Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 77, pp. 1–34. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu KDV TYPE ASYMPTOTICS FOR SOLUTIONS TO HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATIONS PAVEL I. NAUMKIN, ISAHI SÁNCHEZ-SUÁREZ Abstract. We consider the Cauchy problem for the higher-order nonlinear Schrödinger equation i∂tu− a 3 |∂x|3u− b 4 ∂4xu = λi∂x(|u|2u), (t, x) ∈ R+ × R, u(0, x) = u0(x), x ∈ R, where a, b > 0, |∂x|α = F−1|ξ|αF and F is the Fourier transformation. Our purpose is to study the large time behavior of the solutions under the non-zero mass condition ∫ u0(x)dx 6= 0. 1. Introduction We consider the Cauchy problem for the higher-order nonlinear Schrödinger equation i∂tu− a 3 |∂x|3u− b 4 ∂4 xu = i∂x(|u|2u), (t, x) ∈ R+ × R, u(0, x) = u0(x), x ∈ R, (1.1) where a, b > 0, |∂x|α = F−1|ξ|αF and F is the Fourier transformation defined by Fφ = 1√ 2π ∫ R e −ixξφdx, and its inverse by F−1φ = 1√ 2π ∫ R e ixξφ(ξ)dξ. Note that we have the relation u(−t, x) = u(t,−x), so we only consider the case t > 0. Note that for the classical solution of (1.1) we have the conservation laws ∫ R u(t, x)dx =∫ R u0(x)dx and ‖u(t)‖L2 = ‖u0‖L2 . Equation (1.1) arises in the context of high-speed soliton transmission in long- haul optical communication system [13]. Also it can be considered as a particular form of the higher order nonlinear Schrödinger equation introduced by [42] to de- scribe the nonlinear propagation of pulses through optical fibers. This equation also represents the propagation of pulses by taking higher dispersion effects into account than those given by the Schrödinger equation (see [10, 11, 25, 28, 38, 32, 43, 49]). Higher order nonlinear Schrödinger equations have been widely studied recently. For the local and global well-posedness of the Cauchy problem we refer to [5, 6, 40] and references cited therein. The dispersive blow-up was obtained in [2]. The existence and uniqueness of solutions to (1.1) were proved in [1, 19, 30, 33, 34, 39, 47, 50], and the smoothing properties of solutions were studied in [3, 8, 12, 30, 33, 2010 Mathematics Subject Classification. 35B40, 35Q35. Key words and phrases. Nonlinear Schrödinger equation; large time asymptotic behavior; critical nonlinearity; self-similar solutions. c©2020 Texas State University. Submitted June 2, 2018. Published July 22, 2020. 1 2 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 34, 36, 37, 48]. The blow-up effect for a special class of slowly decaying solutions of the Cauchy problem (1.1) was studied in [1]. In this article we are interested in the case of non zero total mass of the initial data ∫ R u0(x)dx 6= 0. Then by (1.1) we obtain the non-zero total mass for the solution ∫ R u(t, x)dx =∫ R u0(x)dx 6= 0 for all t > 0. We develop the factorization technique originated in our previous papers [22, 21, 26, 20, 44, 45, 46]. The case of zero total mass∫ R u(t, x)dx = ∫ R u0(x)dx = 0 is easier and the corresponding results can be ob- tained following the approach in [24]. We denote the Lebesgue space by Lp = {φ ∈ S′ : ‖φ‖Lp <∞}, with norms ‖φ‖Lp = (∫ |φ(x)|pdx )1/p for 1 ≤ p <∞ and ‖φ‖L∞ = supx∈R |φ(x)|. The weighted Sobolev space is Hm,s p = {ϕ ∈ S′ : ‖φ‖Hm,sp = ‖〈x〉s〈i∂x〉mφ‖Lp < ∞}, where m, s ∈ R, 1 ≤ p ≤ ∞, 〈x〉 =√ 1 + x2, 〈i∂x〉 = √ 1− ∂2 x. We also use the notations Hm,s = Hm,s 2 , Hm = Hm,0 shortly, if it does not cause any confusion. Let C(I;B) be the space of continuous functions from an interval I to a Banach space B. Different positive constants might be denoted by the same letter C. In [29], it was proved the existence of the self-similar solutions of the form u = t−1/3fm(xt−1/3) to the reduced equation i∂tu− 1 3 |∂x|3u = i∂x(|u|2u), (1.2) which is defined by the mean value m = ∫ R fm(x)dx 6= 0. Now we state the main result of this article. We show that the asymptotic behavior of the solutions to (1.1) resembles that of the KdV equation, which was studied intensively (see [15, 18, 23]). We denote by µ(x) the root of the equation Λ′(ξ) = ξ|ξ|(a+ b|ξ|) = x for all x ∈ R. Theorem 1.1. Suppose that ∫ R u0(x)dx 6= 0, and that the initial data u0 ∈ H1,1 have a sufficiently small norm ‖u0‖H1,1 ≤ ε. Then there exists a unique global solution u ∈ C([0,∞);H1,1) to the Cauchy problem (1.1). Furthermore there exists W+ ∈ L∞ such that the large time behavior satisfies u(t, x) = M√ itΛ′′(µ(xt )) W+ ( µ( x t ) ) exp ( iµ(xt ) |Λ′′(µ(xt ))| ∣∣W+ ( µ( x t ) )∣∣2 log t ) +O(t−1/3〈t1/3µ( x t )〉−3/4) +O(t− 1 3−δ) on the domain |x| ≥ t 1 3 +2γ and u(t, x) = t−1/3fm(xt−1/3) + O(t− 1 3−δ) on the