive consan C δ such ha he following holds: i If α > and α >α +, we have u e Δ c x α C δ α δ, >,.5 v e Δ c x α C δ α δ, >..6 ii If α > and α = α +, we have u e Δ c x α +g C δ α δ, >,.7 v e Δ c x α C δ α δ, >,.8 where g = e sδ e sδ c x α ds. iii If α =and α >, we have u w C δ δ, >,.9 v w C δ α δ, >,. wih w and w are given by.3.4. Moreover, in all cases he resuling soluion u, v saisfies, for large d α u d α, d α v d α,. where d,d are posiive consans. Remark.4. Using he mehod in,8, we prove ha, if α = α =,we have u w C δ δ, >,. v w C δ δ, >,.3 wih w and w are he self-similar soluion of sysem.3 wih iniial daa c x,c x. I follows from he previous resul ha, he consruced soluion u, v is asympoically self-similar in he L L norm. See Theorem 4. below for general version of he previous Theorem. The case α = α and c = c is proved in,8. Also, for he case c c he resul of,8 can be exended wih minor modificaion. The case α α is new. Noe ha he real par of he complex-valued hea equaion and he real-valued hea equaion behave differenly. In paricular, he behaviors ii and iii can no hold for he realvalued quadraic hea equaion. This shows he influence of he imaginary par

6 A. Chouichi, S. Osmane and S. Tayachi NoDEA Figure. Comparison beween he L -norm of he real and he imaginary pars for large ime. In he shaded region: α > α and α α +, we have Rez Imz. In he Hached region: α < α and α, we have Imz Rez. On he line α = α, we have Imz Rez on he large ime behavior of he real par. For α >, we have ha if α >α he imaginary par v dominaes he real par u, as in he L -norm while if + α α <α he real par u dominaes he imaginary par v Fig.. For blowing up soluions, in 4 a simulaneously blowing up soluion for he sysem.3 is consruced such ha he real par u and he imaginary par v of z blow up simulaneously a x = and such ha he real par dominaes he imaginary par. Le us poin also ha he previous asympoic behavior is no described in 3. Moreover, he convergence resuls in 3 is proved under he assumpions.9,.. Remark.5. By he previous resul we have he following comparison beween he real par and he imaginary par of he soluion: i If α and α = α, for large ime, we have u α and v α, so, he real par Rez and he imaginary par Imz have he same behavior as and we wrie Rez Imz. ii If α > andα >α, here we noe ha he real par Rez dominaes he imaginary par Imz and we wrie Imz Rez. iii If α >, α α andα <α, here we noe ha he imaginary par Imz dominaes he real par Rez and we wrie Rez Imz.

7 Vol. 5 The complex valued nonlinear hea equaion In his work we sudy he global exisence and he differen asympoic behaviors of mild global soluions for he following general parabolic sysem { u =Δu + au + bv,.4 v =Δv + cuv, wih iniial value u,x=u x, v,x=v x,.5 where u, x andv, x are real valued funcions, >, x R N,a,b,c R. Noe ha sysem.4 is a generalizaion of he following equaion z =Δz + az,.6 for >, x R N and wih iniial daa z = u + iv,a R and b = a, c = a. By he well-known Duhamel formulaion, we see ha he problem can be furher reduced ino he following sysem of inegral equaions. u =e Δ u + a v =e Δ v + c e sδ u sds + b e sδ v s ds,.7 e sδ usvs ds,.8 where e Δ is he hea operaor which can be regarded as he convoluion wih he hea kernel. The mehod inroduced by, 6 8 uses he inegral formulaion of he semi-linear equaion or sysem under sudy. So, laer on we will work on he above sysem of inegral equaions insead of.4. The firs sep is o prove he exisence and uniqueness of global soluions for he inegral formulaion.7.8 in some funcional spaces, and allowing o use a suiable conracion mapping argumen on he whole ime inerval,. Second, we will consruc self-similar soluions for he corresponding asympoic sysem of.4: w =Δw + aμw + bνw,.9 w =Δw + cμw w,.3 where w, x andw, x are real valued funcions, >,x R N,a,b,c are he same parameers as in sysem.4. The parameers μ and ν are defined by μ = lim s s α, ν = lim s s α α..3 In reaing self-similar soluions, we shall use he sandard mehod inroduced in 6 see also,7,8. From he scaling principe, i is easy o see ha if u, x,v, x are a soluions of he Cauchy problem.4, hen, he rescaled funcions u λ, x =λ α uλ, λx, v λ, x =λ α vλ, λx, λ >,.3 are also a soluion of he sysem.4 wih iniial value u λ,x=λ α u λx, v λ,x=λ α v λx. Then, a soluion u, v is said o be self-similar when u, x,v, x = λ α uλ, λx,λ α vλ, λx, λ >..33

8 A. Chouichi, S. Osmane and S. Tayachi NoDEA Finally, for he asympoic behavior resuls, we prove ha some of he global soluions of he sysem.7.8 are asympoic, for large ime, o selfsimilar soluions of inegral sysem relaed o.9.3 wih iniial value c x α,c x α. The res of his paper is organized as follows. In he nex secion, we presen some preliminary lemmas which will be needed in he proofs of he heorems. Furhermore, we sae and we prove he main resuls concerning he global exisence and coninuous dependence. In Sec. 3, we prove he exisence of global soluions and self-similar soluions for he asympoic sysems, also we demonsrae Theorems. and.. Finally, in Sec. 4, we sudy he asympoic behaviors for soluions of hese sysems wih small iniial values wih respec o some norm and we prove Theorem.3. In his paper, we will denoe by C a posiive consan which can be differen a differen places and also denoe i by C δ o indicae ha i depends on a real number δ. We someimes denoe u,. byu.. Global exisence Throughou his secion, we give in he firs subsecion he following resuls which will be used laer in he second subsecion for he proof of he exisence of global soluion o Technical resuls In his subsecion, we shall sae some basic facs and obain auxiliary resuls. Firs of all, for r,l r R N denoes he usual Lebesgue space on R N wih he norm. r.lee Δ be he linear hea semigroup which is defined by e Δ ϕx =G ϕx, where G is he hea kernel given by G, x = N e x 4, >, x R N,. 4π and sands for he convoluion produc. We consider α,α wo real number such ha α, α α +,. and N N α >, α >. One can choose a real number r which saisfy < α +α r < N α <r and α < α α +α r < N <r..3 α For r saisfying.3, define he real numbers r,s and s by r = α r, s = α α α r, s = α α r,.4 where α,α be wo real number saisfying α, α α +..5

