Master Thesis. Nguyen Thi Hong Nhung. Asymptotic stability of soliton to the gkdv equation.

Transcription

1 Master Thesis Nguyen Thi Hong Nhung Asymptoti stability of soliton to the gkv equation Avisors: Lu Molinet Defense: Friay June 1 th, 13

2 Contents 1 Introution iii Preliminary results 1 1 Linearize operator aroun Q 1 Deomposition of the solution lose to Q 3 3 Rigiity results 7 31 Linear Liouville property 8 3 Nonlinear Liouville property aroun Q 1 Asymptoti stability 18

3 CONTENTS ii Aknowlegment First an foremost, I woul like to epress my eep love to family an friens of mine, who always enourage me with love an spiritually supporte me throughout my life I woul like to epress my eep an sinere gratitue to my avisor Prof Lu Molinet for the ontinuous support, for his patiene, motivation, enthusiasm, an immense knowlege in all three aaemi months I woul like to aknowlege my issertation ommittee members : Prof Mar Peign, Prof Guy Barles, Prof Emmanuel Chasseigne an Matre e Confrenes Olivier Durieu I woul like to aknowlege Professor Duong Minh Du, an the professor in this ourse, who have inotrinate the neessary knowlege an useful for me The finally, I woul like to aknowlege the finanial, aaemi support of Laboratoire e Mathematiques et Physique Theorique, Universite Franois Rabelais, Tours, FRANCE Signature Nguyen Thi Hong Nhung

4 1 Introution iii 1 Introution We onsier the generalize Korteweg-e Vries gkv equation u t + u + u k =, t, [, T ] R, 1 where interger k: 1 < k < 5, for u = u H 1 R Uner onition >, there eist solution in the energy spae H 1 of equation 1 of the form ut, = Q t These solution is this form are alle solitons In my thesis, we use argument of Kenig, Pone an Vega 11,1, that is The loal Cauhy problem is well-poseness in H 1 Moreover, the following onservation laws hols for H 1 solution, for all t R 1 u t 1 k + 1 u t = u, u k+1 = 1 u,t 1 k + 1 u k+1 Reall that if Q is solution of Q + Q k = Q on R, Q H 1 R, then R, t, = Q t is solution of 1 We all soliton suh nontrivial travelling wave solution of 1 We know that, if Q > is solution of then it is unique positive solution an is given by where Q = 1 k 1 Q, Q = 1 k 1 k + 1 osh k 1 Moreover, there eist C >, δ > suh that Q, Q, Q Ce δ, R Note that Q L = 5 k k 1 Q L In this thesis, we prove that the family of soliton solution aroun Q is asymtotially stable, ie if ut is lose to Q then ut onverges weakly to Q + as t + This result is base on a rigiity property of equation 1 aroun Q relies on the introution of a ual problem The main result of this report is the following in the energy spae whih is prove

5 1 Introution iv Theorem 11 Asymptoti stability Let > be fie There eist α > suh that for any u H 1 satisfies u Q H 1 < α, 3 then the following hol, there eist a real funtion ρt ontinuous an + > suh that ut, + ρt Q + in H 1, as t + Theorem 1 Nonlinear Liouville Property aroun Q Let > be fie There eists α > suh that if ut C R; H 1 is a global solution of 1 satisfying, for some real funtion ρt an some positive real C, σ, t R, u t, + ρt Q H 1 α, 5 t R, u t, + ρt Ce σ 6 then there eists 1 >, an 1 R suh that t, R, ut, = Q t Reall that the first result onerning asymptoti stability for solitons of 1 was prove by Pego an Weinstein 3, for the power ase in some weighte spaes with eponential eay at infinity in spae uner spetral assumptions, heke only for the nonlinearities u an u 3 Organize of this report as the following In setion we present some preliminary results onerning solution of In setion 3, we prove Theorem 1- Nonlinear Liouville theorem, an in setion, we prove Theorem 11- Asymptoti stability In our analyti of this report, we use a property of monotoniity of L mass at the right in spae for solution of the KV equation Appeni

6 Preliminary results 1 Preliminary results 1 Linearize operator aroun Q The first, we efine the linearize operator aroun Q L u = u + u kq k 1 u 7 Then, the linearize equation aroun Q is u t = L u Now, we present the properties of L as Claim 1 Properties of L Let integer k: 1 < k < 5 The operation L efine as 7 satisfies the following properties 1 First eigenfuntion L Q k+1 = k 1k+3 Q k+1 Seon eigenfuntion L Q = 3 L k 1 Q + Q = Q if for all u H 1 satisfies uq = ul Q k+1 λ > = then L u, u λ u, u, where 5 The essential spetrum is [, + [ Proof 1 L Q k+1 = Q k+1 + Q k+1 = k + 1 = k 1 = k 1 = Q k 1 Q kq k 1 + Q k+1 Q k+1 kq k 1 Q k+1 Q k + 1 Q k 1 Q k+1 k 1k + 3 Q k+1 Q Q k + 1 Qk+1 Q k+1 + Q k+1 kq k 1 k Q k+1 L Q = Q + Q kq k 1 + Q = Q Q Q k 1 kq k 1 kq k 1 Q = Q Q k Q kq k 1 Q = Q k+1 Q k+1 Q Q k