do- main |x| ≤ t 1 3 +2γ , where γ, δ > 0 are small and t−1/3fm(xt−1/3) is the self-similar solution to equation (1.2) with m = ∫ R fm(x)dx = ∫ R u0(x)dx 6= 0. We organize the rest of the paper as follows. Section 2 is devoted to the fac- torization formulas and L2-boundedness of pseudodifferential operators. Then in Sections 3 and 4 we obtain estimates for the operator V and V∗, respectively, in EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 3 the uniform norm and the L2-estimates of commutators. In Section 5, we prove a priori estimates of the solution u(t) in the norm ‖u‖XT = sup t∈[1,T ] ( ‖ϕ̂‖L∞ + t−γ‖∂xJ u(t)‖L2 + t−γ‖∂−1 x Ibu(t)‖L2 + t−γ‖∂−1 x Pbu(t)‖L2 + t−γ‖u(t)‖H1 ) , where ϕ̂ = FU(−t)u(t), U(t) = F−1e−itΛ(ξ)F , Λ(ξ) = a 3 |ξ| 3 + b 4ξ 4, J = x − tΛ′(−i∂x), Ib = ∂b + it∂bΛ(−i∂x), Pb = 3t∂t + ∂xx + b∂b. Section 6 is devoted to the proof of Theorem 1.1. 2. Preliminaries 2.1. Factorization techniques. We define the free evolution group U(t) = F−1e−itΛ(ξ)F , Λ(ξ) = a 3 |ξ| 3 + b 4ξ 4. We write U(t)F−1φ = Dt √ |t| 2π ∫ R eit(xξ−Λ(ξ))φ(ξ)dξ, where Dtφ = |t|−1/2φ(xt ) is the dilation operator. There is a unique station- ary point ξ = µ(x) in the integral ∫ R e it(xξ−Λ(ξ))φ(ξ)dξ, which is defined as the root of the equation Λ′(ξ) = ξ|ξ|(a + b|ξ|) = x for all x ∈ R. Thus we ob- tain Λ′(µ(x)) = x for all x ∈ R. Also Λ′′(ξ) = 2|ξ|(a + 3 2b|ξ|) > 0 for all ξ ∈ R. We have Λ′(ξ) = O(ξ2〈ξ〉), therefore µ(x) = O({x}1/2〈x〉1/3). Define the scaling operator (Bφ)(x) = φ(µ(x)). Hence U(t)F−1φ = DtBMVφ, where M = e−it(Λ(η)−ηΛ′(η)), the operator V(t)φ = √ |t| 2π ∫ R e −itS(ξ,η)φ(ξ)dξ and the phase function S(ξ, η) = Λ(ξ) − Λ(η) − Λ′(η)(ξ − η). Denote Ak = M k 1 tΛ′′(η)∂ηM k, k = 0, 1. We have A1 = A0 + iη, also A1V = Viξ, [iη,V] = −A0V, therefore we obtain the commutator ∂ηV = −tΛ′′(η)[iη,V]. Since ∂ξS(ξ, η) = Λ′(ξ)− Λ′(η), we obtain the commutator it[Λ′(η),V]φ = −V∂ξφ. Also we use the representation for the inverse evolution group FU(−t)φ = V∗MB−1D−1 t , where the inverse dilation operator D−1 t φ = |t|1/2φ(xt) and the inverse scaling operator (B−1φ)(η) = φ(Λ′(η)), and the conjugate operator V∗(t)φ = √ |t| 2π ∫ R eitS(ξ,η)φ(η)Λ′′(η)dη. We have iξV∗φ = V∗A1φ. Hence [iξ,V∗] = V∗A0. Define a new dependent variable ϕ̂ = FU(−t)u(t). Since FU(−t)L = ∂tFU(−t) with L = ∂t+iΛ, where Λ = Λ(−i∂x) = F−1Λ(ξ)F , applying the operator FU(−t) to equation (1.1) Lu = ∂x(|u|2u), we obtain ∂tϕ̂ = FU(−t)Lu = FU(−t)∂x(|U(t)F−1ϕ̂|2U(t)F−1ϕ̂) = iξt−1V∗(|Vϕ̂|2Vϕ̂), (2.1) since the nonlinearity is gauge invariant. Also we mention that the operator J = U(t)xU(−t) = x−tΛ′, with Λ′ = Λ′(−i∂x) = F−1Λ′(ξ)F , plays a crucial role in the large time asymptotic estimates. Note that J commutes with L, i.e. [J ,L] = 0. Also we see that the symbol Λ(ξ) = a 3 |ξ| 3 + b 4ξ 4 satisfies the identities ξ∂ξΛ − b∂bΛ = 3Λ and ξ∂ξΛ + a∂aΛ = 4Λ. Hence we have the commutator relations [P̂a, e−itΛ(ξ)] = [P̂b, e−itΛ(ξ)] = 0, with P̂a = 4t∂t−ξ∂ξ−a∂a, P̂b = 3t∂t−ξ∂ξ+b∂b. 4 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 We also use the operators Pa = 4t∂t + ∂xx − a∂a, Pb = 3t∂t + ∂xx + b∂b and Ia = ∂a + it∂aΛ(−i∂x), Ib = ∂b + it∂bΛ(−i∂x), and the commutator relations [L, Ia] = [L, Ib] = 0 and [L,Pa] = 4L, [L,Pb] = 3L hold. Using the relation u(t) = U(t)F−1ϕ̂ = F−1e−itΛ(ξ)ϕ̂, we obtain Pau = F−1P̂ae−itΛ(ξ)ϕ̂ = F−1[P̂a, e−itΛ(ξ)]ϕ̂+ F−1e−itΛ(ξ)P̂aϕ̂ = U(t)F−1P̂aϕ̂, and Pbu = F−1P̂be−itΛ(ξ)ϕ̂= F−1[P̂b, e−itΛ(ξ)]ϕ̂+F−1e−itΛ(ξ)P̂bϕ̂= U(t)F−1P̂bϕ̂. Note that Iau = IaU(t)F−1ϕ̂ = F−1(∂a+it∂aΛ(ξ))Eϕ̂ = F−1E∂aϕ̂ = U(t)F−1∂aϕ̂, and Ibu = IbU(t)F−1ϕ̂ = F−1(∂b + it∂bΛ(ξ))Eϕ̂ = F−1E∂bϕ̂ = U(t)F−1∂bϕ̂. Hence ‖Iau‖L2 = ‖∂bϕ̂‖L2 and ‖Ibu‖L2 = ‖∂bϕ̂‖L2 . Also we have the identities Pa = 4tL+ ∂xJ − aIa and Pb = 3tL+ ∂xJ + bIb. 2.2. Boundedness of pseudodifferential operators. Define the pseudodiffer- ential operator a(x,D)φ ≡ ∫ R eixξa(x, ξ)φ̂(ξ)dξ. There are many papers devoted to the L2-estimates of pseudodifferential operator a(x,D) (see [4, 7, 9, 27]). Below we will use the following results on the L2- boundedness of pseudodifferential operator a(x,D) (see [27]). Lemma 2.1. Let the symbol a(x, ξ) be such that supx,ξ∈R |∂kx∂lξa(x, ξ)| ≤ C for k, l = 0, 1. Then ‖a(x,D)φ‖L2 x ≤ C‖φ‖L2 . Lemma 2.2. Let the symbol a(x, ξ) be such that supx∈R ‖∂lξa(x, ξ)‖L2 ξ ≤ C for l = 0, 1. Then ‖a(x,D)φ‖L2 x ≤ C‖φ‖L2 . Next we consider the time dependent pseudodifferential operator a(t, x,D)φ ≡ ∫ R eixξa(t, x, ξ)φ̂(ξ)dξ. Lemma 2.3. Let the symbol a(t, x, ξ) be such that sup x,ξ∈R,t≥1 |{ξ}−ν〈ξ〉ν(ξ∂ξ) ka(t, x, ξ)| ≤ C for k = 0, 1, 2, where ν ∈ (0, 1). Then ‖a(t, x,D)φ‖L2 