9 Vol. 5 The complex valued nonlinear hea equaion 3 Now, for Φ = ϕ,ϕ S R N S R N, define N by N Φ = β e Δ ϕ r, β e Δ ϕ r, γ e Δ ϕ s, γ e Δ ϕ s, >.6 where β,β,γ and γ are given by β = α N r, γ = α N s,.7 β = α N, γ = α N..8 r s Now, we will give some preliminary lemmas which will be needed in he proofs of he main heorems. Lemma. is a direc consequence of he condiions.3.4 and.7.8. We will need some real inerpolaing number defined in he Lemma.3 which saisfy he condiions saed in Corollary.5. As well as, in he Corollary.5, he Par represen compaibiliy condiions for he hea semigroup, ha is e Δ maps beween he appropriae Lebesgue spaces. The Par is inegrabiliy condiions: o assure ha he various inegrals are convergen. Finally, Par 3 will allow he conracion mapping argumen o be done on he ime inerval, direcly. Lemma.. Le N be an ineger. Le α,α be posiive real numbers which saisfy.. Leα,α be posiive real numbers which saisfy.5. Suppose ha N N >, α α >. Le r be a real number saisfying.3 and le r,s and s be given by.4. Le β,β,γ and γ be given by.7 and.8. Then, we have he following: i <β <β, <γ <γ, ii < αj +α j r j < N α j <r j, < α j +α s j < N j α j iii +αj α j β j <, iv β j +α j α γ j <, for j =,, j N α jr j +αj α j β j +=, γ j N α j sj <s j,forj =,, +α j α γ j +=,forj =,. j Proof. The lemma follows from he expressions of r,r,s and s by a direc consequence of he condiions.3.4 and from he expressions of β,β,γ and γ concluded from he condiions.7.8. Lemma.. Le α,α,α and α be posiive real numbers, hen, hey saisfy respecively. and.5 if and only if he following condiions hold: a <α α and <α α, b α α α+ α α+ c α α α+ α α α, α < α α, α α α+ α α α. α α α + α α + α < α α,

10 4 A. Chouichi, S. Osmane and S. Tayachi NoDEA Lemma.3. Le N be an ineger. Le α,α be posiive real numbers which saisfy.. Leα,α be posiive real numbers which saisfy.5. Suppose ha N N >, α α >. Le r j,s j,β j and γ j for j =, be as in Lemma.. Then, here exiss he inerpolaing real numbers r j,β j,τ j and Γ j,j=, which are defined as follow: r j = α j r j ; β j = +α j β j ; τ j = α j +α j α j +α j s j ; Then, we have he following Γ j = +α j α j γ j. r j r,r s,s, β j β,β γ,γ, j =,. Furhermore, here exiss r j r,r,s j s,s,β j β,β and γ j γ,γ such ha = +, τ j r j s j Γ j = β j + γ j..9 Also, here exiss θ j,θ j, forj =, such ha = θ j + θ j ; r j r r β j = θ j β + θ j β,. = θ j + θ j ; r j s s β j = θ jγ + θ jγ.. Moreover, here exiss θ j,θ j,, for j =, such ha = θ j + θ j ; r j r r β j = θ j β + θ j β ; = θ j + θ j ; s j s s γ j = θ jγ + θ jγ.. Proof. Owing o he condiions.3.4 and.7.8 and Lemma., he proof of he lemma is obvious and lef o he reader. Remark.4. As a choice, we can ake, for j =, r j = s j = α j +α j s j ; β j = γ j = +α j α j γ j. As a corollary of he previous lemmas we have he following resul. Corollary.5. Le N be an ineger. Le α,α be posiive real numbers which saisfy.. Leα,α be posiive real numbers which saisfy.5. Suppose ha N N >, α α >. Le r,r,s,s,β,β,γ and γ be as in Lemma.. Ler j,τ j,β j and Γ j,j=, be as in Lemma.3. Then,forj =,, we have:

11 Vol. 5 The complex valued nonlinear hea equaion 5 < rj <r j, <τ j <s j, N β j <, r j N r j <, Γ j <, τ j s j <, 3 + β j N r j r j β j =, +γ j N τ j s j Γ j =. Proof. Owing o he condiions saisfied by r,r,s, and s in he Lemmas. and.3, and due o he expressions of β,β,γ and γ, he proof of Corollary.5 is simple and can be omied... Global exisence In his subsecion, we give he Theorem and he proof of he global exisence and coninuous dependence of soluions for he sysem.7.8. Theorem.6. Global exisence and coninuous dependence Le N be an ineger. Le α,α,α and α be posiive real numbers saisfying α, α α +, α, α α +. Suppose ha N N > and α α >. Le r be a real number saisfying.3 and le r,s and s be defined in.4 β,γ and β,γ be defined by.7 and.8 respecively. Le M>be such ha C := CM <,.3 where CM is a posiive consan given by.38 below. Choose R> such ha R + CM M..4 If Φ=ϕ,ϕ be an elemen of S R N S R N such ha N Φ R,.5 where N is given by.6, hen, here exiss a unique global soluion U =u, v of he inegral sysem.7.8 such ha > β u r, β u r, γ v s, γ v s M..6 Furhermore, a u e Δ ϕ C,,L τ R N, for αr +α <τ< N α. b v e Δ ϕ C,,L τ R N, for α s +α <τ < N α c u e Δ ϕ L,,L N α R N,v e Δ ϕ L,,L N α R N. d lim u =ϕ and lim v =ϕ in he sense of empered disribuions. Moreover, he global soluion U =u, v saisfies α N r u r, α N r u r, α N s v s, α N s v s <, > for all r r,,r r,,s s, and s s,...7

12 6 A. Chouichi, S. Osmane and S. Tayachi NoDEA In addiion, if U =u,v and U =u,v be respecively he soluions of.7.8 saisfying.6 wih iniial values Φ=ϕ,ϕ and Ψ= ψ,ψ which verify.5, hen, for all r r,,r r,,s s, and s s,, we have > β u u r, β u u r, γ v v s, γ v v s C N Φ Ψ,.8 where C is given in Theorem.6 by.38 below. Proof. We look for global mild soluions of he sysem.7.8 wih iniial daa Φ = ϕ,ϕ via a fixed poin argumen. Le X be he se of Bochner measurable funcions U :, L r R N L r R N L s R N L s R N U :=u,v such ha U X = β u r, β u r, γ v s, γ v s <..9 > Le M be a posiive real number. We denoe by X M he se of u X such ha U X M. Endowed wih he meric du,u = U U X,X M is a complee meric space. Define he mapping F Φ : X M X M by F Φ u, v =F Φ u, v,g Φ u, v,. wih F Φ u, v =e Δ ϕ +a G Φ u, v =e Δ ϕ + c e sδ u s ds + b e sδ v s ds,. e sδ usvs ds,. where Φ = ϕ,ϕ belongs o S R N S R N saisfies.5. We will show ha F Φ given by. is a sric conracion mapping on X M for suiable real numbers M and R. Le Φ = ϕ,ϕ, Ψ=ψ,ψ S R N S R N saisfies.5 andu =u,v,u =u,v be wo elemens of X M. Then, we esimae γj G Φ U G Ψ U sj and βj F Φ U F Ψ U rj, for j =,. Thus, by using he smoohing effec e Δ ϕ q C N p q ϕ p,.3 for all >,ϕ L p R N, C is a posiive consan and p q and due o Corollary.5 par, we apply.3 forp, q =τ j,s j,j =, and ϕ =u sv s u sv s, we wrie γj G Φ U G Ψ U sj γj e Δ ϕ ψ sj +C c γj s N τ j s j u sv s u sv s τj ds,.4