7 1 Linearize operator aroun Q 3 L k 1 Q + Q = = k 1 L Q + L Q [ ] Q + Q kq k Q + L Q k 1 = k 1 Q Q + = k k 1 Q + = k k 1 Q + k 1 Q k 1 Q k k 1 Qk k 1 Q = Q k k 1 Qk Let us efine funtion µ :µ = Q Q Then, µ satisfies the properties that is present in Claim below Claim For all real, µ satisfies µ µ µ = k 1 k + 1 Qk 1, µ µ Q = k 1, = k 1 3 k + 1 Qk 1, Q k 1 k + 1 µ = µ k 1 Q Proof From efine of µ, we an ompute the first erivaties of µ as the following an the seon erivaties Thus, µ µ µ = Q Q + Q Q = 1 Q Q Q Q k = 1 Q k 1 k + 1 Qk+1 = k 1 k + 1 Qk 1, + Q k + 1 Qk+1 µ = k 1 k + 1 Qk 1 k 1 = k + 1 Qk Q = k 1 Q Q Net, we take the thir erivaties of µ µ = = k 1 Q k Q k+ [ k 1 k + 1 k Q k 3 Q + Q k Q k 13 = k + 1 Qk 1 3 k + 1 Qk 1 ]

8 Deomposition of the solution lose to Q 3 Then, from the the first erivaties an thir erivaties of µ, we obtain µ µ = k 1 3 k + 1 Qk 1 The finally, we have Q k 1 µ = Q k 1 Q Q = Q Q k k + 1 = µ k 1 Deomposition of the solution lose to Q Lemma 3 Moulation of the solution lose to Q There eist K > an α > suh that for any < α < α, T > if ut, solution of 1 satisfies for all t [, T ] then there eist real funtions, ρ C 1 [, T ] suh that inf r R ut, + r Q H 1 α, 8 ηt, = ut, + ρt Q t 9 satisfies t [, T ] ηt, Q p+1 t =, 1 Q t ηt, =, 11 t + η H 1 < K α, 1 an t t + ρ t t t K η t, e 1/ 13 Moreover for any t [, T ], t an ρt are the unique, ρ suh that 8 an 9 are satisfies Proof Fi, >, α > Let α = min, α We enote by V α the ball in H 1 R of enter Q an raius α, an onsier the C 1 mapping y : α, + α α, α V α R R, ρ, u y 1, y

9 Deomposition of the solution lose to Q where y 1 = ηt, Q p+1 t, an y = ηt, Q t We see that M =,, Q : y,, Q =, Now, let us ompute the partial erivatives of y at y 1,, Q = y 1 ρ,, Q = Q Q p+1 =, y,, Q = S Q =, y Q ρ,, Q = > S Q p+1 >, where S = 1 p 1 Q 1 Q Therefore, the absolute of the Jaobian of y at M =,, Q is larger than a positive onstant epening only the Thus, using the impliit funtion theorem,there eists α 1 : < α 1 α, an of a C 1 mapping :, ρ : V α 1 α, + α α, α u u, ρu suh that y 1 u, ρu, u = y u, ρu, u =, u V α 1 Moreover, for some onstant K >, if u V α 1 then u + ρu < K α So, if we enote ηu = u t, + ρu Q u then we have ηu, Q p+1 u = ηu, Q u =, u V u α 1 Furthermore, u V α 1, ηu H 1 < K α, where K α, as α by ηu = u + ρu u + u Q + Q Q u Finally, we efine moulation of u, applying these above result with α < min{, α }, suh that K α < α Then, from the orbital stability result, we have >, α 1 >, u t, + rt Q < α 1 for all t It implies u t, + rt V α 1 Net, we efine for all t R t = ut, + rt, ρt = ρ ut, + rt + rt an ηt = u t, + ρt Q t

10 Deomposition of the solution lose to Q 5 Sine u α, + α, an ρu α, +α, so we obtain t < α an ρt rt < α, an we also have ηt, Q p+1 t =, Q t ηt, = The finally, we will prove t + ρ t < K η L We enote Rt, = Q t, implies Rt, = 1 k 1 Q t So t R t = k 1 1 k 1 tq t + t 1 k 1 tq t = t k 1 R + R From efinition of η : ηt, = ut, Q t ρt = ut, Rt,, with ut, is solution of 1, we have t η t + η = u t + u R t R [ = u k t k 1 R + ρtr Or equivalent, ρt R ] R kr k 1 R η t + η + kr k 1 η = t k 1 R ρt R + ρ t ] R [η + R k kr k 1 η R k On the other han, we integrate respet time of equation ηt, Q t ρt =, we obtain = t ηr = η t R + η t R Sine t R = t 1 k 1 Q ρt + t k 1 1 k 1 Q ρt + t = t R + ρt 1 k 1 Q ρt ρ t 1 k 1 Q ρt t k 1 R + t ρtr ρ t R, an ηr =, where R = Q ρt, thus, η t R + t ρtηr ρ t ηr =

