ON THE SUPERCRITICAL KDV EQUATION WITH TIME-OSCILLATING NONLINEARITY arXiv:1106.5961v1 [math.AP] 29 Jun 2011 M. PANTHEE AND M. SCIALOM Abstract. For the initial value problem (IVP) associated the generalized Korteweg-de Vries (gKdV) equation with supercritical nonlinearity, ut + ∂x3 u + ∂x (uk+1 ) = 0, k ≥ 5, numerical evidence [2, 4] shows that, there are initial data φ ∈ H 1 (R) such that the corresponding solution may blow-up in finite time. Also, with the evidence from numerical simulation [1, 17], the physicists claim that a periodic time dependent term in factor of the nonlinearity would disturb the blow-up solution, either accelerating or delaying it. In this work, we investigate the IVP associated to the gKdV equation ut + ∂x3 u + g(ωt)∂x (uk+1 ) = 0, where g is a periodic function and k ≥ 5 is an integer. We prove that, for given initial data φ ∈ H 1 (R), as |ω| → ∞, the solution uω converges to the solution U of the initial value problem associated to Ut + ∂x3 U + m(g)∂x (U k+1 ) = 0, with the same initial data, where m(g) is the average of the periodic function g. Moreover, if the solution U is global and satisfies kU kL5x L10 < ∞, then we prove that the solution uω t is also global provided |ω| is sufficiently large. 1. Introduction Motivated from our earlier work in [5] for the critical KdV equation, we consider the initial value problem (IVP)  ut + ∂ 3 u + g(ωt)∂x (uk+1 ) = 0, x (1.1) u(x, t ) = φ(x), 0 2000 Mathematics Subject Classification. 35Q35, 35Q53. Key words and phrases. Korteweg-de Vries equation, Cauchy problem, local & global well-posedness. M. P. was partially supported by the Research Center of Mathematics of the University of Minho, Portugal through the FCT Pluriannual Funding Program, and through the project PTDC/MAT/109844/2009, and M. S. was partially supported by FAPESP Brazil. 1 2 M. PANTHEE AND M. SCIALOM where x, t, t0 , ω ∈ R and u = u(x, t) is a real valued function, k ≥ 5 is an integer and g ∈ C(R, R) is a periodic function with period L > 0. To simplify the analysis, we translate the initial time t0 to 0 and consider the following IVP  ut + ∂ 3 u + g(ω(t + t0 ))∂x (uk+1 ) = 0, x u(x, 0) = φ(x). (1.2) of the supercritical Korteweg-de Vries (KdV) equation,  ut + ∂ 3 u + ∂x (uk+1 ) = 0, k ≥ 5, x u(x, 0) = φ(x), x, t ∈ R. (1.3) Before analyzing the IVP (1.1) with time oscillating nonlinearity, we discuss some aspects In the literature, the equation (1.3) is known as the supercritical KdV equation because, if one considers the nonlinearity ∂x (uk+1 ), k ∈ Z, then the case when k = 4 is the critical one. As described in [13], the case k = 4 is called critical for three different reasons. First one is that the global solution exists for all data in H 1 (R), whenever k = 1, 2, 3. While for k = 4 the global solution exists only for small data (i.e., data with small H 1 (R)-norm). Second reason is that the index k = 4 is critical for the orbital stability of the solitary wave solutions, see [4]. Finally, the third reason is that the case k = 4 is the only power for which a solitary wave solution cannot have arbitrarily small L2 -norm, see [13]. Well-posedness issues for the IVP (1.3) have been extensively studied in the literature, see for example [9] and [13], [14] and references therein. A detailed account of the recent well-posedness results can be found in Kenig, Ponce and Vega [13], where they proved that, there exists δk > 0 such that the IVP (1.3) is globally well-posed for any data φ ∈ H s (R), s ≥ sk := 1 2 − k 2 satisfying kDxsk φkL2x < δk . They were also able to relax the smallness condition on the given data to obtain local well-posedness result, but paying the price that the existence time now depends on the shape of the data φ and not just in its size. These are the best well-posedness results in the sense that s = sk is the critical exponent given by the scaling argument. However, for data in H s (R), s > sk , they were able to remove the size and shape restriction and got local-well posedness for arbitrary data with life span T of the solution depending on kφkH s (R) . Quite recently, Farah et. al. [8] considered the IVP (1.3) to deal global well-posedness for the data with low Sobolev regularity. In this context, they proved the following local well-posedness result in the function space slightly different from the one used in [13]. In what follows, we state this result, because we will modify it to suit in our context later on. SUPERCRITICAL KDV Theorem 1.1. [8] Let k > 4 and s > sk := 1 2 3 − k2 . Then for any φ ∈ H s (R) there exist T = T (kφkH s (R) ) > 0 (with T (s, ρ) → ∞ as ρ → 0) and a unique strong solution u to the IVP (1.3) satisfying: u ∈ C([0, T ]; H s (R)), (1.4) s k∂x ukL∞ 2 + kDx ∂x ukL∞ L2 < ∞, x L x (1.5) kukL5x L10 + kDxs ukL5x L10 < ∞, (1.6) kDtγk Dxαk Dtβk ukLpxk Lqk < ∞, (1.7) T T T T T where 1 3 6 s − sk − 25k, βk = − , γk = γk (s) = (1.8) 10 10 5k 3 2 1 1 3 4 1 = + , = − . (1.9) pk 5k 10 qk 10 5k Moreover, for any T ′ ∈ (0, T ), there exists a neighborhood V of φ in H s (R) such that the αk = map φ̃ 7→ ũ from V into the class defined by (1.4) to (1.7) with T ′ in place of T is Lipschitz. We recall that, the L2x (R) norm and energy are conserved by the flow of (1.3). More precisely, Z 2 |u(x, t)| dx = Z |φ(x)|2 dx, (1.10) R R and 1 E(u(·, t)) := 2 are conserved quantities. Z {(ux (x, t))2 − ck uk+2 (x, t)}dx = E(φ), (1.11) R As shown with a detailed calculations in [8], these conserved quantities yield an a priori estimate for k∂x u(t)kL2 (R) if the initial data φ is sufficiently small in H 1 (R). This allows to iterate the local solution to get the global one for small data in H 1 (R). However, a numerical study carried out by Bona et. al. [2, 3] (see also [4]) revel the existence of H 1 -data for which the corresponding solution to the supercritical KdV equation may blow-up in finite time. This is the point that motivated us to carry on this work in the light of the recent work of Abdullaev et. al. in [1] and Konotop and Pacciani in [17]. The authors in [1] and [17] investigate the effect of a time oscillating term in factor of the nonlinearity in Bose-Einstein condensates. In [1] the authors investigate solutions which are global for large frequencies, while the authors in [17] study solutions which blow-up in finite time. Their results are numerical. Roughly speaking, they claim that the periodic time dependent term in factor of the nonlinearity would disturb the blow-up solution, either accelerating it or delaying it. Recently, Cazenave and Scialom [6] considered the nonlinear 4 M. PANTHEE AND M. SCIALOM Schrödinger (NLS) equation and got an analytical insight to understand the problem by showing that the solution really depends on the frequency of the oscillating term. They proved that the solution u to the IVP associated to the NLS equation iut + ∆u + θ(ωt)|u|α u = 0, 4 is an H 1 sub-critical (N −2)+ H 1 (RN ) converges as |ω| → ∞ to x ∈ RN , (1.12) where 0 < α < exponent and θ is a periodic function, with initial data φ ∈ the solution U of the limiting equation iUt + ∆U + I(θ)|U |α U = 0, x ∈ RN , (1.13) with the same initial data, where I(θ) is the average of θ. Moreover, they also showed that, if the limiting solution U is global and has a certain decay property as t → ∞, then u is also global if |ω| is sufficiently large. A similar result has been proved for the critical KdV equation in our earlier work [5]. In this work, we are interested in obtaining similar results for the supercritical KdV equation. The numerical evidences for the existence of blow-up solution to (1.3) in H 1 (R) due to Bona et. al. [2, 3] (see also [4]) and the discussion made above strengthen our motivation of studying (1.1) with time oscillating nonlinearity. As discussed above, our interest here is to investigate the behavior in H 1 (R) of the solution of the IVP (1.1) as |ω| → ∞. The natural limiting candidate to think of is the solution to the following IVP  Ut + ∂ 3 U + m(g)∂x (U k+1 ) = 0, x where m(g) := 1 L RL 0 U (x, 0) = φ(x), (1.14) x, t ∈ R, g(t)dt is the mean value of g and is a real number. To this end, we need an appropriate well-posedness result for the supercritical KdV equation in H 1 (R). We recall the local well-posedness result from [8] for arbitrary data in H s (R), s > sk , with life span of solution depending only on the H s (R)-norm of the initial data stated in Theorem 1.1 (See also [13]). The function space used in Theorem 1.1 has additional norm kDtγk Dxαk Dtβk ukLpxk Lqk that T involves time derivatives of the solution. The presence of these norms create extra difficulty to handle the time-oscillating nonlinearity. Therefore, to deal with our case, we need to avoid the presence of the norm that involved time derivatives. Also, it is very important to have an explicit expression that gives the local existence time of the solution. In the literature, we did not find an explicitly written proof of the H 1 (R) well-posedness for the IVP (1.3)that fulfills our requirement. Therefore, we will provide a new proof for the well-posedness of the SUPERCRITICAL KDV 5 IVP (1.3) in H 1 (R). Our proof allows us to extend the result to (1.2) and as a consequence to have an estimate of the local existence time. The only works other than [5] and [6] we did find in the literature that address the wellposedness issue for the equations of the KdV-family and NLS with explicitly time dependent nonlinearity were by Nunes [18, 19] and Damergi and Goubet [7]. The authors in [7] deal with the NLS equation in R2 with nonlinearity cos2 (Ωt)|u|p−1 u in the critical and supercritical cases. The author in [18] considered the transitional KdV with nonlinearity f (t)u∂x u, f a continuous function such that f ′ ∈ L1loc (R) and proved global well-posedness in H s (R), s ≥ 1. The transitional KdV arises in the study of long solitary