13 Vol. 5 The complex valued nonlinear hea equaion 7 where τ j are given by.9 forj =,. Now, we remark ha u v u v τj u v v τj + v u u τj. Due o.9, we use he following Hölder inequaliy fg τj f rj g sj,.5 for all f L rj R N andg L sj R N, o obain u v v τj + v u u τj u rj v v sj + v sj u u rj..6 Injecing.6 in.4, we have γj G Φ U G Ψ U sj γj e Δ ϕ ψ sj +C c γj s N τ j s j u s rj v s v s sj + v s sj u s u s rj ds..7 By Lemma.3, we apply on he second erm of he righ-hand side of he las inequaliy he inerpolaion inequaliy f s f θ r f θ τ, f L r R N L τ R N,.8 where r τ, s = θ r + θ, θ,, τ once wih s, r, τ =r j,r,r,θ= θ j,f = u and s, r, τ =s j,s,s, θ = θ j,f = v v,forj =, and once wih he same riple bu wih f = u u for he firs and f = v for he second, wih he fac ha U and U belongs o X M, we obain γj G Φ U G Ψ U sj γj e Δ ϕ ψ sj +MC c γj s N τ j s j s Γ j ds U U X NΦ Ψ + MC c +γj N τ j s Γ j j s N τ j s j s Γ j ds U U X..9 From properies and 3 of Corollary.5, we deduce γj G Φ U G Ψ U sj NΦ Ψ + C j U U X,.3 where C j = c MC s N τ j s j s βj+γj ds,.3 which is a finie posiive consan from propery of Corollary.5.

14 8 A. Chouichi, S. Osmane and S. Tayachi NoDEA We now esimae βj F Φ U F Ψ U rj. In fac, we have for U,U X M βj F Φ U F Ψ U rj βj e Δ ϕ ψ rj + a βj u e sδ s us ds rj + b βj v e sδ s vs ds. rj We compue, using he smoohing properies of he hea semigroup.3 wih p, q = rj,r j on he second and he hird erm of he righ-hand side of he las inequaliy and par of Corollary.5, forj =,, we have βj F Φ U F Ψ U rj βj e Δ ϕ ψ rj + a C βj s N + b C βj s N r j r j I follows by he Hölder inequaliy, ha for j =, r j u s u s rj/ ds r j v s vs rj/ ds..3 βj F Φ U F Ψ U rj βj e Δ ϕ ψ rj + a C βj s N r j + b C βj s N r j u s rj + u s rj u s u s rj ds r j r j v s rj + v s rj v s v s rj ds..33 Using Lemma.3 and he inerpolaion inequaliy.8 wih s, r, τ =r j, r,r,θ = θ j for f {u,u,u u } on he second erm of he righhand side of he las inequaliy and wih s, r, τ =r j,s,s,θ = θ j for f {v,v,v v } on he hird erm, and he fac ha U and U belongs o X M, we see ha inequaliy.33 becomes β j F Φ U F Ψ U rj NΦ Ψ + CM a + b +β j N s N β r j r j j r j r j s β j ds U U X..34 We deduce, by using he expressions of r j and β j inroduced in Lemma.3, and heir properies given in and 3 of Corollary.5 ha βj F Φ U F Ψ U rj NΦ Ψ + C j U U X,.35

15 Vol. 5 The complex valued nonlinear hea equaion 9 where C j =MC a + b s N r j r j s βj ds,.36 i follows by propery of Corollary.5 ha C j,forj =, is a finie posiive consan. So, inequaliies.3 and.35 wih j =, lead o F Φ U F Ψ U X NΦ Ψ + C U U X,.37 where C := CM = max {C j,c j,j=, }..38 Now if we ake Ψ = and U =, he inequaliy.37 becomes F Φ U X NΦ + C U X..39 If we choose M and R such ha.3 and.4 are saisfied, hen, by.39, F Φ maps X M ino iself. For Φ = Ψ,.37 becomes F Φ U F Ψ U X C U U X, hence inequaliy.3 gives ha F Φ is a sric conracion mapping from X M ino iself. So F Φ has a unique fixed poin U =u, v inx M which is he soluion of he inegral sysem.7.8. This erminaes he proof of he exisence of a unique global soluion of.7.8 inx M. Now, we will prove he saemens a d. Le τ be a posiive real number saisfying we have u e Δ ϕ τ a C α +α r <τ< N α,.4 e sδ us τ ds+ b C e sδ vs τ ds. The smoohing properies of he hea semigroup.3, where p, q = r,τ on he firs and he second erm of he righ-hand side of he las inequaliy and wih j = of Corollary.5, give u e Δ ϕ τ a C + b C s N s N r τ r τ us r ds vs r ds. By using he inerpolaion inequaliy.8 wih s, r, τ =r,r,r,θ = θ,f = u on he firs erm of he righ-hand side of he las inequaliy and for he second erm we ake s, r, τ =r,s,s,θ = θ,f = v and he fac ha U and U belongs o X M, we obain he following esimaes u e Δ ϕ τ a CM s N + b CM s N r τ r τ s β ds s β ds.

16 A. Chouichi, S. Osmane and S. Tayachi NoDEA Thanks o he expressions of r and β given in Lemma.3 and due o.7, we deduce u e Δ ϕ τ CM a + b N τ α s N r τ s β ds..4 Using wih j = of Corollary.5 and he fac ha τ < r, he laer inequaliy gives u e Δ ϕ τ C N τ α,.4 where C is a finie posiive consan. On he oher hand, we se τ be a posiive real number saisfying α +α r <τ < N α..43 We aim o esimae v e Δ ϕ τ, wriing v e Δ ϕ τ c C e sδ usvs τ ds, hen, by using he smoohing effec, where p, q =τ,τ v e Δ ϕ τ c C s N τ τ usvs τ ds. Then, we apply Hölder inequaliy.5, for j =, we find v e Δ ϕ τ c C s N τ τ us r vs s ds. Thus, by he inerpolaion inequaliy.8, wih s, r, τ =r,r,r,θ = θ for f = u and for f = v, we ake s, r, τ =s,s,s,θ = θ, on he erm in he righ-hand side of he las inequaliy and he fac ha U and U belongs o X M, we obain v e Δ ϕ τ c CM s N τ τ s Γ ds..44 Thanks o he expressions of τ and Γ given in Lemma.3 and due o.7, we deduce v e Δ ϕ τ c CM N τ α s N τ τ s Γ ds..45 Using wih j = of Corollary.5 and he fac ha τ <s, he laer inequaliy gives v e Δ ϕ τ C N τ α,.46 where C is a finie posiive consan. Owing o he condiion.4 and.43, he righ-hand sides of.4 and.46 converge o zero as. This proves saemens a,b and d of Theorem.6. Also, i is clear ha.4 and.46 sill hold if τ = N α and τ = N α, his proves saemen c. Finally, we shall demonsrae.7. The proof of he L L esimaes is based on an ieraive argumen as in 8. Le us denoe he real numbers