x ≤ C‖φ‖L2 for all t ≥ 1. Proof. We define the kernel K(t, x, y) = ∫ R e iyξa(t, x, ξ)dξ, then we write a(t, x,D)φ = ∫ R eixξa(t, x, ξ)φ̂(ξ)dξ = ∫ R K(t, x, y)φ(x− y)dy. Integrating two times by parts via the identity eiyξ = H∂ξ(ξe iyξ) with H = (1 + iξy)−1 we obtain K(t, x, y) = ∫ R e iyξξ∂ξ(Hξ∂ξ(Ha(x, ξ)))dξ. By the condition in this lemma, we find |ξ∂ξ(Hξ∂ξ(Ha(t, x, ξ)))| ≤ C{ξ}ν〈ξ〉−ν 〈ξy〉2 for all x, y, ξ ∈ R. Hence we obtain the estimate |K(t, x, y)| ≤ C ∫ 1 0 ξνdξ 〈ξy〉2 + C ∫ ∞ 1 ξ−νdξ 〈ξy〉2 ≤ C|y|−1−ν ∫ |y| 0 ην〈η〉−2dη + C|y|ν−1 ∫ ∞ |y| η−ν〈η〉−2dη ≤ C|y|ν−1〈y〉−2ν . EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 5 Then by Young’s inequality we have ‖a(t, x,D)φ‖L2 = ‖ ∫ R K(t, x, y)φ(x− y)dy‖L2 ≤ C‖ ∫ R |y|ν−1〈y〉−2ν |φ(x− y)|dy‖L2 ≤ C ∥∥|x|ν−1〈x〉−2ν ∥∥ L1‖φ‖L2 ≤ C‖φ‖L2 . The proof is complete. � Similarly, by considering the conjugate operator. Lemma 2.4. Let the symbol a(t, x, ξ) be such that sup x,ξ∈R,t≥1 |{x}−ν〈x〉ν(x∂x)ka(t, x, ξ)| ≤ C for k = 0, 1, 2, where ν ∈ (0, 1). Then ‖a(t, x,D)φ‖L2 x ≤ C‖φ‖L2 for all t ≥ 1. 3. Estimates for the operator V Let χ ∈ C4(R) be such that χ(x) = 1 for |x| ≤ 1 and χ(x) = 0 for |x| ≥ 2. Define the cut off functions χj(z) ∈ C4(R), j = 1, 2, 3, such that χ1 + χ2 + χ3 ≡ 1, χ2(z) = 0 for z ≤ 1 3 or z ≥ 3 and χ2(z) = 1 for 2 3 ≤ z ≤ 3 2 , χ1(z) = 1 − χ2(z) for − 3 2 < z < 2 3 and χ1(z) = 0 for z ≥ 2 3 or z ≤ −3 and χ3(z) = 1 − χ2(z) for z > 3 2 , χ3(z) = 1 − χ1(z) for z < − 3 2 and χ3(z) = 0 for − 3 2 ≤ z ≤ 3 2 . Denote Ψ1(t, ξ, η) = (1 − χ(ηt1/3))χ1( ξη ), Ψ2(t, ξ, η) = (1 − χ(ηt1/3))χ2( ξη ), Ψ3(t, ξ, η) = (1− χ(ηt1/3))χ3( ξη ), Ψ4(t, ξ, η) = χ(ηt1/3)χ( 1 3ξt 1/3) and Ψ5(t, ξ, η) = χ(ηt1/3)(1− χ( 1 3ξt 1/3)) and consider the operators Vj(t)φ = √ |t| 2π ∫ R e−itS(ξ,η)Ψj(t, ξ, η)φ(ξ)dξ for j = 1, 2, 3, 4, 5. 3.1. Estimates for commutators. We first obtain the L2-estimate for the oper- ator V1. Lemma 3.1. Let the weight P ∈ C2(R \ 0) be such that ∂kηP (η) = O(|η|α1−k) for k = 0, 1, 2, with α1 ≤ 2. Assume that the weight Q ≡ 1 if α2 = 0, or Q ∈ C2(R \ 0) satisfies the estimate ∂kξQ(ξ) = O(|ξ|α2−k), k = 0, 1, 2, if α2 ≥ 1. Suppose that 2 ≤ α1 + α2 < 5/2. Then∥∥|Λ′′|1/2PtV1Qφ ∥∥ L2 ≤ C‖∂ξφ‖L2 + C|t| 13 ( 5 2−α1−α2)|φ(0)| for all t ≥ 1. Proof. Integrating by parts we obtain PtV1Qφ = C ( 1− χ(ηt1/3) ) P (η)|t|1/2 ∫ R e−itS(ξ,η)∂ξ (Q(ξ)χ1( ξη ) ∂ξS(ξ, η) φ(ξ) ) dξ = C|t|1/2 ∫ R e−itS(ξ,η)q1(t, η, ξ)φξ(ξ)dξ + C|t|1/2 ∫ R e−itS(ξ,η)q2(t, η, ξ) φ(ξ)− φ(0) ξ dξ + C|t|1/2φ(0) ∫ R e−itS(ξ,η)q3(t, η, ξ)dξ, 6 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 where qk(t, η, ξ) = (1− χ(t1/3η))P (η)(ξ∂ξ) k−1(Q(ξ)χ1( ξη ) 1 ∂ξS(ξ,η) ), k = 1, 2, q3(t, η, ξ) = (1− χ(t1/3η))P (η)∂ξ(Q(ξ)χ1( ξ η ) 1 ∂ξS(ξ, η) ). Next we change η = µ(x) (recalling that the inverse scaling operator (B−1φ)(η) = φ(Λ′(η)) and µ(x) is the root of the equation Λ′(η) = x), we obtain PtV1Qφ = MB−1|t|1/2 ∫ R eitxξq1(t, µ(x), ξ)e−itΛ(ξ)φξ(ξ)dξ +MB−1|t|1/2 ∫ R eitxξq2(t, µ(x), ξ)e−itΛ(ξ)φ(ξ)− φ(0) ξ dξ + φ(0)MB−1|t|1/2 ∫ R eitxξq3(t, µ(x), ξ)e−itΛ(ξ)dξ Also we change the variable of integration ξ = t−1/3ξ′; then PtV1Qφ = MB−1D−1 t2/3 (∫ R eixξq1(t, µ(xt−2/3), ξt−1/3)Dt1/3e−itΛ(ξ)φξ(ξ)dξ + ∫ R eixξq2(t, µ(xt−2/3), ξt−1/3)Dt1/3e−itΛ(ξ)φ(ξ)− φ(0) ξ dξ + φ(0) ∫ R eixξq3(t, µ(xt−2/3), ξt−1/3)Dt1/3e−itΛ(ξ)dξ ) , here D−1 t2/3 φ(x) = |t|1/3φ(xt2/3) and Dt1/3φ(ξ) = |t|−1/6φ(ξt−1/3). Define the pseudodifferential operators ak(t, x,D)φ ≡ ∫ R e ixξak(t, x, ξ)φ̂(ξ)dξ with symbols ak(t, x, ξ) = qk(t, µ(xt−2/3), ξt−1/3). Then we obtain PtV1Qφ = MB−1D−1 t2/3 ( a1(t, x,D)F−1Dt1/3e−itΛ(ξ)φξ(ξ) + a2(t, x,D)F−1Dt1/3e−itΛ(ξ)φ(ξ)− φ(0) ξ + φ(0)a3(t, x,D)F−1Dt1/3e−itΛ(ξ) ) . Let us prove the L2-boundedness of the operators ak(t, x,D), k = 1, 2. We obtain ak(t, x, ξ) = ( 1− χ ( t1/3µ(xt−2/3) )) P (µ(xt−2/3)) × (ξ∂ξ) k−1 (Q(ξt−1/3)χ1(ξt−1/3/µ(xt−2/3)) Λ′(ξt− 1 3 )− xt−2/3 ) . Note that µ(x) = O(|x|1/2) for small |x|, therefore (1 − χ(t1/3µ(xt−2/3))) 6= 0 for |x| ≥ C > 0, t ≥ 1. Also we have χ1(z) 6= 0 for −3 < z < 2/3, hence |Λ′(ξt−1/3)− xt−2/3| = ∣∣ ∫ ξt−1/3 µ(xt− 2 3 ) Λ′′(η)dη ∣∣ ≥ ∣∣ ∫ 2 3µ(xt−2/3) µ(xt−2/3) Λ′′(η)dη ∣∣ ≥ 1 3 |µ(xt−2/3)|Λ′′(2 3 µ(xt−2/3)) ≥ C|Λ′(µ(xt−2/3))| = C|x|t−2/3. Note that |∂kx∂lξak(t, x, ξ)| ≤ C for all x, ξ ∈ R, t ≥ 1,k, l = 0, 1, if 2 ≤ α1 + α2 ≤ 3. Therefore by Lemma 2.1 we have ‖ak(t, x,D)φ‖L2 x ≤ C‖φ‖L2 . Thus the EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 7 pseudodifferential operators ak(t, x,D) are L2-bounded for k = 1, 2. Then we find that ‖|Λ′′| 12MB−1D−1 t2/3 a1(t, x,D)F−1Dt1/3e−itΛ(ξ)φξ(ξ)‖L2 η ≤ ‖a1(t, x,D)F−1Dt1/3e−itΛ(ξ)φξ(ξ)‖L2 x ≤ C‖F−1Dt1/3e−itΛ(ξ)φξ(ξ)‖L2 = C‖∂ξφ‖L2 and by the Hardy inequality∥∥|Λ′′| 12MB−1D−1 t2/3 a2(t, x,D)F−1Dt1/3e−itΛ(ξ)φ(ξ)− φ(0) ξ ‖L2 η ≤ ‖a2(t, x,D)F−1Dt1/3e−itΛ(ξ)φ(ξ)− φ(0) ξ ∥∥ L2 x ≤ C‖F−1Dt1/3e−itΛ(ξ)φ(ξ)− φ(0) ξ ‖L2 ξ = C‖φ(ξ)− φ(0) ξ ‖L2 ξ ≤ C‖∂ξφ‖L2 . Now let us prove the L2 - L∞ boundedness of the pseudodifferential operator a3(t, x,D). We have a3(t, x, ξ) = q3 ( t, µ(xt−2/3), ξt−1/3 ) = ( 1− χ ( t1/3µ(xt−2/3) )) P ( µ(xt−2/3) ) × t1/3∂ξ (Q(ξt−1/3)χ1(ξt−1/3/µ(xt−2/3)) Λ′(ξt−1/3)− xt−2/3 ) . We consider two symbols a3,1(t, x, ξ) = a3(t, x, ξ)χ(ξ) and a3,2(t, x, ξ) = a3(t, x, ξ)(1− χ(ξ)). Note that |〈x〉ν(x∂x)j〈ξ〉δa3,1(t, x, ξ)| ≤ C|t| 13 (3−α1−α2) for all x, ξ ∈ R, t ≥ 1, j = 0, 1, 2 with some ν ∈ (0, 1), if α2 = 0 or α2 ≥ 1, α1 ≤ 2, 2 ≤ α1 + α2 < 5/2. Therefore by Lemma 2.4 we find that ‖a3,1(t, x,D)φ‖L2 x ≤ C|t| 13 (3−α1−α2)‖〈ξ〉−δφ̂‖L2 ≤ C|t| 13 (3−α1−α2)‖φ̂‖L∞ . Also we have |∂kx∂lξ〈ξ〉δa3,2(t, x, ξ)| ≤ C|t| 13 (3−α1−α2) for all x, ξ ∈ R, t ≥ 1, k, l = 0, 1, if 2 ≤ α1 + α2 < 5/2, δ > 1/2. Therefore by Lemma 2.1 we find that ‖a3,2(t, x,D)φ‖L2 x ≤ C|t| 13 (3−α1−α2)‖〈ξ〉−δφ̂‖L2 ≤ C|t| 13 (3−α1−α2)‖φ̂‖L∞ . Then ∥∥|Λ′′|1/2φ(0)MB−1D−1 t 2 3 a3(t, x,D)F−1D t 1 3 e−itΛ(ξ) ∥∥ L2 η ≤ |φ(0)|‖a3(t, x,D)F−1Dt1/3e−itΛ(ξ)‖L2 x ≤ C|t| 13 (3−α1−α2)|φ(0)|‖Dt1/3e−itΛ(ξ)‖L∞ξ ≤ C|t| 13 ( 5 2−α1−α2)|φ(0)|, which yields the result of the lemma. � In the next lemma we estimate the commutator [h,V2]. Lemma 3.2. Let the weights P ∈ C1(R \ 0) and Q ∈ C2(R \ 0) be such that ∂kηP (η) = O(|η|α1−k), k = 0, 1, and ∂kξQ(ξ) = O(|ξ|α2−k), k = 0, 1, 2. Suppose that 8 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 h(ξ) ∈ C4(R \ 0) is such that |∂kξ h(ξ)| ≤ C|ξ|α3−k for ξ ∈ R \ 0, 0 ≤ k ≤ 4. Assume that α1, α2 ∈ R, α3 ≥ 1, 2 ≤ α1 + α2 + α3 < 5/2. Then ‖|Λ′′|1/2Pt[h,V2]Qφ‖L2 ≤ C‖∂ξφ‖L2 + C|t| 13 ( 5 2−α1−α2−α3)|φ(0)| for all t ≥ 1. Proof. Integrating by parts we obtain Pt[h,V2]Qφ = C ( 1− χ ( ηt1/3 )) P (η)|t|1/2 ∫ R e−itS(ξ,η)∂ξ (h(η)− h(ξ) ∂ξS(ξ, η) ) Q(ξ)φ(ξ)χ2( ξ η )dξ = C|t|1/2 ∫ R e−itS(ξ,η)q1(η, ξ)φξ(ξ)dξ + C|t|1/2 ∫ R e−itS(ξ,η)q2(η, ξ) φ(ξ)− φ(0) ξ dξ + Cφ(0)|t|1/2 ∫ R e−itS(ξ,η)q3(η, ξ)dξ, where qk(η, ξ) = (1 − χ(ηt1/3))P (η)(ξ∂ξ) k−1(Q(ξ)χ2( ξη )h(η)−h(ξ) ∂ξS(ξ,η) ), q3(η, ξ) = (1 − χ(ηt1/3))P (η)∂ξ(Q(ξ)χ2( ξη )h(η)−h(ξ) ∂ξS(ξ,η) ). Next we change η = µ(x) and the variable of integration ξ = t−1/3ξ′, then we obtain Pt[h,V2]Qφ = MB−1D−1 t2/3 (∫ R eixξq1(t, µ(xt−2/3), ξt−1/3)Dt1/3e−itΛ(ξ)φξ(ξ)dξ + ∫ R eixξq2 ( t, µ(xt−2/3), ξt−1/3 ) Dt1/3e−itΛ(ξ)φ(ξ)− φ(0) ξ dξ + φ(0) ∫ R eixξq3 ( t, µ(xt−2/3), ξt−1/3 ) Dt1/3e−itΛ(ξ)dξ ) . We define the pseudodifferential operators ak(t, x,D)φ ≡ ∫ R e ixξak(t, x, ξ)φ̂(ξ)dξ with symbols ak(t, x, ξ) = qk(t, µ(xt−2/3), ξt−1/3). Then we obtain Pt[h,V2]Qφ = MB−1D−1 t2/3 ( a1(t, x,D)F−1Dt1/3e−itΛ(ξ)φξ(ξ) + a2(t, x,D)F−1Dt1/3e−itΛ(ξ)φ(ξ)− φ(0) ξ + φ(0)a3(t, x,D)F−1Dt1/3e−itΛ(ξ) ) . We will prove the L2-boundedness of ak(t, x,D). Note that χ2 ( ξt−1/3 µ(xt−2/3) ) 6= 0 for 1/3 < ξ t1/3µ(xt−2/3) < 3. Using the identities Λ′(ξ)−x = (ξ−µ(x)) ∫ 1 0 Λ′′(ξ+(µ(x)− ξ)τ)dτ , and h(µ(x))− h(ξ) = (µ(x)− ξ) ∫ 1 0 h′(ξ + (µ(x)− ξ)τ)dτ , we obtain ak(t, x, ξ) = − ( 1− χ ( t1/3µ(xt−2/3) )) P ( µ(xt−2/3) ) (ξ∂ξ) k−1 ( χ2 ( ξt−1/3/µ(xt−2/3) ) ×Q(ξt−1/3) ∫ 1 0 h′(ξt−1/3 + (µ(xt−2/3)− ξt−1/3)τ)dτ∫ 1 0 Λ′′(ξt−1/3 + (µ(xt−2/3)− ξt−1/3)τ)dτ ) for k = 1, 2. We have |∂kx∂lξak(t, x, ξ)| ≤ C for all x, ξ ∈ R, t ≥ 1, k, l = 0, 1, if α1, α2 ∈ R, α3 ≥ 1, 2 ≤ α1 + α2 + α3 < 5/2. Therefore by Lemma 2.1 we find that EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 9 ‖ak(t, x,D)φ‖L2 x ≤ C‖φ‖L2 . Thus the pseudodifferential operators ak(t, x,D) are L2-bounded for k = 1, 2. Then as above we have∥∥|Λ′′| 12MB−1D−1 t2/3 a1(t, x,D)F−1Dt1/3e−itΛ(ξ)φξ(ξ) ∥∥ L2 η ≤ C‖∂ξφ‖L2 and∥∥|Λ′′| 12MB−1D−1 t2/3 a2(t, x,D)F−1Dt1/3e−itΛ(ξ)φ(ξ)− φ(0) ξ ∥∥ L2 η ≤ C‖∂ξφ‖L2 . Now let us prove the L2 - L∞ boundedness of the pseudodifferential operator a3(t, x,D). We have a3(t, x, ξ) = − ( 1− χ ( t1/3µ(xt−2/3) )) P ( µ(xt−2/3) ) t1/3∂ξ ( Q(ξt−1/3) × χ2 ( ξt−1/3 µ(xt−2/3) ) ∫ 1 0 h′(ξt−1/3 + (µ(xt−2/3)− ξt−1/3)τ)dτ∫ 1 0 Λ′′(ξt−1/3 + (µ(xt−2/3)− ξt−1/3)τ)dτ ) . Note that |∂kx∂lξ〈ξ〉δa3(t, x, ξ)| ≤ C|t| 13 (3−α1−α2) for all x, ξ ∈ R, t ≥ 1, k, l = 0, 1, if α1, α2 ∈ R, α3 ≥ 1, 2 ≤ α1 +α2 +α3 < 5/2, δ > 1/2. Therefore by Lemma 2.1, we have ‖a3(t, x,D)φ‖L2 x ≤ C|t| 13 (3−α1−α2−α3)‖〈ξ〉−δφ̂‖L2 ≤ C|t| 13 (3−α1−α2−α3)‖φ̂‖L∞ . Then ∥∥|Λ′′|1/2φ(0)MB−1D−1 t 2 3 a3(t, x,D)F−1D t 1 3 e−itΛ(ξ) ∥∥ L2 η ≤ |φ(0)|‖a3(t, x,D)F−1Dt1/3e−itΛ(ξ)‖L2 x ≤ C|t| 13 (3−α1−α2−α3)|φ(0)|‖Dt1/3e−itΛ(ξ)‖L∞ξ ≤ C|t| 13 ( 5 2−α1−α2−α3)|φ(0)|, which yields the result of the lemma. � In