waves propagating on the thermocline separating two layers of fluids of almost equal densities in which the effect of the change in the depth of the bottom layer, which the wave feels as it approaches the shore, results in the coefficient of the nonlinear term, for details see [16]. In [19], transitional Benjamin-Ono equation with time dependent coefficient in the nonlinearity has been considered and the main result is the global existence of the solution for data in H s (R), s ≥ 23 . Before stating the main results of this work, we define notations that will be used throughout this work. Notation: We use fˆ to denote the Fourier transform of f and is defined as, Z 1 e−ixξ f (x) dx. fˆ(ξ) = (2π)1/2 R The L2 -based Sobolev space of order s will be denoted by H s with norm Z 1/2 kf kH s (R) = (1 + ξ 2 )s |fˆ(ξ)|2 dξ . R The Riesz potential of order −s is denoted by Dxs = (−∂x2 )s/2 . For f : R × [0, T ] → R we define the mixed Lpx LqT -norm by kf k Lpx LqT = Z Z R T |f (x, t)|q dt 0 p/q dx 1/p , with usual modifications when p = ∞. We replace T by t if [0, T ] is the whole real line R. We use the notation f ∈ H α+ if f ∈ H α+ǫ for ǫ > 0. We define two more spaces XT and YT with norms 2 kf kXT :=kf kL∞ 1 + k∂x f kL∞ L2 + k∂x f kL∞ L2 x x T H T T , + kf kL5x L10 + k∂x f kL5x L10 + k∂x f kL20 L5/2 + kf kL4x L∞ T T T x (1.15) T and kf kYT := k∂x f kL2x L2 + kf kL2x L2 , T T (1.16) 6 M. PANTHEE AND M. SCIALOM respectively. We replace XT by Xt or X(T,∞) , if the time integral is taken in the interval (0, ∞) or (T, ∞) respectively, and similarly for YT . We use the letter C to denote various constants whose exact values are immaterial and which may vary from one line to the next. First, let us state the H 1 -local well-posedness result for the IVP (1.3) in a function space that does not use norms involving time derivatives of the solution. Theorem 1.2. Suppose φ ∈ H 1 (R). Then there exist T = T (kφkH 1 (R) ) > 0 and a unique solution u to the IVP (1.3) satisfying u ∈ C([0, T ]; H 1 (R)), (1.17) 2 k∂x ukL∞ 2 + k∂x ukL∞ L2 < ∞, x x L (1.18) kukL5x L10 + k∂x ukL5x L10 + k∂x ukL20 L5/2 < ∞, (1.19) < ∞. kukL4x L∞ T (1.20) T T T T x T Moreover, for any T ′ ∈ (0, T ), there exists a neighborhood V of (u0 , v0 ) in H 1 (R) such that the map φ̃ 7→ ũ from V into the class defined by (1.17) to (1.20) with T ′ in place of T is Lipschitz. Using Duhamel’s principle, we prove Theorem 1.2 by considering the integral equation associated to the IVP (1.3), u(t) = S(t)φ − Z t S(t − t′ )∂x (uk+1 )(t′ ) dt′ , (1.21) 0 where S(t) is the unitary group generated by the operator ∂x3 that describes the solution to the linear problem. Our interest is to solve (1.21) using the contraction mapping principle in appropriate metric spaces. Remark 1.3. Since the average m(g) of g is a constant, the proof of Theorem 1.2 can be adapted line by line to obtain the similar well-posedness result for the IVP (1.14). The only difference in this case is that, to complete the contraction argument we need to choose T > 0 in such a way that C|m(g)|T 1/2 kφk4H 1 (R) < 12 . So the existence time T depends on |m(g)| and kφkH 1 (R) . We also have the following bound kU kXT ≤ CkφkH 1 (R) , ∀ t ∈ [0, T ]. (1.22) Regarding the well-posedness results for the IVP (1.2), we have the following theorem. SUPERCRITICAL KDV 7 Theorem 1.4. Suppose φ ∈ H 1 (R). Then there exist T = T (kφkH 1 (R) , kgkL∞ ) > 0 and a unique solution uω,t0 ∈ C([0, T ]; H 1 (R)) to the IVP (1.2) satisfying (1.18)–(1.20). Moreover, for any T ′ ∈ (0, T ), there exists a neighborhood V of φ in H 1 (R) such that the map φ̃ 7→ ũω,t0 from V into the class defined by (1.17) to (1.20) with T ′ in place of T is Lipschitz. Now, we state the main results of this work. Theorem 1.5. Fix φ ∈ H 1 (R). For given ω, t0 ∈ R, let uω,t0 be the maximal solution of the IVP (1.2) and U be the solution of the limiting IVP (1.14) defined on the maximal time of existence [0, Smax ). Then, for given any 0 < T < Smax , the solution uω,t0 exists on [0, T ] for all t0 ∈ R and |ω| large. Moreover, kuω,t0 − U kXT → 0, as |ω| → ∞, uniformly in t0 ∈ R. In particular, the convergence holds in C([0, T ]; H 1 (R)) for all T ∈ (0, Smax ). Theorem 1.6. Let φ ∈ H 1 (R) and uω,t0 be the maximal solution of the IVP (1.1). Suppose U be the maximal solution of the IVP (1.14) defined on [0, Smax ). If Smax = ∞ and kU kL5x L10 < ∞, t (1.23) then it follows that uω,t0 is global for all t0 ∈ R if |w| is sufficiently large. Moreover, kuω,t0 − U kXt → 0, when |w| → ∞, (1.24) uniformly in t0 . In particular, convergence holds in L∞ ((0, ∞); H 1 (R)). In view of the numerical prediction in [2, 3] (see also [4]) of existence of blow-up solution for the supercritical KdV equation for in H 1 (R), the Theorem 1.6 is very interesting in the sense that when m(g) = 0 the solution U to the IVP (1.14) will be global for all initial H 1 -data and the solution uω,t0 to the nonlinear problem (1.2) will be global too, for |ω| large enough. Before leaving this section, we discuss the example constructed in [6] in the context of the NLS equation with time oscillating nonlinearity. The authors in [6] showed that for small frequency |ω|, the solution uω,t0 blows-up in finite time or is global depending on t0 , while for the large frequency |ω|, the solution uω,t0 is global for all t0 ∈ R. The same example can be utilized with small modification in the context of the critical KdV equation. We present it here for the convenience of the readers. Example 1.7. Let L > 1, 0 < ǫ < m(g) = 0, L−1 2 and consider a periodic function g defined by  1, |s| ≤ ǫ, and g(s) = (1.25) 0, 1 ≤ s ≤ 1 + ǫ, 8 M. PANTHEE AND M. SCIALOM with period L. Fix φ ∈ H 1 (R) and assume that the solution v of the IVP  vt + vxxx + v k+1 ∂x v = 0, v(x, 0) = φ(x), k ≥ 5, (1.26) blows-up in finite time, say T ∗ . In the light of the numerical evidences presented in [2, 3] (see also [4]) we can suppose that such a solution v(x, t) of (1.26) with t ∈ [0, T ∗ ), exists. From Theorem 1.5, for this particular φ and the periodic function g, we have that the solution uω,t0 to the IVP (1.2) converges, as |ω| → ∞, to the solution U of the linear KdV equation with same initial data φ. So, in view of Theorem 1.6, uω,t0 is global as |ω| → ∞ for all t0 ∈ R. Now we move to analyze the behavior of the solution for |ω| small. Note that g(ωs) = 1 when |ωs| ≤ ǫ. Therefore, if we consider |ω| < (1.26) satisfies (1.2) for t0 = 0 on [0, T ∗ ). ǫ T∗ , then we see that the solution v to the IVP By uniqueness, uω,0 = v. Hence the solution uω,0 of the IVP (1.2) blows-up in finite time, provided |ω| < ǫ T∗ . Let ǫ = ǫ(A) be as in Corollary 3.5 with A = kgkL∞ . From the linear estimate (2.6) we t have that S(·)φ ∈ L5x L10 t , so there exists T > 0 such that = kS(·)φkL5x L10 kS(·)[S(T )φ]kL5x L10 t (T,∞) For ω > 0, if we consider t0 = i.e., for all 0 ≤ s ≤ t0 = 1 ω, ǫ ω. 1 ω, ≤ ǫ. (1.27) we have that g(ω(s + t0 )) = 0 for all 1 ≤ ω(s + t0 ) ≤ 1+ ǫ, Therefore, if we let ω > 0 satisfying ω ≤ ǫ T (i.e., T ≤ ǫ ω ), and choose then g(ω(s + t0 )) = 0 for all 0 ≤ s ≤ T . So, with this choice, uω,t0 solves the linear KdV equation if 0 ≤ t ≤ T . Therefore, for ω ≤ ǫ T , uω,t0 exists on [0, T ] and is given by S(t)φ, in particular uω,t0 (T ) = S(T )φ. From (1.27), kS(·)uω,t0 (T )kL5x L10 ≤ ǫ. Hence, from t Corollary 3.5 we conclude that uω,t0 is global. This paper is organized as follows. In Section 2 we record some preliminary estimates associated to the linear problem and other relevant results. In Section 3 we give a proof of the local well-posedness result for the supercritical KdV equation in H 1 (R) and some other results that will be used in the proof of the main Theorems. Finally, the proof of the main results will be given in Section 4. SUPERCRITICAL KDV 9 2. Preliminary estimates In this section we record some linear estimates associated to the IVP (1.1). These estimates are not new and can be found in the literature. For the sake of clearness we sketch the ideas involved and provide references where a detailed proof can be found. Lemma 2.1. If u0 ∈ L2 (R), then If f ∈ L1x L2t , then ∂x Z t Z t k∂x S(t)u0 kL∞ 2 ≤ Cku0 kL2 . x x Lt (2.1) S(t − t′ )f (·, t′ )dt′ ≤ Ckf kL1x L2t , (2.2) ≤ Ckf kL1x L2t . (2.3) 0 and ∂x2 S(t − t′ )f (·, t′ )dt′ 0 2 L∞ t Lx 2 L∞ x Lt Proof. For the proof of the homogeneous smoothing effect (2.1) and the double smoothing effect (2.3), see Theorem 3.5 in [13] (see also Section 4 in [12]). The inequality (2.2) is the dual version of (2.1).  Now we give the maximal function estimate. Lemma 2.2. If u0 ∈ Ḣ 1/4 (R), then kS(t)u0 kL4x L∞ ≤ CkDx1/4 u0 kL2 (R) . T (2.4) Also, we have ∞ ≤ Cku0 k kS(t)u0 kL∞ x LT 1 H 2 + (R) . (2.5) Proof. For the proof of the estimate (2.4) we refer to Theorem 3.7 in [13] (see also [11] and [15]). The estimate (2.5) follows from Sobolev embedding.  In what follows, we state some more estimates that will be used in our analysis. Lemma 2.3. If u0 ∈ L2 (R), then kS(t)u0 kL5x L10 ≤ Cku0 kL2x . t (2.6) k∂x S(t)u0 kL20 L5/2 ≤ CkDx1/4 u0 kL2x , (2.7) k∂x S(t)u0 kL40/3 L20/7 ≤ CkDx3/8 u0 kL2x . (2.8) Also we have x t and x t 10 M. PANTHEE AND M. SCIALOM Proof. The proof of the estimates (2.6) and (2.7) can be found in Corollary 3.8 and Proposition 3.17 in [13] respectively. To prove (2.8) we consider the analytic family of operators Tz u0 = Dx−z/4 Dx1−z S(t)u0 , with z ∈ C, 0 ≤ ℜz ≤ 1. Now the estimate (2.8) follows by choosing z = 3/40 in the Stein’s theorem of analytic interpolation (see [20]) between the smoothing estimate (2.1) and the maximal function estimate (2.4).  Lemma 2.4. Let u0 ∈ L2x , then for any (θ, α) ∈ [0, 1] × [0, 21 ], we have kDxθα/2 S(t)u0 kLq Lpx ≤ Cku0 kL2x , (2.9) T 6 2 where (q, p) = ( θ(α+1) ). , 1−θ Proof. See Lemma 2.4 in [10].  