17 Vol. 5 The complex valued nonlinear hea equaion r,r,s and s respecively by τ,τ,σ and σ. Choose real numbers τ,τ,σ and σ such ha, for j =,, we have +α j N +αj <τ <τ, <,.47 α j α j τ τ +α j N α j +α j <τ <τ, <,.48 α j α j τ τ and τ = α τ, σ = α α α τ, σ = α α τ..49 Remark ha a such choice is possible hanks o.3. We will concenrae firs on he equaion saisfied by v which is wrien as follows: v =e /Δ v/ + c / α j e sδ usvs ds. Therefore, he condiions.48, for j = and.49 allows us o use he smoohing effec of he hea semigroup.3, wih p, q =σ,σ onhe firs erm and p, q =τ,σ on he second erm of he righ-hand side of he inequaliy, and by applying Hölder inequaliy.5, for j =, we obain α N σ v σ C/ α N σ v/ σ + c C α N σ usvs τ ds, C/ α N σ v/ σ + c C α N σ vs s ds. / / s N s N τ σ τ σ us r Using he inerpolaion inequaliy.8, wih s, r, τ =r,τ,τ, θ = θ and s, r, τ =s,σ,σ, θ = θ respecively for f = u and f = v, and he fac ha U X M we obain α N σ v σ CD + c CD / s N τ σ s Γ ds,.5 where D = α N τ u τ, α N τ u τ, α N σ v σ, α N σ v σ > <..5 Wih he fac ha.48 is saisfied, for j =and.49, we deduce ha he inequaliy.5 leads o α N σ v σ CD <..5 >

18 A. Chouichi, S. Osmane and S. Tayachi NoDEA Now, we esimae α N σ v σ. Indeed, we have α N σ v σ C/ α N σ v/ σ + c C α N σ s N usvs τ ds C/ α N σ v/ σ + c C α N σ / / s N τ σ τ σ us r vs s ds, wherewehaveused.48, for j =,.49 and he smoohing effec of he hea semigroup.3, wih p, q =σ,σ on he firs erm and p, q =τ,σ on he second erm in he firs esimae and Hölder inequaliy.5, for j = in he las one. Now, applying he inerpolaion inequaliy.8, wih s, r, τ =r,τ, τ, θ = θ and s, r, τ =s,σ,σ, θ = θ respecively for f = u and f = v, and due o U X M, we obain α N σ v σ CD + c CD > / s N τ σ s Γ ds, where D is given in.5. By using.48, wih j =, he laer inequaliy leads o α N σ v σ CD <..53 Also, we wrie u =e /Δ u/ + a / e sδ u s ds + b / e sδ v s ds. Then, due o.47, for j = we use he smoohing properies of he hea semigroup.3 wih p, q {τ,τ, r,τ } o obain α N τ u τ C/ α N τ u/ τ + a C α N τ + b C α N τ / / s N s N r r τ us r ds τ vs r ds. Using he inerpolaion inequaliy.8 by aking s, r, τ =r,τ,τ, θ = θ for f = u and s, r, τ =r,σ,σ, θ = θ for f = v, we obain α N τ u τ CD + C a + b D / s N r where D is given in.5. Then, by.47, for j =,wehave > τ s β ds, α N τ u τ CD <..54

19 Vol. 5 The complex valued nonlinear hea equaion 3 Then, o esimae α N τ u τ, we use he smoohing properies of he hea semigroup.3 under he assumpion.47, for j = and.49, wih p, q, ϕ =τ,τ,u on he firs erm, p, q, ϕ = r,τ,u on he second erm and p, q, ϕ = r,τ,v on he hird erm, we obain α N τ u τ C/ α N τ u/ τ + a C α N τ / s N + b C α N τ s N / r τ r τ us r ds vs r ds. We consider s, r, τ =r,τ,τ, θ = θ and s, r, τ =r,σ,σ, θ = θ in he inerpolaion inequaliy.8 o apply i in he las inequaliy respecively for f = u and f = v and so, we obain α N τ u τ CD + C a + b D > / s N r τ s β ds, where D is given in.5. Then, by.47, for j =and.49, we have α N τ u τ CD <..55 The inequaliies.5,.53,.54 and.55 imply ha D = α N τ u τ, α N τ u τ, α N σ v σ, α N σ v σ > <. We ierae his procedure. For he nex sep, we consider τ,τ,σ and σ verifying, for j =, +α j N +αj <τ <τ, <,.56 α j α j τ τ +α j N α j +α j <τ <τ, <,.57 α j α j τ τ and τ = α τ, σ = α α α τ, σ = α α τ..58 Using similar esimaes as he previous sep wih he following inerpolaing numbers and r,j = θ j σ r,kj = θ kj τ + θ j σ ; α j + θ kj τ, k, j =,, s,j = θ j σ + θ j σ, j =,,

20 4 A. Chouichi, S. Osmane and S. Tayachi NoDEA wih θ kj and θ kj are as in Lemma.3 for k, j =, and τ,j = r,j + s,j. Remark ha we se r,j = r j and τ,j = r,j + s,j, where r,j = r j,s,j = s j,forj =, for he firs sep. We obain D = > α N τ u τ, α N τ u τ, α N σ u σ, α N σ u σ CM <. Therefore, we consruc he sequences {τ n } n which saisfies for j =, he following condiions +α j N +αj <τ n <τ n+, <, α j α j τ n τ n+ +α j N α j +α j <τ n <τ n+, α j α j α j <, τ n τ n+ for all ineger n. Then, we se he sequences {τ n} n, {σ n } n and {σ n} n such ha for all n =,,,... τ n = α τ n, σ n = α α α τ n, σ n = α α τ n,.59 One can check ha we can consruc a suiable sequence {τ n } n such ha i can reach, for some finie n, ha is τ n =, his prove.7, for r = s =. The inequaliy.7 follows by an argumen inerpolaion. The coninuous dependence inequaliy.8, for r, s =r,s ofhe soluion on he iniial daa resuls by considering F Φ U =U and F Φ U = U in he esimae.37. Formula.8, for r = s = can be proved saring wih.8, for r, s =r,s and using an ieraive procedure similar o he proof of.7, for r = s =. By Lemma of inerpolaion, we conclude.8. This achieves he proof of Theorem Self-similar soluions This secion is concerned wih he mild soluions of he sysem.9.3 which have a self-similar srucure wih power α, α. The inegral sysem relaed o.9.3 is given by w =e Δ ϕ + aμ w =e Δ ϕ + cμ e sδ w s ds + bν e sδ w s ds, 3. e sδ w sw s ds, 3. where a, b, c are he same parameers appearing in sysem.4 andμ, ν are given by.3. We firs esablish he global exisence only wih he norm N s. This is needed o prove he exisence of self-similar soluions in Theorem 3.3. In his secion, we will use α and α saisfying., r saisfying.3 and s be given by.4, β and γ be given by.7.