11 Deomposition of the solution lose to Q 6 On the other han, sine [ ] 1 p 1 R + 1 ρtr R =, an η + kr k 1 η R = η + kr k 1 η η R ηr = η L R kk 1 ηr k R η R = kk 1 ηr k R + ηr Thus, we obtain = ρ t ] R [η + R k kr k 1 η R k R η + kr k 1 η R + t ρtηr ρ t ηr = ρ t R ρ t ηr + t ρtηr ] kk 1 ηr k R [η + R k kr k 1 η R k R We able write again the following ρ t ηr ρ t R = t We have ρt ηr kk 1 [ ] η + R k kr k 1 η R k Kη, ηr k R η + R kr k 1 η R k R therefore η + R k kr k 1 η R k R K Net, we have the following estimates η R K η e ρt ηr k R K K η e ρt K 1 η e ρt K 1 η e ρt η e ρt, ρ t R K ρ t, ρ t ηr K ρ t ηe ρt K ρ t t ρt ηr K t ηe ρt 1 t η e ρt 1 η e ρt,

12 3 Rigiity results 7 Thus, we obtain K ρ t ρ t 1 η e ρt t + K 1 η e ρt + K η e ρt It implies, Or equivalent ρ t K t ρ t t 1 η e ρt + K 1 η e ρt + K η e ρt η e ρt Similary, we also have = ηr k+1 = η t R k+1 + k + 1 t ηr k 1 Rt = t k + 1 R k+3 + t k + 1 ρtηr R k 1 ρ t k + 1 k 1k + 3 [ ] ] η kr k 1 η η R k+1 [R + η k kr k 1 η R k R k+1 = k + 1 t R k+3 ρ t k + 1 η R k 1 R k 1 [ ] ] η kr k 1 η η [R + η k kr k 1 η R k R k+1 We also write k + 1 t k 1 R k+1 R k+3 = ρ t k + 1 We have estimate the followings ρ t η R k 1 R K ρ t [ ] η R k 1 R η kr k 1 η η ] [R + η k kr k 1 η R k R k+1 ηe ρt K ρ t R k+1 1 η e ρt, η R k 1 R we obtain t ρ t Thus, t + ρ t K η e ρt 1 η e ρt + K 1 η e ρt 3 Rigiity results This setion present the proof of Theorem 1 onsier setion 3 First, in setion 31, we give a linear version of the result to present the main iea in the simplest ase

13 31 Linear Liouville property 8 31 Linear Liouville property In this setion, uner assumption of Theorem 1, we prove a rigiity result for H 1 solutions of the following linearize equation η t = L η on R R, where L η = η + η kq k 1 η 1 Reall that, the Cauhy problem for 1 is global well- pose in H 1 R, by H 1 solution, we mean a solution onstrute in this way Any suh solution an be approahe by regular solutions whih allows to justify formal omputations Proposition 31 Linear Liouville property Let > Let η C R, H 1 R be solution of 1 Assume that there eist K >, δ > suh that t, R R, ηt, Ce δ 15 Then, there eist b R suh that t R, ηt b Q Remark 3 If ηt b Q then η is solution of 1, 15 Inee, sine Q is verifies L Q =, hene η t = an L η = b L Q =, so ηt satisfies 1 On the other han, by eponential eay of Q, we obtain 15 Let t be an H 1 solution of 1 satisfying 15 Now, we introue a ual problem relate to Lemma 33 Introution of the ual problem Let vt, = L ηt, + αtq where αt = Then, v C R, H 1 R an vt satisfies 1 Equation of v ηl Q p+1 p+1 Q Q t v = L v + α tq, t, R R 16 Eponential eay There eist K > suh that t, R R, vt, Ke 8 17

14 31 Linear Liouville property 9 3 Orthogonality relations t R, vt, Q p+1 = vt, Q = 18 Virial ientity the ual problem 1 v t, µ = t Proof 1 We have vl vµ, µ = Q Q 19 η t t, = L η = vt, αtq = v αtq, an vt, = L ηt, + αtq It implies v t t, = L η t t, + α tq = L v αtq + α tq, by η t t, = v αtq = L v + α tq, by L Q = Thus, v t = L v + α tq Eponential eay From monotoniity argument on ηt an on vt, we laim that there eists K >, suh that t R, v + v t ep K On the other han, we have vt, ep 8 vt, ep C 8 vt, ep L 8 It implies, vt, ep 8 K, or equivalent H 1 So, is prove vt, K ep 8

15 31 Linear Liouville property 1 3 Orthogonality relations For all t R vt, Q p+1 = = = L η + αtq Q p+1 η L Q p+1 η L Q p+1 + αt Q Q p+1 η L Q p+1 Q Q p+1 Q Q p+1 = An vt, Q = L η + αtq Q = Virial ientity on the ual problem 1 t v t, µ = = vv t µ = v L vµ + α t v L v µ + α t Q µ v Q v = v L vµ Proof of Proposition 31 We have t v µ = L v vµ Taking part integration, we have L v vµ = v + v kq k 1 v vµ = v vµ + v vµ k = v v µ + v vµ = 3 v µ = 3 v µ + v vµ Set wt, = vt, µ = vt, Q p 1 We have w = v µ + 1 v µ µ, Q k 1 v µ + k v µ + k v [ µ µ + k Q k 1 µ ] v vµ Q k 1 µ v Q k 1 µ v