We state next the Leibniz’s rule for fractional derivatives whose proof is also given in [13], Theorem A.8. Lemma 2.5. Let α ∈ (0, 1), α1 , α2 ∈ [0, α], α1 + α2 = α. Let p, p1 , p2 , q, q1 , q2 ∈ (1, ∞) be such that 1 p = 1 p1 + 1 1 p2 , q = 1 q1 + 1 q2 . Then kDxα (f g) − f Dxαg − gDxα f kLpx LqT ≤ CkDxα1 f kLpx1 Lq1 kDxα2 gkLpx2 Lq2 . T T (2.10) Moreover, for α1 = 0 the value q1 = ∞ is allowed. Definition 2.6. Let 1 ≤ p, q ≤ ∞, − 14 ≤ α ≤ 1. We say that a triple (p, q, α) is an admissible triple if 1 1 1 + = p 2q 4 and α= 2 1 − . q p (2.11) Proposition 2.7. For any admissible triples (pj , qj , αj ), j = 1, 2, the following estimate holds Z t α1 Dx S(t − t′ )f (·, t′ )dt′ p1 q1 ≤ CkDx−α2 f k p′2 q2′ , (2.12) 0 Lx Lt Lx Lt where p′2 , q2′ are the conjugate exponents of p2 , q2 . Proof. For the proof we refer to Proposition 2.3 in [14]. The following results will be used to complete the contraction mapping argument.  SUPERCRITICAL KDV 11 Lemma 2.8. Let XT and YT be the spaces defined earlier and S be the unitary group associated to the operator ∂x3 , then we have kS(t)u0 kXT ≤ C0 ku0 kH 1 (R) , Z t S(t − t′ )f (t′ )dt′ XT 0 (2.13) ≤ CT 1/2 kf kYT . (2.14) Proof. The estimate (2.13) follows from the linear estimates in Lemmas 2.1, 2.2 and 2.3. For the proof of the estimate (2.14), we refer to our earlier work in [5].  Lemma 2.9. The following estimate holds, k∂x (uk+1 )kYT ≤ Ckukk+1 XT . (2.15) Proof. The idea of the proof is similar to the one we used in [5] for the critical KdV equation. Using Hölder’s inequality and the fact that H 1 (R) ֒→ L∞ (R), we get k−2 2 ∞ ku ∂x uk 2 2 ≤ Ckuk ∞ kuk2L4x L∞ k∂x ukL∞ k∂x (uk+1 )kL2x L2 ≤ Ckuk−2 kL∞ 2 . (2.16) Lx L x LT L H 1 (R) x L T T T T T Similarly   k∂x2 (uk+1 )kL2x L2 ≤ C kuk−1 (∂x u)2 kL2x L2 + kuk ∂x2 ukL2x L2 T T T   2 k−2 ∞ ∞ ∞ ku(∂x u) k 2 2 + ku ≤ C kuk−2 kL∞ kLx LT ku2 ∂x2 ukL2x L2 Lx LT x LT T   k−2 k−2 k∂x ukL5x L10 k∂x ukL20 L5/2 +kukL4 L∞ k∂x2 ukL∞ ≤ CkukL∞ H 1 (R) kukL4x L∞ 2 . x L T T T x T x T T (2.17) In view of definitions of XT -norm and YT -norm, the estimates (2.16) and (2.17) yield the required result (2.15).  The following result from [6] will also be useful in our analysis. Lemma 2.10. Let T > 0, 1 ≤ p < q ≤ ∞ and A, B ≥ 0. If f ∈ Lq (0, T ) satisfies kf kLq (0,t) ≤ A + Bkf kLp (0,t) , (2.18) for all t ∈ (0, T ), then there exists a constant K = K(B, p, q, T ) such that kf kLq (0,T ) ≤ KA. (2.19) 12 M. PANTHEE AND M. SCIALOM 3. Proof of the well-posedness results We start this section by proving the well-posedness results for the IVP (1.3) announced in Theorem 1.2. Proof of Theorem 1.2. For a > 0, consider a ball in XT defined by BTa = {u ∈ C([0, T ] : XT (R)) : kukXT < a}. Our aim is to show that, there exist a > 0 and T > 0, such that the application Φ defined by Φ(u) := S(t)φ − Z t S(t − t′ )∂x (uk+1 )(t′ )dt′ , (3.1) 0 maps BTa into BTa and is a contraction. Using the estimates (2.14) and (2.15), we obtain kΦkXT ≤ C0 kφkH 1 + CT 1/2 k∂x (uk+1 )kYT ≤ C0 kφkH 1 + CT 1/2 kukk+1 XT . (3.2) Hence, for u ∈ BTa , kΦkXT ≤ C0 kφkH 1 + CT 1/2 ak+1 . (3.3) Now, choose a = 2C0 kφkH 1 and T such that CT 1/2 ak < 1/2. With these choices we get, from (3.3), kΦkXT ≤ a a + . 2 2 Therefore, Φ maps BTa into BTa . With the similar argument, one can prove that Φ is a contraction. The rest of the proof follows standard argument.  Remark 3.1. From the choice of a and T in the proof of Theorem 1.2 it is clear that the local existence time is given by . T ≤ Ckφk−2k H 1 (R) (3.4) Moreover, we have the following bound, kukXT ≤ CkφkH 1 (R) . (3.5) In what follows, we sketch a proof for the local well-posedness result for the IVP (1.2). SUPERCRITICAL KDV 13 Proof of Theorem 1.4. As in the proof of Theorem 1.2, this theorem will also be proved by considering the integral equation associated to the IVP (1.2), Z t S(t − t′ )g(ω(t′ + t0 ))∂x (uk+1 )(t′ ) dt′ , u(t) = S(t)φ − (3.6) 0 and using the contraction mapping principle. First of all, notice that the periodic function g is bounded, say kgkL∞ ≤ A, for some positive t constant A. Since the norms involved in the space Y permit us to take out kgkL∞ -norm as t a coefficient, the proof of this theorem follows exactly the same argument as in the proof of Theorem 1.2. Moreover, as the initial data φ is the same, the choice of the radius a of the ball is exactly the same. However, to complete the contraction mapping argument, we must T 1/2 a4 < 12 , which implies that the existence T is given by select T > 0 such that CkgkL∞ t T = T (kgkL∞ , kφkH 1 (R) ) = t C kgk2L∞ kφk2k H 1 (R) t . (3.7) Furthermore, in this case too, from the proof, one can get kukXT ≤ CkφkH 1 (R) . (3.8)  In sequel, we present some results that play a central role in the proof of the main theorems of this work. We begin with the following lemma whose proof can be found in [5]. Lemma 3.2. Let XT and YT be spaces as defined in (1.15) and (1.16). Let f ∈ YT , then we have the following convergence Z t Z t S(t − t′ )f (t′ )dt′ , g(ω(t′ + t0 ))S(t − t′ )f (t′ )dt′ → m(g) (3.9) 0 0 whenever |ω| → ∞, in the XT -norm. With the similar argument as in the case of the critical KdV equation (see [5]), we have the following convergence result. Lemma 3.3. Let the initial data φ ∈ H 1 (R). Let uω,t0 be the maximal solution of the IVP (1.1). Suppose U be the maximal solution of the IVP (1.14) defined in [0, Smax ). Let 0 < T < Smax and let uω,t0 exists in [0, T ] for |ω| large and that < ∞, lim sup sup kuω,t0 kL∞ 1 T H (R) |ω|→∞ t0 ∈R (3.10) 14 M. PANTHEE AND M. SCIALOM and lim sup sup kuω,t0 kL4x L∞ < ∞. T (3.11) |ω|→∞ t0 ∈R Then, for all t ∈ [0, T ], sup kuω,t0 − U kXT → 0, as |ω| → ∞. (3.12) t0 ∈R In particular, uω,t0 → U as |ω| → ∞, in H 1 (R). Proof. Since uω,t0 and U have the same initial data φ, from Duhamel’s formula, we have t uω,t0 − U = Z t = Z ′ g(ω(t + t0 ))S(t − t 0 0 ′ ′ )∂x (uk+1 ω,t0 )dt − m(g) Z t S(t − t′ )∂x (U k+1 )dt′ 0 k+1 )dt′ g(ω(t′ + t0 ))S(t − t′ )∂x (uk+1 ω,t0 − U + Z t (3.13) [g(ω(t′ + t0 )) − m(g)]S(t − t′ )∂x (U k+1 )dt′ 0 =: I1 + I2 . We note that |uk+1 − v k+1 | ≤ C(|u|k + |v|k )|u − v| (3.14) and   |∂x (uk+1 − v k+1 )| ≤ C (|u|k + |v|k )|∂x (u − v)| + (|∂x u| + |∂x v|)(|u|k−1 + |v|k−1 )|u − v| . (3.15) Let kgkL∞ ≤ A. Use of (2.2), (3.14), Hölder’s inequality and the assumptions (3.10) and T (3.11), yield k+1 k+1 kI1 kL∞ kL1x L2 2 ≤ CkgkL∞ kuω,t − U 0 T T Lx T ≤ CAkukω,t0 (uω,t0 ≤ CAkukω,t0 kL2x L∞ kuω,t0 T − U )kL1x L2 + kU k (uω,t0 − U )kL1x L2 T T k − U kL2x L2 + kU kL2x L∞ kuω,t0 − U kL2x L2 T T T i h k−2 ∞ ∞ 2 2 k−2 ∞ ∞ kuω,t0 − U kL2 L2x + kU kLx LT kU kL2x L∞ ≤ CA kuω,t0 kLx LT kuω,t0 kL2x L∞ T T T i h k−2 k−2 2 2 ≤ CA kuω,t0 kL∞ H 1 (R) kuω,t0 kL4x L∞ + kU kL∞ H 1 (R) kU kL4x L∞ kuω,t0 − U kL2 L2x T T T T T ≤ CAkuω,t0 − U kL2 L2x . T (3.16) SUPERCRITICAL KDV 15 Again, using (2.2) and (3.15), one can obtain k+1 k+1 )kL1x L2 k∂x I1 kL∞ 2 ≤ CAk∂x (uω,t − U 0 T Lx T h ≤ CA k(|uω,t0 |k + |U |k )∂x (uω,t0 − U )kL1x L2 T + k(|∂x uω,t0 | + |∂x U |)(|uω,t0 |k−1 + |U |k−1 )(uω,t0 − U )kL1x L2 T i (3.17) =: CA[J1 + J2 ]. With the same argument as in (3.16) J1 ≤ Ck∂x (uω,t0 − U )kL2 L2x . (3.18) T k−1 ∂ u (uω,t0 − U )kL1x L2 in J2 , the estimates Now we move to estimate the first term, kuω,t 0 x ω,t0 T for the other terms are similar. We have, k−3 k−1 ∂ u (uω,t0 − U )kL2x L2 ∂ u (uω,t0 − U )kL1x L2 ≤ Cku2ω,t0 kL2x L∞ kuω,t kuω,t 0 x ω,t0 0 x ω,t0 T T T ≤ k−3 ∞ ∞ k∂x uω,t0 kL∞ k Ckuω,t0 k2L4x L∞ kuω,t 2 k(uω,t0 0 LT Lx T Lx T − U )kL2 L∞ x T k−2 ≤ Ckuω,t0 k2L4x L∞ kuω,t0 kL ∞ H 1 (R) k(uω,t0 − U )kL2 H 1 (R) T T T ≤ Ck(uω,t0 − U )kL2 H 1 (R) . T (3.19) Inserting (3.18) and (3.19) in (3.17), we get k∂x I1 kL∞ 2 ≤ CAk(uω,t0 − U )kL2 H 1 (R) . T Lx T (3.20) Combining (3.16) and (3.20), we obtain ≤ CAk(uω,t0 − U )kL2 H 1 (R) . kI1 kL∞ 1 T H (R) T (3.21) From Lemma 3.2, we have 1 ≤ Cω → 0, kI2 kL∞ T H (R) as |ω| → ∞. (3.22) Therefore, we have ≤ CAk(uω,t0 − U )kL2 H 1 (R) + Cω . kuω,t0 − U kL∞ 1 T H (R) T (3.23) Applying Lemma 2.10 in (3.23), we get ≤ KCω → 0, kuω,t0 − U kL∞ 1 T H (R) as |ω| → ∞. (3.24) From (3.23) and (3.24), it is easy to conclude that k(uω,t0 − U )kL2 H 1 (R) → 0, T as |ω| → ∞. (3.25) 16 M. PANTHEE AND M. SCIALOM Now, we move to estimate the other norms involved in the definition of XT . Let, 2 L1 := k∂x (uω,t0 − U )kL∞ 2 + k∂x (uω,t0 − U )kL∞ L2 + kuω,t0 − U kL5 L10 + kDx (uω,t0 − U )kL5 L10 x L x x x T T T T and . L2 := k∂x (uω,t0 − U )kL20 L5/2 + kuω,t0 − U kL4x L∞ T x T Use of (2.2), (2.3), the estimate (2.12) from Proposition 2.7 with admissible triples (p1 , q1 , α1 ) = (5, 10, 0), and (p2 , q2 , α2 ) = (∞, 2, 1) in (3.13), yields k+1 k+1 )kL1x L2 + CAkuk+1 kL1x L2 + kI2 kXT . L1 ≤ CAk∂x (uk+1 ω,t0 − U ω,t0 − U T T (3.26) Therefore, with the same argument as in (3.16)-(3.20), we can obtain L1 ≤ CAkuω,t0 − U kL2 H 1 + Cω . (3.27) T Hence, using Lemma 3.3 and (3.25) we get from (3.27) that L1 |ω|→∞ → 0. (3.28) Finally, to estimate L2 we use Proposition 2.7 with admissible triples (p1 , q1 , α1 ) = (20, 5/2, 3/4) and (p2 , q2 , α2 ) = (20/3, 5, 1/4), to get Z t S(t − t′ )f (·, t′ )dt′ ∂x 0 5/2 L20 x LT ≤ Ckf kL20/17 L5/4 , x (3.29) T and with admissible triples (p1 , q1 , α1 ) = (4, ∞, −1/4), and (p2 , q2 , α2 ) = (20/3, 5, 1/4), to have Z t S(t − t′ )f (·, t′ )dt′ 0 L4x L∞ T ≤ Ckf kL20/17 L5/4 . x (3.30) T Using (3.29), (3.30), and the definition of XT , we get from (3.13) that k+1 )kL20/17 L5/4 + kI2 kXT L2 ≤ CAk∂x (uk+1 ω,t0 − U x (3.31) T Using (3.15), we can obtain h k+1 ≤ C k(|uω,t0 |k + |U |k )∂x (uω,t0 − U )kL20/17 L5/4 − U )k k∂x (uk+1 5/4 20/17 ω,t0 L L x x T k−1 + k(|∂x uω,t0 | + |∂x U |)(|uω,t0 | T k−1 + |U | )(uω,t0 − U )kL20/17 L5/4 x =: C[J˜1 + J˜2 ]. (3.32) T i SUPERCRITICAL KDV 17 Hölder’s inequality, the fact that 20/13 > 10/7, Sobolev immersion and the assumption (3.10), imply that J˜1 ≤ Ck∂x (uω,t0 − U )kL5x L10 {kukω,t0 kL20/13 L10/7 + kU k kL20/13 L10/7 } T ≤ Ck∂x (uω,t0 − x T ≤ Ck∂x (uω,t0 − U )kL5x L10 T 7/10 T ≤CT 7/10 x T U )kL5x L10 {kukω,t0 kL10/7 L20/13 T x T k + kU kL10/7 L20/13 } {kuω,t0 kkL∞ H 1 T T x (3.33) + kU kkL∞ H 1 } T k∂x (uω,t0 − U )kL5x L10 . T k−1 ∂ u (uω,t0 − U )kL20/17 L5/4 An in (3.17), we give details in estimating the first term, kuω,t 0 x ω,t0 x T in J˜2 , the estimates for the other terms are similar. Here too, Hölder’s inequality, the fact that 20/3 > 5, Sobolev immersion and the assumption (3.10), yield k−1 k−1 kuω,t ∂ u (uω,t0 − U )kL20/17 L5/4 ≤ Ckuω,t k 20/3 L5 k∂x uω,t0 kL2x L2 kuω,t0 − U kL5x L10 0 x ω,t0 0 L x x T T T T k−1 ≤ Ckuω,t k 20/3 k∂x uω,t0 kL2 L2 kuω,t0 − U kL5 L10 0 L5 L x x T T x T ≤ C T 7/10 kuω,t0 kkL∞ H 1 kuω,t0 − U kL5x L10 T T ≤CT 7/10 (3.34) kuω,t0 − U kL5x L10 . T In view of (3.32), (3.33) and (3.34), we get from (3.31) that L2 ≤ CA T 7/10 {k∂x (uω,t0 − U )kL5x L10 + kuω,t0 − U kL5x L10 } + Cω . T T (3.35) Therefore, Lemma 3.3 and (3.28), imply L2 |ω|→∞ → 0. Now, the proof of the Lemma follows by combining (3.24), (3.28) and (3.36). (3.36)  In what follows, as we did in our earlier work [5], we consider the supercritical KdV equation with more general time dependent coefficient on the nonlinearity. Given h ∈ L∞ we consider  ut + uxxx + h(t)∂x (uk+1 ) = 0, x, t ∈ R, k ≥ 5 (3.37) u(x, 0) = φ(x). The results for the IVP (3.37) and their proofs that we are going to present here are quite similar to the ones we have for the critical KdV equation in [5]. For the sake of clarity, we reproduce them here. Proposition 3.4. Given any A > 0, there exist ǫ = ǫ(A) and B > 0 such that if khkL∞ ≤ A and if φ ∈ H 1 (R) satisfies kS(t)φkL5x L10 ≤ ǫ, t (3.38) 18 M. PANTHEE AND M. SCIALOM then the corresponding solution u of (3.37) is global and satisfies , ≤ 2 kS(t)φkL5x L10 kukL5x L10 t t (3.39) kukXt ≤ BkφkH 1 (R) . (3.40) Conversely, if the solution u of (3.37) is global and satisfies kukL5x L10 ≤ ǫ, t (3.41) kS(t)φkL5x L10 ≤ 2kukL5x L10 . t t (3.42) then ≤ A, as in Theorem 1.4 we can prove the local well-posedness for the IVP Proof. Since khkL∞ t (3.37) in H 1 (R) with time of existence T = T (kφkH 1 (R) , khkL∞ ). Let u ∈ C([0, Tmax ); H 1 (R)) be the maximal solution of the IVP (3.37). For 0 ≤ t < Tmax , we have that u(t) = S(t)φ + w(t), where w(t) = − Z t (3.43) S(t − t′ )h(t′ )∂x (uk+1 )(t′ ) dt′ . 0 Using (2.12) from Proposition 2.7 for admissible triples (5, 10, 0) and (∞, 2, 1), we obtain 5 ∞ ku k 1 2 kwkL5x L10 ≤ CAkuk+1 kL1x L2 ≤ CAkuk−4 kL∞ Lx L x LT T T ≤ T k−4 5 CAkukL ∞ H 1 kukL5 L10 x T T ≤ CAkuk5L5 L10 . x T (3.44) From (3.43) and (3.44) it follows that | kukL5x L10 − kS(t)φkL5x L10 | ≤ CAkuk5L5 L10 . T T x T (3.45) Thus, for all T ∈ (0, Tmax ) one has kukL5x L10 ≤ ǫ + CAkuk5L5 L10 . (3.46) CA(2ǫ)4 < 1/2, (3.47) T x T Choose ǫ = ǫ(A) such that and suppose that the estimate (3.38) holds. As the norm is continuous on T and vanishes at T = 0, using continuity argument, the estimate (3.46) and the choice of ǫ in (3.47), imply that kukL5x L10 Tmax ≤ 2ǫ. (3.48) SUPERCRITICAL KDV 19 Moreover, from (3.45) kukL5x L10 Tmax ≤ kS(t)φkL5x L10 Tmax + CAkuk5L5 L10 x Tmax (3.49) 4 ≤ kS(t)φkL5x L10 Tmax + CA(2ǫ) kukL5x L10 Tmax . Therefore, with the choice of ǫ satisfying (3.47), the estimate (3.49) yields kukL5x L10 Tmax ≤ 2kS(t)φkL5x L10 Tmax . (3.50) In what follows, we will show that Tmax = ∞. The inequalities (2.2), (2.3), (2.12) with admissible triples (5, 10, 0) and (∞, 2, 1), and Hölder’s inequality imply 2 4 kwkL∞ 1 + k∂x wkL∞ L2 + k∂x wkL∞ L2 + kwkL5 L10 + k∂x f kL5 L10 ≤ CAkuk 5 10 kukXT . (3.51) L L x x x x T H T T T T x T Now using (3.29), (3.30) and Hölder’s inequality, we have k∂x wkL20 L5/2 + kwkL4x L∞ ≤CAk∂x (uk+1 )kL20/17 L5/4 T x x T T k ≤CAku kL5/4 L5/2 k∂x ukL20 L5/2 x k−4 ≤CAku x T T 4 ∞ ku k 5/4 5/2 k∂x uk kL∞ 5/2 x LT L L L20 L x T x T k−4 4 ≤CAkukL ∞ H 1 kukL5 L10 k∂x uk 20 5/2 Lx LT x T T ≤CAkuk4L5 L10 k∂x ukL20 L5/2 . x T x (3.52) T Combining (3.51) and (3.52), we obtain kwkXT ≤ CAkuk4L5 L10 kukXT . x T (3.53) This estimate with (3.47) and (3.48) gives 1 kwkXT ≤ CA(2ǫ)4 kukXT < kukXT . 2 (3.54) 1 kukXT ≤ kS(t)φkXT + kwkXT ≤ CkφkH 1 (R) + kukXT , 2 (3.55) Using (3.43) we obtain for all T ∈ (0, Tmax ). Therefore, we have kukXTmax ≤ 2 CkφkH 1 (R) . (3.56) Hence, from the definition of kukXTmax , we have that kukL∞ T max H 1 (R) ≤ Cku(0)kH 1 (R) . (3.57) 20 M. PANTHEE AND M. SCIALOM Now, combining the local existence from Theorem 1.4 and the estimate (3.57), the blow-up alternative implies that Tmax = ∞. Finally, the estimates (3.50) and (3.56) yield (3.39) and (3.40) respectively with B = 2C. Conversely, let Tmax = ∞ and (3.41) holds. With the similar argument as in (3.45), we can get | kukL5x L10 − kS(t)φkL5x L10 | ≤ CAkuk5L5 L10 . t t x t (3.58) Thus, from (3.58) in view of (3.41) and (3.47), one has kS(t)φkL5x L10 ≤ kukL5x L10 + CAǫ4 kukL5x L10 ≤ 2kukL5x L10 . t t t t (3.59)  Corollary 3.5. Let h ∈ L∞ (R) satisfy khkL∞ ≤ A and ǫ and B be as in Proposition 3.4. Given φ ∈ H 1 (R), let u be the solution of the IVP (3.37) defined on the maximal interval [0, Tmax ). If there exists T ∈ (0, Tmax ) such that kS(t)u(T )kL5x L10 ≤ ǫ, t then the solution u is global. Moreover kukL5x L10 (T,∞) ≤ 2ǫ, and kukX(T,∞) ≤ Bku(T )kH 1 (R) . Proof. The proof follows by using a standard extension argument. For details we refer to the proof of Corollary 2.4 in [6].  