21 Vol. 5 The complex valued nonlinear hea equaion 5 For Φ = ϕ,ϕ S R N S R N, we define N s by N s Φ := β e Δ ϕ r, γ e Δ ϕ s. 3.3 > For <δ<δ, where δ is given by δ = N +α max, +α maxα,α r α α, 3.4 we define N δ by N δ Φ := β +δ e Δ ϕ r, γ+δ e Δ ϕ s. 3.5 > We firs give examples of homogeneous iniial values wih finie norm N defined by.6. We firs recall he following resul due o,8. Proposiion 3..,8 Le N be an ineger. Le α, α be wo real numbers such ha N α α, >. α Le r,r be wo real numbers such ha Le r = α α r, r > N α. β = α N r, β = α N r. Le x ϕx =ω x α, x where ω L r S N. Then, for any cu-off funcion η saisfying η near he origin and having a compac por, 3.6 we have he following: i > β e Δ ϕ r <, ii > β+δ e Δ ηϕ r <, for <δ< N α, iii > β e Δ ηϕ r < ; iv > β e Δ ηϕ r <. Also, we give examples of iniial values wih finie norm N s given by 3.3. These iniial values are used in Theorems 3.3 and 4. below. Proposiion 3.. 6, Proposiion.3 Le N be an ineger. Le α,α be posiive real numbers which saisfy.. Leα,α be posiive real numbers which saisfy.5. Suppose ha N N >, >. α α

22 6 A. Chouichi, S. Osmane and S. Tayachi NoDEA Le r be a real number saisfying.3 and r,s and s be given by.4. Le β,γ and β,γ be given by.7 and.8 respecively. Le ϕ and ψ be wo empered disribuions homogeneous of degree α, α respecively. If ϕx =ω x x α and ψx =ω x x α, 3.7 where ω L r S N and ω L s S N are homogeneous of degree, hen, N s ϕ, ψ <, 3.8 where N s is given by 3.3. Also, for any cu-off funcion η saisfying η near he origin and having a compac por, 3.9 we have he following: i > β+δ e Δ ηϕ r <, for <δ< N α ; > γ+δ e Δ ηψ s <, for <δ< N α ; ii > β e Δ ηϕ r < ; > γ e Δ ηψ s < ; iii N ηϕ, ηψ <, wheren is given by.6. We esablish he global exisence wih he norm N s which is given by 3.3 and wih he norm N δ is given by 3.5. This is needed o prove he exisence of self-similar soluions. Theorem 3.3. Le N be an ineger. Le α and α be posiive real numbers saisfying.. Suppose ha N N > and α α >. Le r be a real number saisfying.3 and s be given in.4. Leβ and γ be given by.7 and define β r and γ s by β r =α N r, r r,, 3. γ s =α N s, s s,. 3. Le M > be such ha C := CM <, where C is given by 3.3 and 3.34 below. Choose R > such ha R + CM M. If Φ = ϕ,ϕ S R N S R N such ha N s Φ R, 3. where N s is given by 3.3, hen, he inegral sysem has a unique global soluion W =w,w such ha > βr w r, γs w s M, r r,, s s,. 3.3 In addiion, if W =w,w and W = w, w respecively are he soluions of he sysem wih iniial values Φ=ϕ,ϕ and Ψ=ψ,ψ which saisfy 3., hen, we have > β w w r, γ w w s C N s Φ Ψ, 3.4

23 Vol. 5 The complex valued nonlinear hea equaion 7 where N s is given by 3.3. Also, for some <δ<δ, where δ is given by 3.4, we have β +δ w w r, γ+δ w w s > C δ N δ Φ Ψ, 3.5 where N δ is given by 3.5, C δ is given by 3.8 and 3.36 below. The posiive consan M is chosen small enough so ha C δ <. Furhermore, If Φ=ϕ,ϕ and ϕ,ϕ be as in he Proposiion 3., hen, he resuling soluion of wih iniial daa Φ is self-similar, denoe i by W =w,w. LeW =w,w be he soluion of wih iniial daa ηφ, η is a cu-off funcion saisfying 3.9. Then,we have w w r C δ βr δ, >, r r,, 3.6 w w s C δ γs δ, >, s s,. 3.7 The real number M is smaller. A his sage, we conclude ha he soluion W is asympoically self-similar in he L L -norm. Remark The saemens a d of Theorem.6 are also saisfied by global soluion W =w,w given by Theorem The soluion W given by Theorem 3.3 saisfy W, x =w, x, w, x = α w,x/, α w,x/. 3.8 By using he simple dilaaion argumen, he resuls is given as follow: α w,. w,. r C δ δ, >, r r,, 3.9 α w,. w,. C δ δ, >, s s,. 3. s Proof of Theorem 3.3 Le X be he se of Bochner measurable funcions: W :. L r R N L s R N W :=w,w such ha W X := β w r, γ w s <, 3. > where r is a real number saisfying.3, s is given by.4, β and γ are given by.7. Le M be a posiive real number. Define X M = {W X, W X M}. In his proof, we do no need o rea all he cases described by μ and ν. In fac, i if μ = ν =, hen he asympoic sysem is he linear hea sysem: w =Δw, w =Δw. This case can be reaed as in 6, ii if μ = ν =, hen he asympoic sysem is he nonlinear hea sysem.7.8. This case is sudied in 8.

24 8 A. Chouichi, S. Osmane and S. Tayachi NoDEA So, we are ineresed only by he cases μ =,ν=andμ =,ν=. Case : μ =and ν =. We have, by Lemma.3, r = r,β = β, τ = r + s and Γ = β + γ. Le us consider he mapping FΦ s defined by F Φ s W =F Φ sw,gs Φ W, where W =w,w X M, Φ=ϕ,ϕ S R N S R N and F s Φ W =eδ ϕ + a G s Φ W =eδ ϕ + c e sδ w s ds, e sδ w sw s ds. Through calculaions similar o he proof of Theorem.6, wehaveforw, W X M FΦW s FΦ s W XM N s Φ Ψ + C W W XM, 3. where N s is defined by 3.3, C is a finie posiive consan given by such ha C =M a C C =M c C C = maxc,c, 3.3 σ N r σ β dσ, 3.4 σ N r σ γ β dσ. 3.5 So, he mapping FΦ s has a unique fixed poin in X M and 3.4 follows similarly as in he proof of Theorem.6. Now,leΦ, Ψ, W =w,w and W = w, w. We esimae he difference W W wih he norm W X,δ := β +δ w r, γ+δ w s <, 3.6 > o obain > wih where β +δ w w r, γ+δ w w s Cδ N δ Φ Ψ, C δ =M a C C δ =M c C 3.7 C δ = maxc δ,c δ, 3.8 σ N r σ β δ dσ, σ N r σ γ β δ dσ. Since <δ<δ, where δ is given by 3.4 one can see ha β + δ <,β + γ + δ<andc δ is a finie posiive consan. This proves 3.5.