16 31 Linear Liouville property 11 whih also rea Thus, where t w = v µ + 1 µ v 1 µ v µ = v t, µ = 3 A = 3 = 3 = 3 v µ, w 1 µ v bigg + 1 µ µ µ v µ + v µ [ ] v µ µ + k Q k 1 µ µ w 3 µ w µ µ w µ + 3 w [ µ µ + µ k ] Q k 1 µ w µ w A, + 1 µ Q k k µ µ µ Using Claim above, we obtain A = 3 Q k µ kk + 1 µ + 1 Q µ k 1 µ = 3 Q k 1 + 3k + 1 µ Q k µ = 3 Q k 1 + 3k Q k + 1 Qk k + 6k 1 = + 1 3kQ k 1 Thus, 1 3 t On the other han, v t, µ = w t, + 1 3k + 6k 1 3 w = L w, w w + k + 1 w k w Q k 1, w Q k 1 therefore, 1 3 t 3k + 6k 1 v t, µ = L w, w w = L w, w + 1 3k + 6k k + 6k w, w w

17 3 Nonlinear Liouville property aroun Q 1 sine L w, w λ w By µ is boune, the funtion v t, µ is uniformly boune time, an + w t, t < + It follow that for some t n + we have + w t n, t, as n + Thus, from vt, C ep we obtain v t n,, as n + Similarly, we have for a sequene s n + v s n,, as n + we integrate over s n, t n for estimates of an the boune of µ we obtain, n N tn w t C v t n, + v s n, s n By passing to the limit as n +, we obtain + w t =, giving wt, an thus vt, Therefore,L η = αtq,an by Claim 1, implies ηt, = αt k + 1 Q + αt Q + btq = αt k + 1 Q + Q + btq But from the equation of ηt,, we have η t = αtq Hene, α t =, b t = αt Sine bt an αt are boune, we eue αt an bt b Thus, η b Q 3 Nonlinear Liouville property aroun Q In this setion, we present proof of Theorem 1 the same as proof of Proposition 31 Let ut as in Theorem 1 The first, we eompose ut, as in Lemma 3, using moulation theory We obtain, for all t, ηt, = u t, + ρt Q t, 1

18 3 Nonlinear Liouville property aroun Q 13 where t, ρt are C 1 funtion hosen so that ηt, Q p+1 = ηt, Q t = As in the ase linear, we also introue ual prolem Lemma 3 Dual problem for the nonlinear equation Let Then, v C R; H R an vt satisfies 1 Equation of v vt, = η + η Q + η k Q k = L η Q + η k Q k kq k 1 η v t = v + v kv Q + η k 1 + ρ t v + t Q + η = L v kv Q + η k 1 Q k 1 + ρ t v + t Q + η 3 Eponential eay There eists K > suh that, t, R R, ηt, + vt, K ep 8 3 Estimates an almost orthogonality relations There eist K > suh that, t R, t + ρ t K η L, vq + vq p+1 K η L, η L K v L 5 Virial type estimates There eists λ 3, B > suh that, t R, 1 v µ λ 3 v µ 1 v H η t λ 1 L, v v λ 3 t + v 1 v 7 λ 3 B Proof We start proof of Lemma 3 1 From equation of η : ηt, = ut, + ρt Q t, implies u = η + Q = η + Q Q k,

19 3 Nonlinear Liouville property aroun Q 1 an sine v = n + η Q + η k Q k,we have η t = u t + ρ t u t Q = u + u k + ρ tu t S by 1 = η + Q Q k + η + Q k + ρ t Q + η t Q = η + η Q + η k Q k + ρ t Q C + η t S = v + ρ t Q + η t S, 8 From equation of v, we take the erivative respet to, then we obtain [ ] v = η + η k Q + η k 1 Q + η Q k 1 Q = η + η kη Q + η k 1 kq Q + η k 1 Q k 1 9 Now, we ompute v t v t = η t + η t kη t Q + η k 1 + t η k t S Q + η k 1 Q k 1 ] ] = [v + ρ t Q + η t S + [v + ρ t Q + η t S ] k [v + ρ t Q + η t S Q + η k 1 [ + t η k t S Q + η k 1 Q k 1 [ ] = v + v kv Q + η k 1 t S + S ks Q + η k 1 [ ] + ρ t Q + η + Q + η kq + η Q + η k 1 + t η [ k t S Q + η k 1 Q k 1 ] = v + v kv Q + η k 1 + ρ t v + L Q t L S + t η Thus, by L Q = an L S = Q, we obtain v t = v + v kv Q + η k 1 + ρ t v + t Q + t η So, 3 is prove Eponential eay = Lv kv Q + η k 1 Q k 1 + ρ t v + t Q + η By monotoniity arguments, we laim that there eists K > inepenent of α suh that for all t R, v + v t ep K 3 Proof of 3 see in Appeni B Similary proof of Lemma 33 ]

20 3 Nonlinear Liouville property aroun Q 15 3 Estimates an almost orthogonality relations Net, [ ] vq = L η Q + η k Q k kq k 1 η Q, an sine L Q = an Q + η k Q k kq k 1 η Kη, we obtain vq vq Q + η k Q k kq k 1 η Q K η, by eponential eay of Q Similarly, sine L Q p+1 = an eponential eay of Q, we also obtain p+1 vq K η Therefore, we have 7 By laim 3 an, we have L η, η λ 1 η Thus, v, η = L η, η Q + η k Q k kq k 1 η η λ 1 η K η L η 1 λ 1 η Thus So, η L K v L 1 λ 1 η L v, η η L v L Virial type estimates Proof of 6 By the equation of v 1 t L v µ = v vµ + k ρ t = L µ v + R, v vµ t [ v vµ Q + η k 1 Q k 1 ] Q + ηvµ t v µ where R = k [ v vµ Q +η k 1 Q k 1 ] 1 + ρ t v µ + t vq ηvµ v µ By argument of Proposition 31 we have, 3k + 6k 3 v L vµ v µ 31 Now we estimate of R

21 3 Nonlinear Liouville property aroun Q 16 We have kq + η k 1 kq k K η, so kq + η k 1 kq k µ v v K v L η v K η L v L v L K η L v H 1 K η L v L Net, we have these following estimates ρ t v µ ρ t v µ K η L v L, t vq t vq K η L η L η L v L, t ηvµ K t η L v L K η L v L, t v µ K η L v L Thus, we implie R K η L v L 3 From 31 an 3, e obtain 1 v 3k + 6k 3 µ + 1 t 8 Proof of 7 v µ K η L v L By the equation of v we have 1 v = tv t = v L v + t R where R = k [ Q + η k 1 Q k 1 ] v v ρ t v v t vq + η = k [ Q + η k 1 Q k 1 ] v v + ρ t v v + t vq t By straighforwar alulation, we have L v = v v + v kq k 1 v, an v L v = v v + v v v + k Q k 1 vv = 3 v + v kk 1Q k v kq k 1 v 3v + v K v ep k 1 3v + v K v ep Now, we estimate for R From these following estimate k [ Q + η k 1 Q k 1 ] v v K v L η v K v L v L η L 1 K v H ηη K v 1 H 1 η 1 L η 1 L, vη

22 3 Nonlinear Liouville property aroun Q 17 ρ t v K η L v L, t vη t η v K η L v L η L K v L η 1 L η 1 L, t vq t v Q K η L v ep we are obtiane R K v H 1 η 1 L η 1 L + K η L v L + K η L v ep Therefore, 1 t v 3v + v K v ep 1 v+v K v ep Now, we hose B > suh that K ep B 1 Then, we obtain 1 t 1 v + v K v ep 1 v + v 1 B v Proof of Theorem 1 Consier ut satisfies Theorem with α, σ > small enough Lemma apply to t Let V t = 1 µ + ɛ v, where, ɛ = λ inf{µ : B} > By erivatis respet of t,an then using result of Lemma 3, we have for all t V t = 1 µ v 1 ɛ v t t λ 3 v µ v K η L v L + ɛ λ 3 + v 1 ɛ λ 3 λ 3 ɛ v + v K η L v L v B We hoose α > small enough so that K η L Kα 1 λ 3ɛ Thus, V t 1 λ 3ɛ v + v An by eponential eay of v, V t is boune on R, lim t + V t = V +, an lim t V t = V, thus + v + v 1 ɛ V + V 33

23 Asymptoti stability 18 Thus, there eists t n + suh that vt n, as n + an from eponential eay, V + = lim t + V t n =, Similarly, V = lim t + V t n = We obtain t, R, vt, From estimates of paramer, we have t =, ρ t = t Thus, by equation of η, ut, = Q t ρ Asymptoti stability This proof of the asymptoti stability is base on the nonlinear Liouville property Proposition 1 Convergene to a ompat solution Uner the assumption of Theorem 11, for any sequene t n +, there eists a subsequene t nk an ũ H 1 R suh that for all A >, u t nk, + ρt nk k ũ in H 1 as n +, 3 where t, ρt are assoiate to the eomposition of ut as in Lemma 3 Moreover, the solution ũt of 1 orresponing to ũ = ũ,there eist K > suh that t R, ũ t, + ρt Q H 1 α, t, R, ũ t, + ρt K ep, 35 where t, ρt are assoiate to the eomposition of ũt as in Lemma 3 an ρt The first, assuming that Proposition 1 has prove, we prove theorem 11, an then we return proof of Proposition 1 Proof of Theorem 11 From Proposition 1, for any sequene t n +, there eists a subsequene t nk an suh that t nk, an ũ H 1 R satisfies u t nk, + ρt nk ũ, in H 1 Moreover, the solution of ũt assoiate to ũ satisfies 35 an =, ρ = ρ Now, we apply Theorem 1 to the solution ũt It follow that ũt = Q t By the unique of the eomposition in Lemma 3 applie ũ, we have 1 = an 1 =

24 Asymptoti stability 19 Therefore, u t nk, + ρt nk Q in H 1 or equivalently, u t nk, + ρt nk Q tnk in H 1 Thus, this being true for any sequene t n +, it follow that, u t, + ρt Q t in H 1, as t + Convergene of t For K >, we onsier the funtion Ψ K as in Lemma Reall efine of I,t, for α >, >, an t, t R,It : Iut = u t,, I,t t = u t, Ψ ρt + t t Thus, from monotoniity on ut see Lemma, t t, we obtain I,t t I,t t + K ep, t t We introue another trunate energy, for > B We efine I R, I R t = u t, Ψ ρt Corollary Let a 3, B >, C > an K > be as in Proposition above If u Q H 1 < a 3, then the following is satisfie for all real t, for all t t, an for all real suh that > B I R t I R t + C ep Proof of Corollarry From efinition of I,t t implies I,tt = I R t an, by t t, it implies Therefore, I,t t I,t t + C ep, I,tt I,tt + C ep, t t I R t I,tt + C ep, t t On the other han, sine Ψ is not ereasing, so Ψ ρt + t t Ψ ρt +, t t Hene, we have I,tt I R t Thus, I R t I R t + C ep, t t

25 Asymptoti stability We able write I R t = = u t, + Ψ ρt u t, + ρt + Ψ + We laim that for all ɛ >, there eists R > suh that for all ma B, R, the following is satisfie for all t I Rt u t, + ρtψ < ɛ 36 < Now, we enote v H 1 R, we set I lo v = v t, + ρtψ + < Reall that, we take R suffiiently large so that for all t R I lo Q t I Q t ɛ 37 Now,sine 36 an Corolarry an R large enough, we have I lo ut, + ρt I R t + ɛ I R t + C ep + ɛ I lo ut, + ρt + C ep I lo ut, + ρt + ɛ So, taking R large enough, we have t R, t, t t + ɛ, I lo ut, + ρt Ilo ut, + ρt + ɛ 38 Then, from ηt, in H 1 R, we eue that ηt in L lo R Sine ut, + ρt = Q t + ηt,, so from 38 we eue that there eist t > suh that for all real t an t suh that t t t, I lo Q t I lo Q t + 3ɛ 39 The estimates 37, 39 imply that for all ɛ >, there eist t > suh that for all real t an t, with t t t we have, I Q t Ilo Q t + ɛ I lo Q t + ɛ I Q t + 5ɛ

26 Asymptoti stability 1 On the other han, I Q t = Q 5 k t = k 1 Q = f t Q Thus, from, we have f t f t + ɛ An, by f >,, so ft is inreasing funtion respet So, t t, t t t Thus, t is ereasing funtion On the other han, t is boune, so t is onvergene, ie there eists + > suh that t + Thus, From inequality 1 in Lemma 3, let us for any ɛ >, we obtain + < ɛ u t, + ρt Q + in H 1, as t + That s all proof of Theorem 11 Proof of Proposition 1 Let ut satisfying the assumption of asymptoti stability result in Theorem 11 Then by the orbital stability result, for all t inf r R u t, + r Q H 1 < α 1 Let t, ρt be the moulation parameters given by Lemma 3, we obtain η H 1 K α, t It hols that t, u t, + ρt Q t H 1 K α Let t n + From 3, u t n, +ρt n is boune in H 1, thus there eists a subsequene of t n an ũ H 1 R suh that u t n, + ρt n ũ in H 1 From the ontinuity of the solution-map with respet to initial ata for the H 1 weak topology well- poseness result We infer that T > u n tn +, + ρt n ũ in C [ T, T ]; Hweak 1, where ũ is the solution of gkv with ũ = ũ Moreover, sine Hlo 1 L lo ompatly, we have u n tn +, + ρt n ũ in C [ T, T ]; L lo R

27 Asymptoti stability From the estimates in Lemma 3 an Asoli s theorem then there eist are funtion an ρ from [ T, T ] to R suh that t n + in C [ T, T ]; R, an ρt n + ρt n ρ in C [ T, T ]; R Inee, we have K α,ɛ >, is not epen on t Hene, t n + M, with M >, is not epen on n, t Thus, is equi-boune We must have is equiontinuous From estimates t + ρ t K η L, we have t K η L On the other han, η L < η H 1 < K α So, t < K α It implites t n + t t n + h < K α t h, where t h < δ Choose δ = ɛ K α, we obtain t n + t t n + h < ɛ Therefore, is equiontinuous Thus, using Asoli s theorem, there eists > suh that t n + t in C T, T ], R Similary, from how eomposition of u we have ρ K α, implies ρt n + t ρt n K 1, so ρt n + ρt n is equi-boune We also have ρ t K η L < K η H 1 < K 1, by, η H 1 < K α Hene, we have equivalent ρ t < K 1, ρt n + t ρt n + h t t h < K 1, if t h < δɛ, ρt n + t ρt n + h t t h < K 1 t h, t h < δɛ We have ρt n + t ρt n + h < K 1 t h + t t h t h < δɛ < K 1 + αk t h < ɛ, by δɛ = Hene, we have ρt n + t ρt n ρt n + h + ρt n < ɛ ɛ K 1 + αk Thus, ρt n + t ρt n is equiontinuous Using Asoli s theorem, there eists ρ suh that ρt n + t ρt n ρ in C T, T ], R From the onvergene result we eue that t R u t n + t, + ρt n + t ũ t, + ρt in H 1 weak From we thus get t R ũ t, + ρt Q t H 1 < α, an sine Lemma 3, t t < Kα, we obtain that t R, t < Kα That leas to ũ t, + ρt Q H 1 < Kα Now, we prove eponential eay of ũ