4. Proof of the main results The argument in the proof of the main results, Theorem 1.5 and Theorem 1.6, is quite similar to the one used in the case of the critical KdV equation [5]. As mentioned earlier, Lemma 3.3 and the local existence Theorem 1.4 are used in the proof of Theorem 1.5. While, Proposition 3.4 and Theorem 1.5 are crucial in the proof of Theorem 1.6. He we adapt the techniques used in [5] and [6] to complete the proofs. Proof of Theorem 1.5. Let A = kgkL∞ , T ∈ (0, Smax ) fixed and set M0 = 2 sup kU (t)kH 1 (R) . (4.1) t∈[0,T ] In particular, for t = 0, (4.1) gives kφkH 1 (R) ≤ M0 /2. From Theorem 1.4, we have that for all ω, t0 ∈ R, uω,t0 exists on [0, δ]. Using (3.7) we have that the existence time δ, is given by δ= C A2 M08 . (4.2) SUPERCRITICAL KDV 21 Moreover, from (3.8) 1 ≤ CkφkH 1 (R) lim sup sup kuω,t0 kL∞ δ H (R) (4.3) ≤ CkφkH 1 (R) . lim sup sup kuω,t0 kL4x L∞ 1 δ H (R) (4.4) |w|→∞ t0 ∈R and |w|→∞ t0 ∈R From Lemma 3.3, we have that supt0 ∈R kuω,t0 − U kXT |w|→∞ sup kuω,t0 (δ) − U (δ)kH 1 (R) → 0, in particular |w|→∞ → 0. (4.5) t0 ∈R Combining (4.1) and (4.5), for |w| sufficiently large, we deduce that sup kuω,t0 (δ)kH 1 (R) ≤ M0 . (4.6) t0 ∈R We suppose δ ≤ T , otherwise we are done. Using Theorem 1.4 we can extend the solution uω,t0 (as in the proof of Corollary 3.5) on the interval [0, 2δ], with kũω,t0 kL∞ 1 ≤ t (0,δ)H (R) Ckũω,t0 (0)kH 1 (R) , where ũω,t0 (t) = uω,t0 (t + δ) i.e., kuω,t0 kL∞ 1 ≤ Ckuω,t0 (δ)kH 1 (R) ≤ t (δ,2δ)H (R) C 2 kφkH 1 (R) . Therefore, (4.3) gives lim sup sup kuω,t0 kL∞ 1 ≤ C(1 + C)kφkH 1 (R) . t (0,2δ)H (R) (4.7) |w|→∞ t0 ∈R Similarly, from (4.4), ≤ C(1 + C)kφkH 1 (R) . lim sup sup kuω,t0 kL4x L∞ 1 2δ H (R) (4.8) |w|→∞ t0 ∈R So, we can again apply the Lemma 3.3. Iterating this argument at a finite number of times with the same time of existence in each iteration, we see that lim sup sup kuω,t0 kL∞ ≤ CkφkH 1 (R) 1 T H (R) |w|→∞ t0 ∈R and ≤ CkφkH 1 (R) . lim sup sup kuω,t0 kL4x L∞ T |w|→∞ t0 ∈R The result is therefore a consequence of Lemma 3.3.  Proof of Theorem 1.6. Let ǫ ∈ (0, ǫ(A)), where ǫ(A) is as in Proposition 3.4. If T is sufficiently large, from (1.23), we have that ǫ . (4.9) 4 Applying Proposition 3.4 to the global solution Ũ (t) = U (t + T ), the inequality (3.42) gives kU kL5x L10 (T,∞) ≤ = 2kU kL5x L10 ≤ 2kŨ kL5x L10 = kS(t)Ũ (0)kL5x L10 kS(t)U (T )kL5x L10 t t t (T,∞) ǫ ≤ . 2 (4.10) 22 M. PANTHEE AND M. SCIALOM From this inequality and Corollary 3.5 we get kU kX(T ;∞) ≤ BkU (T )kH 1 (R) . (4.11) From Theorem 1.5 it follows that sup sup kuω,t0 (t) − U (t)kH 1 (R) → 0, as |ω| → ∞. (4.12) t0 ∈R 0≤t≤T Thus, if |w| is sufficiently large, the triangular inequality along with (4.12) gives + kS(t)U (T )kL5x L10 ≤ kS(t)uω,t0 (T ) − S(t)U (T )kL5x L10 kS(t)uω,t0 (T )kL5x L10 t t t ǫ ≤ kuω,t0 (T ) − U (T )kL2x + 2 (4.13) ≤ ǫ. Therefore, Corollary 3.5 implies that uω,t0 is global. Moreover, sup kuω,t0 kL5x L10 t0 ∈R (T,∞) ≤ 2ǫ, and kuω,t0 kX(T,∞) ≤ Bkuω,t0 (T )kH 1 (R) , (4.14) for |w| sufficiently large. Let M0 = sup0≤t≤T kU (t)kH 1 (R) , as in (4.1). Now, we move to prove (1.24). The inequalities (4.12) and (4.14) show that there exists L > 0 such that sup sup sup kuω,t0 (t)kH 1 (R) ≤ (1 + M0 ) + Bkuω,t0 (T )kH 1 (R) = M1 < ∞. (4.15) |w|≥L t0 ∈R t≥0 In what follows, we prove that uω,t0 → U in the k · kXt -norm, when |ω| → ∞. Using Duhamel’s formulas for uω,t0 and U we have uω,t0 (T + t) − U (T + t) = S(t)(uω,t0 (T ) − U (T )) Z t ′ ′ S(t − t′ )g(ω(T + t′ + t0 ))∂x (uk+1 − ω,t0 )(T + t )dt 0 + m(g) Z t ′ S(t − t )∂x (U k+1 ′ )(T + t )dt (4.16) ′ 0 =: I1 + I2 + I3 . Using properties of the unitary group S(t) we have by (4.12) that kI1 kXt = kS(t)(uω,t0 (T ) − U (T ))kXt ≤ Ckuω,t0 (T ) − U (T )kH 1 (R) |ω|→∞ → 0. (4.17) With the same argument as in (3.53), we have kI2 kXt ≤ CAkuω,t0 k4L5 L10 x (T,∞) kuω,t0 kX(T,∞) , (4.18) SUPERCRITICAL KDV 23 From (4.18), with the use of (4.14) and (4.15), we have kI2 kXt ≤ CA(2ǫ)4 BM1 . (4.19) As in I2 , using (4.9) and (4.11), we get kI3 kXt ≤ CAkU k4L5 L10 x (T,∞) kU kX(T,∞) (4.20)  ǫ 4 BM0 . ≤ CA 4 Now given β > 0, we choose ǫ > 0 sufficiently small (T sufficiently large) such that   CA(2ǫ)4 BM0 + BM1 < β/3 and |ω| sufficiently large, so that (4.16), (4.17), (4.19) and (4.20) imply kuω,t0 (t) − U (t)kX(T,∞) = kuω,t0 (T + t) − U (T + t)kXt ≤ kI1 kXt + kI2 kXt + kI3 kXt (4.21) < β. On the other hand, from Theorem 1.5, we have kuω,t0 (t) − U (t)kX(0,T ) = kuω,t0 (t) − U (t)kXT |ω|→∞ → 0. Therefore, from (4.21) and (4.22), we can conclude the proof of the theorem. (4.22)  References [1] F. K. Abdullaev, J. G. 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