25 Vol. 5 The complex valued nonlinear hea equaion 9 Le Φ = ϕ,ϕ be as in Proposiion 3. andψ= ηφ. I follows from formula 3.5, ha β +δ w w r, γ+δ w w s CNδ Φ ηφ > = CN δ η Φ. 3.9 Using par i of Proposiion 3., we have ha N δ η Φ <, for<δ<δ. Thus, we ge 3.6 and 3.7, for r, s =r,s. The L L asympoic self-similar esimaes: Here we wrie w w =e Δ w / w / +c e sδ w sw s w sw s ds, hus, by he smoohing properies of he hea semigroup.3, for p, q = s, andϕ = w / w /, we obain α +δ w w C / α +δ N s w / w / s + c C α +δ Now, we compue ha, for arbirary T> / w sw s w sw s ds. α +δ w w C / γ+δ w / w / s + MC c α+ s α α δ ds / α+δ w w,,t α +δ w w, where M is as in Theorem.6. By3.7, for s = s and he fac ha α =, he las inequaliy becomes α +δ w w C δ + c CM,δ α+δ w,t,t Similarly, we wrie w w =e Δ w / w / + a w, α +δ w w. 3.3 e sδ w s w s ds.

26 3 A. Chouichi, S. Osmane and S. Tayachi NoDEA Thus, by he smoohing properies of he hea semigroup.3, for p, q = r, andϕ = w w, for arbirary T>and < T, we compue ha α+δ w w C / α+δ N r w / w / r +M C a α+ s α δ ds / α+δ w w,,t where M is as in Theorem.6. By3.6, for r = r and he fac ha α =, he las inequaliy becomes α+δ w w,t C δ + a CM,δ α+δ w w, α +δ,t w w. 3.3 Since CM,δ ifm, i is clear ha if M is sufficienly small hen 3.3 and 3.3 imply ha α+δ w w, α +δ w w C δ. 3.3,T Since he las consan C δ does no depend on he arbirary real number T>, he esimaes 3.6 and 3.7, for r = s = are valid for all >. The general case for 3.6 and 3.7 follows by inerpolaion. Case : μ =and ν =. We have, by Lemma.3, r = s and β = γ. Le us consider he mapping FΦ s defined by F Φ s W =F Φ sw,gs Φ W, where W =w,w X M, Φ= ϕ,ϕ S R N S R N and F s Φ W =eδ ϕ + b e sδ w s ds, G s Φ W =eδ ϕ. Through calculaions similar o he proof of Theorem.6, wehaveforw, W X M F s ΦW FΦ s W N s Φ Ψ + C W W XM, 3.33 XM where N s is defined by 3.3, C is a finie posiive consan given by C =M b C σ N s r σ γ dσ We conclude ha he mapping F s Φ has a unique fixed poin in X M and 3.4 follows as in Theorem.6.

27 Vol. 5 The complex valued nonlinear hea equaion 3 Similarly, We esimae he difference W W wih he norm defined by 3.6 andweseφ, Ψ, W=w,w and W = w, w. We find β +δ w w r, γ+δ w w s Cδ N δ Φ Ψ, > wih C δ =M b C σ N s 3.35 r σ γ δ dσ Since <δ<δ, where δ is given by 3.4, noe ha γ + δ<, C δ is a finie posiive consan and 3.5 follows. Le Φ = ϕ,ϕ be as in Proposiion 3. andψ= ηφ. I follows from he formula 3.5, ha > β +δ w w r, γ+δ w w s CNδ Φ ηφ = CN δ η Φ Using par i of Proposiion 3., we have ha N δ η Φ <, for<δ<δ. Thus, we obain 3.6 and 3.7, for r, s =r,s. The L L -asympoic self-similar esimaes: here we wrie w w =e Δ w / w /. We compue, by he smoohing properies of he hea semigroup.3 and 3.7 fors = s α +δ w w C / γ+δ w / w / s C δ Now, we esimae α+δ w w. We have w w =e Δ w / w / + b e sδ w s w s ds. Thus, by he smoohing properies of he hea semigroup.3, for p, q = r, on he second and he hird erm of he righ-hand side of he las inequaliy, for arbirary T> and < T, we compue ha α+δ w w C / α+δ N r w / w / r +M C b α α + s α δ ds / α +δ w w, 3.39,T where M is as in Theorem.6. By3.6, for r = r, and he fac ha α = α +, he las inequaliy becomes

28 3 A. Chouichi, S. Osmane and S. Tayachi NoDEA α+δ w w C δ + b CM,δ,T α+δ w w, α +δ,t w w. 3.4 Since CM,δ ifm, i is clear ha if M is sufficienly small hen 3.38 and 3.4 imply ha α+δ w w, α +δ w w C δ. 3.4,T Since he las consan C δ does no depend on he arbirary real number T>, he esimaes 3.6 and 3.7 forr = s = are valid for all >. The general case for 3.6 and 3.7 will be deduced from inerpolaion. ProofofTheorem. Le u,v be as in he assumpions of he Theorem.. Then, by Par i of Proposiion 3., we have N u,v <. Le ε> be such ha max c, c <εand u,v saisfies.5. Then, by Theorem.6, here exiss u, v global mild soluion of.4. Since u,v L R N CR N, hen, u, v is he soluion of.4 and is global. Proof of Theorem. Le u,v be as in he assumpions of he Theorem.. Then by 3.8, we have N u,v <. Leε> be such ha max c, c <εand u,v saisfies 3.. Then by Theorem 3.3, here exiss w,w global mild soluion of.. Since u,v L R N CR N, hen, w,w is he soluion of. and is global. Using he Theorem 3.3, we conclude ha he resuling soluion w,w is self-similar. 4. Asympoically self-similar soluions In his secion, we are mainly concerned wih he proof of he asympoic behavior resuls for some global soluions of he sysem.7.8 wih small iniial daa wih respec o he norm N. These allowed iniial daa ϕ,ϕ decrease in space as x α, x α, where α,α are any real numbers such ha hey saisfy. andα,α < N. We fulfil he proof of Theorem 3.3 hrough he following lemma. Lemma 4.. Le N be an ineger. Le α,α and α,α be posiive real numbers which saisfy respecively. and.5. Suppose ha N N >, α α >. Le r j,s j,β j and γ j,forj =, be as in Lemma.. Leμ, ν be given by.3 and δ>. Ler and β be as in Lemma.3 given also by.. We define he inerpolaing numbers r and β as follow: = μδ +, β = β + r +α r μδ +α.