28 Asymptoti stability 3 Definition 3 We say that an solution of gkv is L ompat if for any ɛ >, there eists R ɛ >, suh that for all real t >R ɛ ut, + ρt < ɛ 3 We set I,t t = ũ t, + ρtψ + t t + Sine Ψ is even, we have t R, I,t t 1 ũt, + ρt L > From Lemma, we obtain, I,t t I,t t K ep We laim that I,t t = ũ t, + ρtψ + K ep 5 Inee, by 51, for t t we have I,t t I,t t + K ep 6 Let R 1 > Sine Ψ inreasing, by onservation law of L norm, we have I,t t ũ t, + ρt + Ψ R 1 >R 1 t t ũ Let ɛ >, by hoosing R 1 = R 1 ɛ large enough, by 6, we have t R, ũ t, + ρt ɛ >R 1 Thus R 1 being fie, Sine lim Ψ =, so ɛ >, there eists tɛ t so that Ψ R 1 t t ɛ, therefore for all > This prove lim t I,t t = Ψ R 1 t t Ψ R 1 t t ɛ Now, it follows from an that ũt, + ρt L > K ep 7 Therfore, passing to the limit as t in I,t t I,t t + K ep, we obtain 5 The invariane of gkv uner the transformation t, t, obtain that ũt, + ρt L < K ep 8

29 Asymptoti stability Thus, t R, R ũ t, + ρt L K ep, an Sobolev embeing theorem permits to onlue the proof of Thus, Proposion 1 is prove ũt, + ρt K ũt, + ρt L 1 K ũt, + ρt L K ep

30 Appeni A: Monotoniity results Define Ψ = π artan ep, so that lim+ Ψ = 1, lim Ψ = By iret alulations Ψ 1 = osh > ; Ψ 1 16 Ψ, 9 δ 1 >, <, Ψ δ 1 ep, Ψ δ 1 ep 5 A1 - Monotoniity argument on ut Let ut be a solution of 1 satisfying the assumption of Lemma 3 for t [, T ] Let > We efine, t t t T : Ψ t, = Ψ ρt + t t +, an I,t = J,t = u t, Ψ t,, u p + 1 up+1 + u t, Ψ t, Lemma There eist K = K > suh that for α small enough, for all t t T Proof For Φ : R R of lass C 3, we have I,t t I,t t K ep, 51 J,t t J,t t K ep 5 t u Φ = u t uφ = u + u k uφ = u + u k u Φ + uφ = u u Φ + u uφ + u k u Φ + u k+1 Φ = u Φ u Φ + uu Φ u k+1 Φ + k + 1 = 3 u Φ uu Φ + k u k+1 Φ k + 1 = 3u + k k + 1 uk+1 Φ + u Φ u k+1 Φ

31 Asymptoti stability 6 u t k + 1 uk+1 t, Φ = u tu u k u t Φ = u t u Φ u k u t Φ u k Φu t = u t u Φ u t u Φ = u t u + u k Φ u t u Φ = u + u k u + u k Φ + u u + u k Φ = u + u k Φ + u u Φ + u k u Φ = u + u k Φ u Φ u u Φ + k u u k 1 Φ = u + u k Φ u Φ + u Φ + k u u k 1 Φ = u + u k u + ku u k 1 Φ + u Φ So, t u Φ = 3u + k k + 1 uk+1 Φ + u Φ, an u t k + 1 uk+1 Φ = u + u k u + ku u k 1 Φ + where Φ : R R We obtain from the previous alulations an t u Φ, u Ψ = u t uψ + u t Ψ, where = ρt + t t + = u t uψ ρ t u Ψ = 3u ρ t u + k k + 1 uk+1 Ψ + u Ψ 3u 16ρ t 8 1 u + k 16 k + 1 uk+1 Ψ Let R > be hosen later We put I 1 = ρt R ut, k+1 Ψ, an I = ρt R ut, k+1 Ψ For t, suh that ρt R, we have I 1 = u k+1 Ψ ρt R R +ρt R +ρt ut, + ρt k 1 L R ut, k+1 Ψ u t, Ψ

32 Asymptoti stability 7 On the other han, ut, is L ompat, so there eist R > suh that t R, ut, + ρt L R < ɛ Now R be fie to suh value By the following Gagliaro-Nirenberg type inequality, w L R w 1 L R w 1 L R By ut H 1 K, we hose ɛ an R suh that k ut, + ρt k 1 k + 1 L R 16ρ t u t, Ψ Then t u Ψ In the ase ρt R, e have 3u Ψ + k k + 1 I k k + 1 I t t + = ρt + t t + + ρt + ρt + R it implies R + t t +, sine t t Therfore, by the eay properties of Ψ we have Ψ δ 1 ep δ 1 e R ep 8 t t ep On the other han, we have I ρt R u k+1 Ψ K 1 u k 1 L ep 8 t t ep u t, Sine, u L K, therefore, t t, By integration between t, t we obtain t t I,t t K ep 8 t t ep I,t ττ K ep t t ep 8 t τ τ K ep Thus, I,t t I,t t K ep