29 Vol. 5 The complex valued nonlinear hea equaion 33 Moreover, by he formula. of r and β, we se he inerpolaing numbers s and γ which be wrien as follow: s = νδ +, γ νδ = β +. r +α +α Furhermore, from he numbers τ and Γ cied in Lemma., we inroduce he inerpolaing numbers τ and Γ as follow: τ = μδ +, τ +α Γ μδ =Γ + +α. Also, here exiss r r,r,s s,s,β β,β and γ γ,γ such ha = +, Γ = β + γ. 4. τ r s Then, here exiss a real number δ >, which can be compued, such ha for all <δ<δ he numbers defined recenly verify he following assumpions: i There exiss ζ kj,ζ kj, fork, j {,,, } such ha and r kj = ζ kj r + ζ kj r, s kj = ζ kj s + ζ kj s, β = ζ β + ζ β, γ = ζ γ + ζ γ, β = ζ β + ζ β, γ = ζ γ + ζ γ, ii < r <r, < s <r, <τ <s, iii β + μδ <, γ + νδ <, Γ + μδ <, iv N r N r <, s N r <, τ s <, v + β N r r β + μδ =, +β N s r γ + νδ =, +γ N τ s Γ + μδ =. Proof. Owing o he condiions saisfied by r,r,s,s,β,β,γ and γ in he Lemmas. and.3 and in he Corollary.5, he proof of Lemma 4. is simple and can be omied. Theorem 4.. Le N be an ineger. Le α,α,α and α be posiive real numbers which saisfy α, α α + α, α α +. Suppose ha N N > and α α >. Le r be a real number saisfying.3 and s defined in.4. Leβ and γ be given by.7. LeΦ and η be as in Proposiion 3.. Assume ha.3.4 are saisfied.

30 34 A. Chouichi, S. Osmane and S. Tayachi NoDEA Le W given by 3.8 be he self-similar soluion of.9.3 wih iniial daa Φ given by Theorem 3.3 and le U be he global soluion of.7.8 wih iniial daa ηφ consruced by Theorem.6 we muliply Φ by a small consan so ha.5 and hen 3. are saisfied wih boh Φ and ηφ. Then, here exiss a real number δ >, which can be compued, such ha for all δ saisfying <δ<δ here exiss a posiive consan C δ such ha for r r,,s s,, u w r C δ βr δ, >, 4. v w s C δ γs δ, >, 4.3 where β r and γ s are given respecively by 3. and 3.. Hence U =u, v is asympoically self-similar in he L L -norm. Furhermore, here exiss wo posiive consans d and d such ha, for large ime, we have d α u d α, d α v d α. 4.4 We require ha he real number M appearing in Theorem.6 mus be smaller so ha C δ < where C δ is given by 3.8 and Remark 4.3. Remark ha, by he dilaaion argumen, we can prove he following inequaliies: α u,. w,. r C δ δ, >, r r,, 4.5 α v,. w,. C δ δ, >, s s,. 4.6 s Proof. Over he proof, we use he noaions esablished in Lemma 4.. The sysem.7.8 coincides wih he sysem in he paricular case where μ = ν =. As well as he esimaes 4. and 4.3 follow from hose esablished in he Theorem 3.3. We prove he resuls of Theorem 4. only for wo cases μ =,ν=andμ =,ν=, since he cases μ = ν = and μ = ν = are reaed as well as in 6,8 respecively. Case : μ =and ν =. Sep : L r L s -asympoic resuls. LeΨ=ψ,ψ S R N S R N saisfying he condiion.5. Le U =u, v be he soluion of.7.8 wih iniial daa Ψ consruced by Theorem.6. LeW =w,w behe soluion of he sysem: w =e Δ ψ + a w =e Δ ψ + c which given by Theorem 3.3. We have u w r a + b e sδ w s ds, 4.7 e sδ w sw s ds, 4.8 e sδ u s w s r ds e sδ v s r ds, 4.9

31 Vol. 5 The complex valued nonlinear hea equaion 35 v w s c e sδ usvs w sw s s ds. 4. Le δ be such ha <δ<δ, where δ is a small posiive real number. We firs deal wih he inequaliy 4.. By Lemma 4. and he smoohing properies of he hea semigroup.3 wih p, q =τ,s andϕ = usvs w sw s, we have v w s C c s N τ s usvs w sw s τ ds, 4. where τ is given in Lemma.3. In fac, since μ =wehaveα =andwe wrie τ = +α r = r + s.so,byhölder inequaliy, we ge uv w w τ v s u w r + w r v w s. 4. Le T be an arbirary posiive number. Injecing 4. in4., we obain, for < T where γ+δ v w s,t C δ γ+δ N r β γ δ+ β+δ u w r, γ+δ v w s, C δ =CM c s N r s β γ δ ds, he posiive consans C δ is finie. Noe ha, in his case we have α =,so Then, we obain γ+δ v w s C δ γ + δ N r β γ δ +=.,T β+δ u w r, γ+δ v w s. 4.3 Now we esimae he inequaliy 4.9. By Lemma 4. and he smoohing properies of he hea semigroup.3 wih p, q, ϕ = r,r,u w on he firs erm and wih p, q, ϕ = s,r,v on he second erm of he righ-hand side of he inequaliy 4.9, we have u w r C a +C b s N s N r r u s w s r/ ds s r v s s / ds. 4.4 Le T > be an arbirary real number. Using Lemma 4., Eq..8 wih s, r, τ, θ =s,s,s,ζ on he second erm of he righ-hand side of he las inequaliy, we ge, for < T β+δ u w r C δ β +δ u w r + Cδ,,T

32 36 A. Chouichi, S. Osmane and S. Tayachi NoDEA where C δ =MC a C δ = M C b s N r s β δ ds, s N s r s γ ds. By Lemma 4., we find ha he posiive consans C δ,c δ are finie. Thus, we obain β+δ u w r C δ C δ, 4.5 for a given consan C δ, since i does no depend on T he pervious inequaliies hold on he whole inerval,. I follows ha, for all δ saisfying, <δ<δ, here exiss a posiive consan C δ, such ha u w r C δ β δ, >, 4.6 v w s C δ γ δ, >. 4.7 Sep : L L -asympoic resul. Le Φ = ϕ,ϕ be as in Proposiion 3., hen using he resuls , for r, s =r,s of Theorem 3.3, one obains ha, for all <δ<δ, here exiss a posiive consan C δ >, such ha w w r C δ β δ, >, 4.8 w w s C δ γ δ, >, 4.9 where δ is given by 3.4. Hence, W =w,w is soluion of wih iniial daa ηφ, where η is a cu-off funcion saisfying 3.9. Then by Proposiion 3. iii we have ha N ηφ is finie. Thus, he inequaliies 4. and 4.3 follow by wriing U W = U W + W W and using Le δ> be sufficienly small and recall he following numbers given in i of Lemma 4., ζ = θ ; ζ = θ + δ α α. 4. In paricular, δ is chosen small enough so ha ζ and ζ belong o, and <δ<δ. Wrie now v w =e Δ v/ w / +c e sδ usvs w sw s ds. Then, by.3, 4. and 4.7, we have v w e Δ v/ w / + c e sδ usvs w sw s ds.

33 Vol. 5 The complex valued nonlinear hea equaion 37 Tha is α +δ v w C / γ+δ v/ w / s + c C α +δ usvs w sw s ds C δ + C c α +δ / + w s vs w s ds. Using.7, we ge for arbirary T>, we obain α +δ v w C δ + CM α +δ,t vs us w s / s α α δ ds α+δ u w, α +δ v w C δ + CM α +δ α α δ+ / s α α δ ds α+δ u w, α +δ v w.,t Thus, we obain, from α =, ha α +δ v w C δ +CM,δ α+δ u w, α +δ v w. 4.,T Le us wrie now u w =e Δ u/ w / + a +b e sδ v s ds. e sδ u s w s ds We have α+δ u w α+δ e Δ u/ w / + a α+δ e sδ u s ws ds + b α+δ e sδ v s ds. Therefore by.3 and 4.6, we obain α+δ u w C / β+δ u/ w / r + C a α+δ / u s w s ds

34 38 A. Chouichi, S. Osmane and S. Tayachi NoDEA +C b α+δ / v s ds C δ + C a α+δ us + w s / us w s ds +C b α+δ vs ds. Due o.7, and an inerpolaion argumen, we ge for arbirary T>, α+δ u w C δ +MC a α+δ s α ζ + ζ α δ ds Hence,,T / / αζ+ ζα+δ u w +CM b α+δ α s ζ + ζ α / ds. α+δ u w C δ +MC a +α αζ ζα s α ζ + ζ α δ ds / αζ+ ζα+δ u w,t +M C b +α α ζ + ζ α +δ ds. α s ζ + ζ α / From he expressions of ζ,ζ given by 4., we obain α+δ u w C δ + CM,δ+CM αζ+ ζα+δ u w. 4.,T We subsiue ζ = θ = and obain αζ+ ζα+δ u w = α+δ u w.,t,t 4.3 Case : μ =and ν =. Sep : L r L s -asympoic resuls. LeΨ=ψ,ψ S R N S R N saisfying he condiion.5. Le U =u, v be he soluion of.7.8 wih iniial daa Ψ consruced by Theorem.6. LeW =w,w behe

35 Vol. 5 The complex valued nonlinear hea equaion 39 soluion of he sysem: w =e Δ ψ + b e sδ w s ds, 4.4 w =e Δ ψ, 4.5 which is given by Theorem 3.3. Wehave u w r b + a v w s c e sδ v s w s r ds e sδ u s r ds, 4.6 e sδ usvs s ds. 4.7 Le δ be such ha <δ<δ, where δ is a small posiive real number. We firs deal wih he inequaliy 4.7. By Lemma 4. and he smoohing properies of he hea semigroup.3 wih p, q =τ,s andϕ = usvs, we have v w s C c s N τ s usvs τ ds. 4.8 where τ is given by 4.. Using 4., Hölder inequaliy, we ge uv τ v s u r. 4.9 Le T be an arbirary posiive number. Injecing 4.9 in4.8 and applying.8 wih s, r, τ, θ = r,r,r,ζ for f = u and s, r, τ, θ = s,s,s,ζ forf = v and using Lemma 4., we obain, for < T where C δ = CM c γ+δ v w s C δ, 4.3 s N τ s s Γ ds. By pars iii, iv of Lemma 4. and by he fac <δ<δ, he posiive consans C δ is finie. Now we esimae he inequaliy 4.6. By Lemma 4. and he smoohing properies of he hea semigroup.3 wih p, q, ϕ = s,r,v wand p, q, ϕ = r,r,u respecively on he firs and he second erm of he righ-hand side of he inequaliy 4.6, we have u w r C a s N r r u s r / ds +C b s N s r v s ws s/ ds. 4.3 Le T > be an arbirary real number. Using Lemma 4., Eq..8 wih s, r, τ, θ =r,r,r,ζ on he firs erm of he righ-hand side of he las

36 4 A. Chouichi, S. Osmane and S. Tayachi NoDEA inequaliy, we ge, for < T β+δ u w r C δ + C δ β+δ v w s, where C δ = M C a s N r r s β ds, C δ =MC b s N s s γ δ ds. By Lemma 4., we find ha he posiive consans C δ,c δ are finie. Thus, we obain for a given consan C δ, since i does no depend on T he pervious inequaliies hold on he whole inerval,. I follows ha, for all δ saisfying, <δ<δ, here exiss a posiive consan C δ, such ha 4.6 and 4.7 hold. Sep : L L -asympoic resul. Le Φ = ϕ,ϕ be as in Proposiion 3., hen using he inequaliies , for r, s =r,s of Theorem 3.3, one obains ha, for all <δ<δ, here exiss a posiive consan C δ >, such ha 4.8 and 4.9 hold. Hence, W =w,w is soluion of wih iniial daa ηφ, where η is a cu-off funcion saisfying 3.9. Then, by Proposiion 3. iii, we have ha N ηφ is finie. Thus, he inequaliies 4. and 4.3 follow by wriing U W = U W + W W and using Le δ> be sufficienly small and recall he following numbers given in i of Lemma 4., ζ = θ + δ α α ; ζ = θ, 4.3 ζ α α +ζ α α =θ α α +θ α α +δ In paricular, δ is chosen small enough so ha ζ,ζ,ζ and ζ belong o, and <δ<δ. Wrie now v w =e Δ v/ w / + c e sδ usvs ds. We have α +δ v w α +δ e Δ v/ w / + c α +δ e sδ usvs ds. Tha is, hen by.3 we obain α +δ v w C / γ+δ v/ w / s + c C α +δ us vs ds. Using.7, 4.7 and an inerpolaion argumen, we ge for arbirary T>, we obain

37 Vol. 5 The complex valued nonlinear hea equaion 4 α +δ v w C δ + C c M α +δ α s ζ + ζ α +α ζ + ζ α / C δ + C c M +α αζ+ ζα+α ζ + ζ α δ ds. α s ζ + ζ α +α ζ + ζ α / ds Now hanks o he relaion beween ζ and ζ given by 4.33, we obain Le us wrie now We have α +δ v w C δ + C c M u w =e Δ u/ w / + a +b e sδ v s w s ds. e sδ u s ds α+δ u w α+δ e Δ u/ w / + a + b Therefore by.3 and 4.6, we obain e sδ u s ds e sδ v s w s ds. α+δ u w C / β+δ u/ w / r +C a α+δ u s ds / +C b α+δ / v s w s ds C δ + C a α+δ us ds / +C b α+δ vs + w s / vs w s ds. Due o.7 and an inerpolaion argumen, we ge for arbirary T>, α+δ u w C δ + C α+δ M a s / α ζ + ζ α ds