33 Asymptoti stability 8 Similarly, we have u t k + 1 uk+1 Ψ = = u tu u k u t Ψ + u [ u + u k ] u + ku u k 1 ρ t 1 u k + 1 uk+1 Ψ [ u + u k + u ku u k+1 Ψ ρ t + k + 1 k + 1 uk+1 Ψ + Ψ t u Φ u k 1 16ρ t u ] Ψ We also separate in regions ρt R, ρt R as before, we also ontrol the nonlinear terms, an then we integrating between t an t we obtain J,t t J,t t K ep A - Monotoniity argument on the linearize problem Claim 5 There eists K > suh that, for all t R, t η + η + η e 1 t t t K 53 Proof Let t R, > an = t t Then, by similarly alulations as in Lemma 6, using in partiular 8, we have t η Ψ = η t ηψ + η Ψ = L ηηψ + η Ψ = η + η kq k 1 η η Ψ + ηψ + η Ψ = 3 ηψ η Ψ + η Ψ + k η Q k 1 Ψ k 1Q k Q Ψ By inequality Ψ 1 16 Ψ, thus η Ψ 3 η t Ψ η Ψ 16 + k η Q k 1 Ψ k 1Q k Q Ψ

34 Asymptoti stability 9 By similarly alulations, using inequality Ψ 1 16 Ψ we also have η t Ψ 3 ηψ η 16 Ψ 5 kq k 1 η η Ψ Using arguming in [1], proof of Lemma 5, we obtain for >, η Ψ 3 η t Ψ η Ψ 16 + K ep 8 t t We integrate between t an t, we obtain, for all t < t η t Ψ η t Ψ 1 16 t t t t η + η Ψ + K ep or equivalent η t Ψ + 1 t η + η Ψ 16 t, η t Ψ + K ep t t Passing to the limit t +,using 15 an propertie of Ψ: lim Ψ =, we obtain, for all t, t η + η Ψ t K ep An using inequality Ψ δ 1 ep, we fin t η + η 1 ep < + t t 1 δ 1 ep Let +, we fin, for all t R, t η + η ep 55 t t t t η + η Ψ t K 1 t t t K 56 Now, we use 5 We epan the nonlinear term as follows: kq k 1 η η Ψ = kq k 1 η + kk 1Q k Q η η Ψ + η Ψ = η kk 1Q k Q Ψ + kq k 1 Ψ + kk 1Q k Q ηη Ψ + kk 1Q k Q ηη Ψ = I + II + III

35 Asymptoti stability 3 We have kk 1Q k Q Ψ + kq k 1 Ψ K ep k 1 Ψ KΨ 57 Thus, I K η Ψ Now, we estimate III III = kk 1Q k Q ηη Ψ K ep k 1 ηη Ψ K η Ψ 58 Therefore, I + III η + η Ψ, an II K ηη Ψ ηψ + K η Ψ From these results above, we have t ηψ + η Ψ K η + η Ψ 59 We integrate 59 between t an t n an passing to the limit as n +, we obtain, for all t R, It implies t t From 56 an 61, we obtain Claim 5 ηψ t K ep 6 1 η ep t t t K 61 A3 - Proof of Proof Setting ṽ = vt αtq, using equation of vt an L Q =, we see that ṽt satisfies, ṽ t = L ṽ By the efinition of ṽt an Claim see 55,6, we have t ṽ tψ t K ep 6 ṽ t n Ψ t 63

36 Asymptoti stability 31 By the equation of ṽ, we have as in proof of the Claim 5 ṽ Ψ 1 ṽ + ṽ Ψ t t ṽ Ψ 1 ṽ kq k 1 Ψ + kk 1Q k Q Ψ ṽ + ṽ Ψ ṽ kq k 1 Ψ kk 1Q k Q Ψ Integrating on, t an ombining 6, 63, arguming as in the proof of Claim 5, we obtain for all t R, ṽ + ṽ ep K using the transform, t t, is prove A - Proof of 3 Claim 6 There eists K > suh that u t R, + u + u 1 t ep ρt K 6 Proof From 6, letting t in 88, we have t By 85, an then letting +, t or equivalent t < +ρt t t < +ρt t t u + u Ψ t K ep u + u 1 t ep ρt + u + u 1 t ep 1 δ 1 ep 1 δ 1 t ρt + t 65 t t t u + u Ψ t, t t t u + u Ψ t Thus, from inequality above an 65, we are obtaine t u + u 1 t ep ρt + t t t K 66

37 Asymptoti stability 3 From 66, there eists a sequene t n suh that u t n + u t n Ψ t n as n + Thus, using 89 with t = t n, an passing to the limit as n +, we obtain t u + u + ou Ψ t K ep We argument similarly with 85, we get, for all t, t u + u + ou 1 ep ρt + t t t K 67 Arguming as before, we obtain the following onlusion, for all t R, u + u + ou 1 t ep ρt K 68 Sine equation 1 is invariant by the transformation, t t, the laim is prove Proof of 3 Estimates 6 an the aay on Q t imply v t, ep K From this estimate, using the equation of v an 68, an arguming as for the linear ase proof of, we obtain 3

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