the next lemma we estimate the operator V3. Lemma 3.3. Let the weights P ∈ C1(R \ 0) and Q ∈ C2(R \ 0) be such that ∂kηP (η) = O(|η|α1−k), k = 0, 1, and ∂kξQ(ξ) = O(|ξ|α2−k), k = 0, 1, 2. Assume that α1 ≥ 0, α2 ∈ R, 2 ≤ α1 + α2 ≤ 5/2. Then ∥∥|Λ′′|1/2PtV3Qφ ∥∥ L2 ≤ C‖∂ξφ‖L2 + C|t| 13 ( 5 2−α1−α2)|φ(0)| for all t ≥ 1. Proof. Integrating by parts we obtain PtV3Qφ = CP (η)|t|1/2 ∫ R e−itS(ξ,η)∂ξ 1 ∂ξS(ξ, η) Q(ξ)φ(ξ)Ψ3(t, ξ, η)dξ = C|t|1/2 ∫ R e−itS(ξ,η)q1(t, η, ξ)φξ(ξ)dξ + C|t|1/2 ∫ R e−itS(ξ,η)q2(t, η, ξ) φ(ξ)− φ(0) ξ dξ + Cφ(0)|t|1/2 ∫ R e−itS(ξ,η)q3(t, η, ξ)dξ, 10 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 where qk(t, η, ξ) = P (η) ( ξ∂ξ )k−1 (Q(ξ)Ψ3(t, ξ, η) 1 ∂ξS(ξ,η) ), q3(t, η, ξ) = P (η)∂ξ(Q(ξ)Ψ3(t, ξ, η) 1 ∂ξS(ξ,η) ). Next we change η = µ(x), and the variable of integration ξ = t−1/3ξ′, then we obtain PtV3Qφ = MB−1D−1 t2/3 (∫ R eixξq1(t, µ(xt−2/3), ξt−1/3)Dt1/3e−itΛ(ξ)φξ(ξ)dξ + ∫ R eixξq2(t, µ(xt− 2 3 ), ξt−1/3)Dt1/3e−itΛ(ξ)φ(ξ)− φ(0) ξ dξ + φ(0) ∫ R eixξq3 ( t, µ(xt−2/3), ξt−1/3 ) Dt1/3e−itΛ(ξ)dξ ) . Define the pseudodifferential operators ak(t, x,D)φ ≡ ∫ R e ixξak(t, x, ξ)φ̂(ξ)dξ with symbols ak(t, x, ξ) = qk ( t, µ(xt−2/3), ξt− 1 3 ) . Then we obtain PtV3Qφ = MB−1D−1 t2/3 ( a1(t, x,D)F−1Dt1/3e−itΛ(ξ)φξ(ξ) + a2(t, x,D)F−1Dt1/3e−itΛ(ξ)φ(ξ)− φ(0) ξ + φ(0)a3(t, x,D)F−1Dt1/3e−itΛ(ξ) ) . Let us prove the L2-boundedness of the operators ak(t, x,D), k = 1, 2. We have ak(t, x, ξ) = ( 1− χ ( t1/3µ(xt−2/3) )) P (µ(xt−2/3)) × (ξ∂ξ) k−1 (Q(ξt−1/3)χ3(ξ/(t1/3µ(xt−2/3))) Λ′(ξt−1/3)− xt−2/3 ) . Note that µ(x) = O(|x|1/2) for small |x|, therefore χ(t1/3µ(xt−2/3)) 6= 0 for |x| ≤ C, t ≥ 1. Also χ3(ξt−1/3/µ(xt−2/3)) 6= 0 for ξ t1/3µ(xt−2/3) > 3/2, hence |Λ′(ξt−1/3)− xt−2/3| = | ∫ ξt−1/3 µ(xt−2/3) Λ′′(η)dη| ≥ | ∫ ξt−1/3 2 3 ξt −1/3 Λ′′(η)dη| ≥ 1 3 |ξt−1/3|Λ′′ (2 3 ξt−1/3 ) ≥ C|Λ′(ξt−1/3)|. Note that |∂kx∂lξak(t, x, ξ)| ≤ C for all x, ξ ∈ R, t ≥ 1, k, l = 0, 1, if 2 ≤ α1 +α2 ≤ 3. Therefore by Lemma 2.1 we find that ‖ak(t, x,D)φ‖L2 x ≤ C|t|‖φ‖L2 . Thus the pseudodifferential operators ak(t, x,D) are L2-bounded for k = 1, 2. Then as above we have∥∥|Λ′′| 12MB−1D−1 t2/3 a1(t, x,D)F−1Dt1/3e−itΛ(ξ)φξ(ξ) ∥∥ L2 η ≤ C‖∂ξφ‖L2 ,∥∥|Λ′′| 12MB−1D−1 t2/3 a2(t, x,D)F−1Dt1/3e−itΛ(ξ)φ(ξ)− φ(0) ξ ∥∥ L2 η ≤ C‖∂ξφ‖L2 . Now let us prove the L2 - L∞ boundedness of the pseudodifferential operator a3(t, x,D). We have a3(t, x, ξ) = ( 1− χ ( t1/3µ(xt−2/3) )) P ( µ(xt−2/3) ) t1/3 × ∂ξ (Q(ξt−1/3)χ3(ξ/(t1/3µ(xt−2/3))) Λ′(ξt−1/3)− xt−2/3 ) . EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 11 Note that |∂kx∂lξ〈ξ〉δa3(t, x, ξ)| ≤ C|t| 13 (3−α1−α2) for all x, ξ ∈ R, t ≥ 1, k, l = 0, 1, if 2 ≤ α1 + α2 + δ ≤ 3, δ > 1/2. Therefore by Lemma 2.1 we have ‖a3(t, x,D)φ‖L2 x ≤ C|t| 13 (3−α1−α2)‖〈ξ〉−δφ̂‖L2 ≤ C|t| 13 (3−α1−α2)‖φ̂‖ L∞ . Thus as above we obtain ‖|Λ′′|1/2φ(0)MB−1D−1 t 2 3 a3(t, x,D)F−1D t 1 3 e−itΛ(ξ)‖L2 η ≤ C|t| 13 ( 5 2−α1−α2)|φ(0)|. The proof is complete. � Next we obtain the L2-estimate for the operator V4. Lemma 3.4. Let the weight P be such that P (η) = O(|η|α1), with α1 > −1. Assume that the weight Q satisfies the estimate Q(ξ) = O(|ξ|α2), if α2 > −1. Then∥∥|Λ′′|1/2PV4Qφ ∥∥ Lp ≤ C|t|− 1 3 ( 1 2 +α1+α2)− 1 3p (‖∂ξφ‖L2 + |t| 16 |φ(0)|) for all t ≥ 1, where 1 ≤ p ≤ ∞. Proof. Using the estimate |φ(ξ)− φ(0)| ≤ C|ξ|1/2‖∂ξφ‖L2 , we have ‖|Λ′′|1/2PV4Qφ‖Lp ≤ C|t|1/2‖|Λ′′|1/2P (η)χ(ηt1/3) ∫ R e−itS(ξ,η)χ( 1 3 ξt1/3)Q(ξ)φ(ξ)dξ‖Lp ≤ C|t|1/2‖|Λ′′|1/2P (η)χ(ηt1/3)‖Lp‖χ( 1 3 ξt1/3)Q(ξ)φ(ξ)‖L1 ≤ C|t|1/2‖∂ξφ‖L2‖|η| 12 +α1‖Lp(|η|≤2t−1/3)‖|ξ| 1 2 +α2‖L1(|ξ|≤6t−1/3) + C|t|1/2|φ(0)|‖|η| 12 +α1‖Lp(|η|≤2t−1/3)‖|ξ|α2‖L1(|ξ|≤6t−1/3) ≤ C|t|− 1 3 ( 1 2 +α1+α2)− 1 3p (‖∂ξφ‖L2 + |t|1/6|φ(0)|). The proof is complete. � In the next lemma we estimate the operator V5. Lemma 3.5. Let the weights P and Q ∈ C3(R \ 0) be such that P (η) = O(|η|α1), and ∂kξQ(ξ) = O(|ξ|α2−k), k = 0, 1, 2, 3. Assume that α1 ≥ 0, α2 ∈ R, −1 ≤ α1 + α2 ≤ 2. Then∥∥|Λ′′|1/2PV5Qφ ∥∥ L2 ≤ Ct− 1+α1+α2 3 (‖∂ξφ‖L2 + t 1 6 |φ(0)|) for all t ≥ 1. Proof. Integrating by parts we obtain PV5Qφ = Cχ(ηt1/3)P (η)|t|1/2 ∫ R e−itS(ξ,η)∂ξ 1 ∂ξS(ξ, η) Q(ξ) ( 1− χ( 1 3 ξt1/3) ) φ(ξ)dξ = C|t|1/2 ∫ R e−itS(ξ,η)q1(t, η, ξ)φξ(ξ)dξ + C|t|1/2 ∫ R e−itS(ξ,η)q2(t, η, ξ) φ(ξ)− φ(0) ξ dξ + Cφ(0)|t|1/2 ∫ R e−itS(ξ,η)q3(t, η, ξ)dξ, 12 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 where qk(t, η, ξ) = t−1χ(ηt1/3)P (η)(ξ∂ξ) k−1(Q(ξ)(1−χ( 1 3ξt 1/3)) 1 ∂ξS(ξ,η) ), q3(t, η, ξ) = t−1χ(ηt1/3)P (η)∂ξ(Q(ξ)(1−χ( 1 3ξt 1/3)) 1 ∂ξS(ξ,η) ). Next we change η = µ(x), and the variable of integration ξ = t−1/3ξ′, then we obtain PV5Qφ = MB−1D−1 t2/3 (∫ R eixξq1(t, µ(xt−2/3), ξt−1/3)Dt1/3e−itΛ(ξ)φξ(ξ)dξ + ∫ R eixξq2(t, µ(xt−2/3), ξt−1/3)Dt1/3e−itΛ(ξ)φ(ξ)− φ(0) ξ dξ + φ(0) ∫ R eixξq3(t, µ(xt−2/3), ξt−1/3)Dt1/3e−itΛ(ξ)dξ ) . Define the pseudodifferential operators ak(t, x,D)φ ≡ ∫ R e ixξak(t, x, ξ)φ̂(ξ)dξ with symbols ak(t, x, ξ) = qk(t, µ(xt−2/3), ξt− 1 3 ). Then we obtain PV5Qφ = MB−1D−1 t2/3 ( a1(t, x,D)F−1Dt1/3e−itΛ(ξ)φξ(ξ) + a2(t, x,D)F−1Dt1/3e−itΛ(ξ)φ(ξ)− φ(0) ξ + φ(0)a3(t, x,D)F−1Dt1/3e−itΛ(ξ) ) . Let us prove the L2-boundedness of the operators ak(t, x,D), k = 1, 2. We have ak(t, x, ξ) = t−1χ ( t1/3µ(xt−2/3) ) P (µ(xt−2/3))(ξ∂ξ) k−1 (Q(ξt−1/3)(1− χ( 1 3ξ)) Λ′(ξt−1/3)− xt−2/3 ) , Note that µ(x) = O(|x|1/2) for small |x|, therefore χ(t1/3µ(xt−2/3)) 6= 0 for |x| ≤ C, t ≥ 1. Also χ3(ξt−1/3/µ(xt−2/3)) 6= 0 for ξ t1/3µ(xt−2/3) > 3/2, hence |Λ′(ξt−1/3)− xt−2/3| = | ∫ ξt−1/3 µ(xt−2/3) Λ′′(η)dη| ≥ | ∫ ξt−1/3 2 3 ξt −1/3 Λ′′(η)dη| ≥ 1 3 |ξt−1/3|Λ′′(2 3 ξt−1/3) ≥ C|Λ′(ξt−1/3)|. Note that |〈ξ〉ν(ξ∂ξ) jak(t, x, ξ)| ≤ Ct− 1+α1+α2 3 for all x, ξ ∈ R, t ≥ 1, j = 0, 1, 2 with some ν ∈ (0, 1), if α1 ≥ 0, α2 ∈ R, −1 ≤ α1 + α2 < 2. Therefore by Lemma 2.3 we find ‖ak(t, x,D)φ‖L2 x ≤ Ct− 1+α1+α2 3 ‖φ‖L2 . Then as above we have∥∥|Λ′′|1/2MB−1D−1 t2/3 a1(t, x,D)F−1Dt1/3e−itΛ(ξ)φξ(ξ) ∥∥ L2 η ≤ Ct− 1+α1+α2 3 ‖∂ξφ‖L2 and ∥∥|Λ′′|1/2MB−1D−1 t2/3 a2(t, x,D)F−1Dt1/3e−itΛ(ξ)φ(ξ)− φ(0) ξ ∥∥ L2 η ≤ Ct− 1+α1+α2 3 ‖∂ξφ‖L2 . Now let us prove the L2 - L∞ boundedness of the pseudodifferential operator a3(t, x,D). We have a3(t, x, ξ) = t−1χ ( t1/3µ(xt−2/3) ) P (µ(xt−2/3))t1/3∂ξ (Q(ξt−1/3)(1− χ( 1 3ξ)) Λ′(ξt−1/3)− xt−2/3 ) , Note that |〈ξ〉ν(ξ∂ξ) j〈ξ〉δa3(t, x, ξ)| ≤ C|t|− 1 3 (α1+α2) for all x, ξ ∈ R, t ≥ 1, j = 0, 1, 2 with some small ν ∈ (0, 1), δ > 1/2. Therefore by Lemma 2.3 we find EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 13 ‖a3(t, x,D)φ‖L2 x ≤ C|t|− 1 3 (α1+α2)‖〈ξ〉−δφ̂‖L2 ≤ Ct− 1+α1+α2 3 t 1 6 ‖φ̂‖ L∞ . Thus we ob- tain∥∥|Λ′′|1/2φ(0)MB−1D−1 t2/3 a3(t, x,D)F−1Dt1/3e−itΛ(ξ) ∥∥ L2 η ≤ Ct− 1+α1+α2 3 t 1 6 |φ(0)|. The proof is complete. � Applying the above lemmas we obtain estimates for the derivatives, ∂ηVk = |Λ′′(η)|[η,Vk] + Vk∂η ( (1 − χ(ηt1/3))χk(ξ/η) ) , k = 1, 2, ∂ηV4 = |Λ′′(η)|[η,V4] + V4∂η(χ(ηt1/3)χ( 1 3ξt 1/3)). Note that ∂η((1 − χ(ηt1/3))χk(ξ/η)) ≤ Cη−1 ≤ Ctη2 since ηt1/3 ≥ 1 and ∂η(χ(ηt1/3)χ( 1 3ξt 1/3)) ≤ Cη−1 ≤ Ctη2. We choose in Lemma 3.1 and 3.4 P (η) = η1−j |Λ′′(η)|, Q(ξ) = ξj , or P (η) = η−j |Λ′′(η)| and Q(ξ) = ξ1+j . Also we choose in Lemma 3.2 h(ξ) = ξ, P (η) = η−j |Λ′′(η)|, Q(ξ) = ξj . Then we obtain Corollary 3.6. The following estimates hold∥∥|Λ′′|1/2η−j∂ηVkξjφ∥∥L2 ≤ C ( ‖∂ξφ‖L2 + t 1 6 |φ(0)| ) for all t ≥ 1, j ≥ 0, k = 1, 4; and∥∥|Λ′′|1/2η−j∂ηV2ξ jφ ∥∥ L2 ≤ C ( ‖∂ξφ‖L2 + t 1 6 |φ(0)| ) for all t ≥ 1, j ∈ Z. Choosing in Lemma 3.3 P (η) = η1−j |Λ′′(η)| andQ(ξ) = ξj , or P (η) = η−j |Λ′′(η)| and Q(ξ) = ξ1+j , and since ∂ηV3 = |Λ′′(η)|[η,V3]+V3(∂η(χ(ηt1/3)(1−χ( 1 3ξt 1/3)))), we note that ∂η ( (1− χ(ηt1/3))χ3(ξ/η) ) ≤ Cη−1 ≤ Ctη2 since ηt1/3 ≥ 1. Corollary 3.7. It holds∥∥|Λ′′|−1/2η1−j∂ηV3ξ jφ ∥∥ L2 ≤ C ( ‖∂ξφ‖L2 + t1/6|φ(0)| ) for all t ≥ 1, j = 0, 1. Choosing in Lemma 3.5 P (η) = η1−j |Λ′′(η)| andQ(ξ) = ξj , or P (η) = η−j |Λ′′(η)| and Q(ξ) = ξ1+j , since ηt1/3 ≤ 2 in V5 and ∂ηV5 = |Λ′′(η)|[η,V5] + V5(∂η((1 − χ(ηt1/3))χ3(ξ/η))), we note that ∂η(χ(ηt1/3)(1− χ( 1 3ξt 1/3))) ≤ Cη−1 ≤ Ctη2. Corollary 3.8. It holds ‖|Λ′′|−1/2∂ηV5ξ jφ‖L2 ≤ Ct 1−j 3 (‖∂ξφ‖L2 + t 1 6 |φ(0)|) for all t ≥ 1, j = 0, 1. 3.2. Asymptotic behavior. Define the kernel Aj(t, η) = √ |t| 2π ∫ R e−itS(ξ,η)Ψ2(t, ξ, η)ξjdξ. With the changing of variables ξ = ηy, we obtain Aj(t, η) = (1− χ(ηt1/3))|η|ηj √ |t| 2π ∫ 3 1/3 e−it|η| 3G(y,η)χ2(y)yjdy, 14 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 where S(ηy, η) = Λ(ηy)− Λ(η)− ηΛ′(η)(y − 1) = |η|3G(y, η) and G(y, η) = (1 3 (y + 2) + b 4 |η|(y 3 + 2y2 + 3))(y − 1)2, y > 0. To study the asymptotic behavior of the kernel Aj(t, η) for large t we apply the stationary phase method (see [14, p. 110])∫ R eizg(y)f(y)dy = eizg(y0)f(y0) √ 2π z|g′′(y0)| ei π 4 sgn g′′(y0) +O ( z−3/2 ) (3.1) for z → +∞, where the stationary point y0 is defined by the equation g′(y0) = 0. By (3.1) with g(y) = −G(y, η), f(y) = χ2(y)yj and y0 = 1, we obtain Aj(t, η) = t1/2|η|ηj√ i〈tη3〉Λ ′′(η) |η| +O ( t1/2η1+j〈η〉−1〈tη3〉−1 ) for t1/3|η| → ∞. Also since Λ′′(η) = O(|η|〈η〉) we have the estimate |Aj(t, η)| ≤ Ct 1 2 |η|j+1〈tη3〉−1/2〈η〉−1/2. By the Sobolev embedding theorem and Lemma 3.1 we have ‖|Λ′′|1/2〈tη3〉η−1−jV1ξ jφ‖L2 ≤ C(t 1 6 |φ(0)|+ ‖∂ξφ‖L2) and using Corollary 3.6, ‖|Λ′′|1/2η−j∂ηV1ξ jφ‖L2 ≤ C(t 1 6 |φ(0)|+ ‖∂ξφ‖L2) we obtain ‖〈tη3〉1/2η−jV1ξ jφ‖L∞ ≤ C‖|Λ′′|1/2〈tη3〉η−1−jV1ξ jφ‖1/2L2 ‖|Λ′′|1/2η−j∂ηV1ξ jφ‖1/2L2 + C‖|Λ′′|1/2〈tη3〉η−1−jV1ξ jφ‖L2 ≤ C ( t1/6|φ(0)|+ ‖∂ξφ‖L2 ) . for j ≥ 0, Similarly by the Sobolev embedding theorem and Lemma 3.3 we have ‖|Λ′′|1/2〈tη3〉|η|−1−jV3ξ jφ‖L2 ≤ C‖∂ξφ‖L2 + Ct 1 6 |φ(0)|. Then by Corollary 3.7, ‖|Λ′′|1/2η−j∂ηV3ξ jφ‖L2 ≤ C‖∂ξφ‖L2 + Ct 1 6 |φ(0)|, we have ‖〈tη3〉 12− j 3V3ξ jφ‖L∞ ≤ Ct−j/3‖|Λ′′|1/2〈tη3〉|η|−1−jV3ξ jφ‖1/2L2 ‖|Λ′′|1/2η−j∂ηV3ξ jφ‖1/2L2 + Ct−j/3‖|Λ′′|1/2〈tη3〉|η|−1−jV3ξ jφ‖L2 ≤ Ct−j/3(t 1 6 |φ(0)|+ ‖∂ξφ‖L2) for j = 0, 1. Similarly by the Sobolev embedding theorem and Lemma 3.5 we have ‖|Λ′′|1/2V5ξ jφ‖L2 ≤ Ct− 1+j 3 (‖∂ξφ‖L2 + Ct 1 6 |φ(0)|). Then by Corollary 3.8, ‖|Λ′′|−1/2∂ηV5ξ jφ‖L2 ≤ Ct 1−j 3 (‖∂ξφ‖L2 + t1/6|φ(0)|) for j = 0, 1, and ‖V5ξ jφ‖L∞ ≤ C‖|Λ′′|1/2V5ξ jφ‖1/2L2 ‖|Λ′′|−1/2∂ηV5ξ jφ‖1/2L2 ≤ Ct−j/3(t1/6|φ(0)|+ ‖∂ξφ‖L2) for j = 0, 1. In the next lemma we estimate the operators V2 in the uniform norm. We denote {η} = |η|〈η〉−1. Lemma 3.9. The estimate 〈t1/3η〉3/4|V2ξ jφ−Ajφ| ≤ C‖∂ξφ‖L2 holds for all t ≥ 1 if j ≥ 0. EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 15 Proof. We integrate by parts via identity e−itS(ξ,η) = H1∂ξ((ξ − η)e−itS(ξ,η)) with H1 = (1− it(ξ − η)∂ξS(ξ, η))−1, to obtain V2ξ jφ−Ajφ = Ct1/2 ∫ R e−itS(ξ,η)(φ(ξ)− φ(η))(ξ − η)∂ξ(H1Ψ2(t, ξ, η)ξj)dξ + Ct1/2 ∫ R e−itS(ξ,η)(ξ − η)H1Ψ2(t, ξ, η)ξj∂ξφ(ξ)dξ. Note that Λ′′(ξ) > 0 for all ξ > 0, and ∂kξΛ(ξ) = O({ξ}3−k〈ξ〉4−k), k = 0, 1, 2. Also ∂ξS(ξ, η) = Λ′(ξ)− Λ′(η). Hence we have |H1Ψ2(t, ξ, η)ξj |+ |(ξ − η)∂ξ(H1Ψ2(t, ξ, η)ξj)| ≤ C|η|j 1 + t|η|〈η〉(ξ − η)2 on the domain 1/3 < ξ η < 3. Therefore we obtain |V2ξ jφ−Ajφ| ≤ Ct1/2|η|j ∫ 1/3<ξ/η<3 |φ(ξ)− φ(η)| |ξ − η| |ξ − η|dξ 1 + t|η|〈η〉(ξ − η)2 + Ct1/2|η|j ∫ 1/3<ξ/η<3 |ξ − η||∂ξφ(ξ)|dξ 1 + t|η|〈η〉(ξ − η)2 . By the Hardy inequality we have ∫ 1/3<ξ/η<3 |φ(ξ)−φ(η)|2 |ξ−η|2 dξ ≤ C‖∂ξφ‖2L2 , then by the Cauchy-Schwarz inequality, |V2ξ jφ − Ajφ| ≤ Ct1/2|η|j‖∂ξφ‖L2I1/2, where I =∫ 1/3<ξ/η<3 (ξ−η)2dξ (1+t|η|〈η〉(ξ−η)2)2 . Changing ξ = ηy we have I ≤ C|η|3 ∫ 3 1/3 (y − 1)2dy (1 + |t||η|3〈η〉(y − 1)2)2 ≤ C|η|3〈t|η|3〈η〉〉−3/2. Thus we obtain |V2ξ jφ − Ajφ| ≤ Ct1/2‖∂ξφ‖L2 |η|j+ 3 2 〈t|η|3〈η〉〉−3/4. The proof is complete. � 4. Estimates for the operator V∗ 4.1. Asymptotic behavior. Define the kernel A∗(t, ξ) = √ |t| 2π ∫ R eitS(ξ,η)χ2(ηξ−1)|Λ′′(η)|dη for ξ 6= 0. Changing variables, η = ξy, we obtain A∗(t, ξ) = |ξ| √ |t| 2π ∫ 3 1/3 eit|ξ| 3G̃(y,ξ)χ2(y)|Λ′′(ξy)|dy, where S(ξ, ξy) = Λ(ξ)−Λ(ξy)−ξΛ′(ξy)(1−y) = |ξ|3G̃(y, ξ) with G̃(y, ξ) = (a3 (2y+ 1) + b 4 |ξ|(3y 2 + 2y + 1))(y − 1)2, y > 0. Then by (3.1) with g(y) = G̃(y, ξ), f(y) = χ2(y)|Λ′′(ξy)|, y0 = 1, we obtain A∗(t, ξ) = t1/2ξ2 √ iΛ′′(ξ) 〈tξ3〉|ξ| +O ( t1/2{ξ}2〈ξ〉3〈tξ3〉−1 ) for t|ξ|3 →∞. Since Λ′′(ξ) = O(ξ〈ξ〉) we have |A∗(t, ξ)| ≤ Ct1/2ξ2〈tξ3〉−1/2〈ξ〉1/2. In the next lemma we study the asymptotic behavior of V∗. We denote ξ̂ = ξt1/3 and define ‖φ‖Iα,β = ‖{η}α〈η̂〉−β〈η〉σ∂ηφ‖L2 + ‖{η}α−1〈η̂〉−β〈η〉σφ‖L2 , 16 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 where σ = 3 4 + 2β. Lemma 4.1. Let 1 4 + 2β ≤ α < 5 2 − 2β, 0 ≤ β ≤ 11 16 . Then ‖〈ξ̂〉β(V∗φ−A∗φ)‖L∞ ≤ C max(t 2β 3 − 1 2 , t− 1 4 , t α−1 3 )‖φ‖Iα,β for all t ≥ 1. Proof. We write V∗φ−A∗φ = √ |t| 2π ∫ R eitS(ξ,η)(φ(η)− φ(ξ))Λ′′(η)χ2(ηξ−1)dη + √ |t| 2π ∫ R eitS(ξ,η)φ(η)Λ′′(η)(1− χ2(ηξ−1))dη = I1 + I2 for ξ 6= 0. In the integral I1 we use the identity eitS(ξ,η) = H3∂η((η − ξ)eitS(ξ,η)) with H3 = (1 + it(η − ξ)∂ηS(ξ, η))−1, ∂ηS(ξ, η) = −Λ′′(η)(ξ − η), and integrate by parts I1 = Ct1/2 ∫ R eitS(ξ,η)φ(η)− φ(ξ) η − ξ (η − ξ)2∂η(H3Λ′′(η)χ2(ηξ−1))dη + Ct1/2 ∫ R eitS(ξ,η)(η − ξ)H3Λ′′(η)χ2(ηξ−1)∂ηφ(η)dη. Then applying the estimates Λ′′(η) = O({η}〈η〉2), we obtain |(η − ξ)H3|Λ′′(η)|{η}−α〈η̂〉β〈η〉−σχ2(ηξ−1)| + |(η − ξ)2〈η̂〉β〈η〉−σ∂η ( H3|Λ′′(η)|{η}−αχ2(ηξ−1) ) | ≤ C{ξ}1−α〈ξ〉2−σ〈ξ̂〉β |η − ξ| 1 + t{ξ}〈ξ〉2(η − ξ)2 in the domain 1/3 ≤ η ξ ≤ 3. If |κ(x)| ≤ C|κ(yx)| for all y ∈ (0, 1) then we find the Hardy inequality ‖κ(x) x (φ(x)− φ(0))‖L2 = ‖κ(x) x ∫ x 0 φ′(y)dty‖L2 = ‖κ(x) ∫ 1 0 φ′(xy)dy‖L2 ≤ C‖ ∫ 1 0 κ(xy)φ′(xy)dy‖L2 ≤ ∫ 1 0 ‖κ(xy)φ′(xy)‖L2dt ≤ ∫ 1 0 y−1/2dy‖κφ′‖L2 ≤ 2‖κφ′‖L2 . Hence ∫ 1/3≤η/ξ≤3 |φ(η)−φ(ξ)|2 (η−ξ)2 {η}2α〈η̂〉−2β〈η〉2σdη ≤ C‖φ‖2Iα,β . Therefore 〈ξ̂〉β |I1| ≤ Ct1/2{ξ}1−α〈ξ〉2−σ〈ξ̂〉2β‖φ‖Iα,βI 1/2 3 , where I3 = ∫ 1/3≤η/ξ≤3 (η−ξ)2dη (1+t{ξ}〈ξ〉2(η−ξ)2)2 . Changing η = yξ, we have I3 ≤ C|ξ|3 ∫ 3 1/3 (1− y)2dy( 1 + t{ξ}3〈ξ〉4(1− y)2 )2 ≤ C|ξ|3〈t|ξ|3〈ξ〉〉−3/2. Hence 〈ξ̂〉β |I1| ≤ Ct1/2{ξ} 5 2−α〈ξ〉2+ 3 2−σ〈ξt1/3〉2β〈t|ξ|3〈ξ〉〉−3/4‖φ‖Iα,β EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 17 ≤ C max(t− 1 4 , t α−1 3 )‖φ‖Iα,β if 1 4 +2β ≤ α ≤ 5/2. In the integral I2, using the identity eitS(ξ,η) = H4∂η(ηeitS(ξ,η)) with H4 = (1 + itη∂ηS(ξ, η))−1, ∂ηS(ξ, η) = −Λ′′(η)(ξ − η), we integrate by parts I2 = Ct1/2 ∫ ∞ −∞ eitS(ξ,η) {η}α〈η〉σφ(η) η〈η̂〉β η2∂η ( H4(1− χ2(ηξ−1)) Λ′′(η)〈η̂〉β {η}α〈η〉σ ) dη + Ct1/2 ∫ ∞ −∞ eitS(ξ,η)ηH4(1− χ2(ηξ−1)) Λ′′(η)〈η̂〉β {η}α〈η〉σ ∂η ({η}α〈η〉σ 〈η̂〉β φ(η) ) dη. Then using the estimates ∂ηS(ξ, η) = O(|η|〈η〉(ξ + |η|)) in the domains η ξ < 2 3 and η ξ ≥ 3 2 , we obtain∣∣η2∂η ( H4(1− χ2(ηξ−1)) Λ′′(η)〈η̂〉β {η}α〈η〉σ )∣∣+ ∣∣ηH4(1− χ2(ηξ−1)) Λ′′(η)〈η̂〉β {η}α〈η〉σ ∣∣ ≤ C{η}2−α〈η̂〉β〈η〉3−σ 1 + tη2〈η〉(|ξ|+ |η|) . Therefore by Hardy’s inequality 〈ξ̂〉β |I2| ≤ Cβt1/2‖∂η({η}α〈η̂〉−β〈η〉σφ(η))‖L2I 1/2 3 ≤ Ct1/2‖φ‖Iα,βI 1/2 3 , where I3 = ∫ R 〈ξ̂〉2β〈η̂〉2β{η}4−2α〈η〉6−2σdη (1+tη2〈η〉(|ξ|+|η|))2 . We have I3 ≤ C ∫ 1 0 〈ξ̂〉2βη4−2α〈η̂〉2βdη (1 + tη2(|ξ|+ |η|))2 + Ct 2β 3 −2 ∫ ∞ 1 〈ξ̂〉2βη2β−2σ(|ξ|+ |η|)−2dη ≤ C ∫ 1 0 η4−2α(1 + t 2β 3 η2β)dη (1 + tη3)2 + Ct 2β 3 ∫ 1 0 |ξ|2β(1 + t 2β 3 η2β)η4−2αdη (tη2|ξ|)2β(1 + tη3)2−2β + Ct 4β 3 −2 ∫ ∞ 1 η4β−2σ−2dη ≤ Ct 2α−5 3 ∫ t1/3 0 η4+2β−2α〈η〉−6dη + Ct 2α−5 3 ∫ t1/3 0 η4−2β−2α〈η〉6β−6dη + Ct 4β 3 −2 ≤ C max(t 4β 3 −2, t 2α−5 3 ) if − 1 2 + 2β < α < 5 2 − 2β with 0 ≤ β ≤ 1. Then 〈ξ̂〉β |I2| ≤ C max(t 2β 3 − 1 2 , t α−1 3 )‖φ‖Iα,β , and the proof is complete. � 4.2. Estimates for commutators. In the next lemma we estimate the commu- tator [|ξ|,V∗]. Lemma 4.2. t‖[|ξ|,V∗]φ‖L2 ≤ C‖|Λ′′|−1/2∂ηφ‖L2 +C‖|Λ′′|−1/2η−1φ‖L2 holds for all t ≥ 1. Proof. Integrating by parts and using ∂ηS(ξ, η) = −|Λ′′(η)|(ξ − η), we obtain t[|ξ|,V∗]φ = Ct|t|1/2 ∫ R eitS(ξ,η)(|ξ| − |η|)φ(η)|Λ′′(η)|dη = C|t|1/2 ∫ R eitS(ξ,η)∂η ( |ξ| − |η| ξ − η φ(η) ) dη = C|t|1/2 ∫ R eitS(ξ,η)q1(η, ξ)∂ηφ(η)dη + C|t|1/2 ∫ R eitS(ξ,η)q2(η, ξ) φ(η) η dη, 18 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 where q1(η, ξ) = |ξ|−|η| ξ−η = ξ+η |ξ|+|η| and q2(η, ξ) = η∂η ( |ξ|−|η| ξ−η ) = |ξ|η (|ξ|+|η|)2 . Next we define the pseudodifferential operators a∗k(ξ,D)φ = C ∫ R e −ixξak(x, ξ)φ̂(x)dx, with symbols ak(x, ξ) = qk(µ(x), ξ), then changing the variable of integration η = µ(x), we obtain t[|ξ|,V∗]φ = a∗1(ξ,D)F−1DtB ( M |Λ′′| ∂ηφ ) + a∗2(ξ,D)F−1DtB ( M |Λ′′| φ η ) . We prove the L2-boundedness of the pseudodifferential operators a∗k(ξ,D) by con- sidering the adjoint operators ak(x,D)φ = ∫ R eixξak(x, ξ)φ̂(ξ)dξ. Define the symbols a1,1(x, ξ) = ξ |ξ|χ(x)(1 − χ(ξ)), a2,1(x, ξ) = 0, ak,2(x, ξ) = ak(x, ξ)χ(x)(1− χ(ξ))− ak,1(x, ξ), ak,3(x, ξ) = ak(x, ξ)(1− χ(x)), and ak,4(x, ξ) = ak(x, ξ)χ(x)χ(ξ). Applying the Plancherel theorem we see that ‖ak,1(x,D)φ‖L2 x ≤ C‖φ‖ L2 . Note that the symbols a1,2(x, ξ) = χ(x)(1− χ(ξ)) µ(x)− |µ(x)| |ξ|+ |µ(x)| , a2,2(x, ξ) = χ(x)(1− χ(ξ)) |ξ|µ(x) (|ξ|+ |µ(x)|)2 are such that |{ξ}−1〈ξ〉(ξ∂ξ)lak,2(x, ξ)| ≤ C for all x, ξ ∈ R, l = 0, 1, 2. Then by Lemma 2.3 we have the estimate ‖ak,2(x,D)φ‖L2 x ≤ C‖φ‖L2 for all t ≥ 1. The symbol ak,3(x, ξ) satisfies the estimate supx,ξ∈R |∂jx∂lξak,3(x, ξ)| ≤ C for j, l = 0, 1, then by Lemma 2.1 we have ‖ak,3(x,D)φ‖ L2 x ≤ C‖φ‖L2 . By the Cauchy-Schwarz inequality we find that | ∫ R eixξak,4(x, ξ)φ̂(ξ)dξ| ≤ |χ(x)|‖χ(ξ)‖L2‖φ̂‖L2 = C|χ(x)|‖φ‖L2 . Then we obtain ‖ak,4(x,D)φ‖L2 x = ‖ ∫ R eixξak,4(x, ξ)φ̂(ξ)dξ‖L2 ≤ C‖χ(x)‖L2‖φ‖L2 ≤ C‖φ‖L2 . Therefore the pseudodifferential operator ak(x,D) = ∑4 j=1 ak,j(x,D) is L2-bounded. Hence t‖[|ξ|,V∗]φ‖L2 ≤ ∥∥a∗1(ξ,D)F−1DtB ( M |Λ′′| ∂ηφ )∥∥ L2 + ∥∥a∗2(ξ,D)F−1DtB ( M |Λ′′| φ η )∥∥ L2 ≤ C‖B( M |Λ′′| ∂ηφ)‖L2 + C‖B( M |Λ′′| φ η )‖L2 ≤ C‖|Λ′′|−12∂ηφ‖L2 + C‖|Λ′′|−1/2η−1φ‖L2 . The proof is complete. � In the next lemma we estimate the operator V∗Kj , V∗Kjφ = √ |t| 2π ∫ ∞ −∞ eitS(ξ,η)Kj(ξ, η)φ(η)Λ′′(η)dη, EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 19 where the symbols K3(ξ, η) = ((iξ)2−(iη)2)η2 Z(ξ,η) and K4(ξ, η) = (|ξ|(iξ)−|η|(iη))η2 Z(ξ,η) with Z(ξ, η) = (Λ(ξ)− 2Λ(η)) ( 1− χ( ξ 4η ) ) +2(Λ(ξ) ξ − Λ(η) η )ηχ( ξ 4η ). Lemma 4.3. The estimate ‖V∗Kjφ‖L2 ≤ C‖|Λ′′|1/2ηφ‖L2 holds for all t ≥ 1 and j = 3, 4. Proof. We define the operators a∗j (ξ,D)φ = C ∫ R e −ixξaj(x, ξ)φ(x)dx, with symbols aj(x, ξ) = 1 µ(x)Kj(ξ, µ(x)), then changing the variable of integration η = µ(x), we obtain V∗Kjφ = a∗j (ξ,D)F−1DtBMηφ. We prove the L2-boundedness of the pseu- dodifferential operators a∗j (ξ,D) by considering the adjoint operators aj(x,D)φ =∫ R e ixξaj(x, ξ)φ(ξ)dξ. As above we define the symbols aj,1(x, ξ) = aj(x, ξ)χ(x)(1 − χ(ξ)), aj,2(x, ξ) = aj(x, ξ)(1− χ(x)), and aj,3(x, ξ) = aj(x, ξ)χ(x)χ(ξ). We have aj(x, ξ) = ((iξ)2 − (iµ(x))2)µ(x) Z(ξ, µ(x)) , aj(x, ξ) = (|ξ|(iξ)− |µ(x)|(iµ(x)))µ(x) Z(ξ, µ(x)) , where Z(ξ, µ(x)) = (Λ(ξ)− 2Λ(µ(x))) ( 1−χ( ξ 4µ(x) ) ) + 2 (Λ(ξ) ξ − Λ(µ(x)) µ(x) ) µ(x)χ( ξ 4µ(x) ). Note that |{ξ}−1〈ξ〉(ξ∂ξ)laj,1(x, ξ)| ≤ C for all x, ξ ∈ R, l = 0, 1, 2. Then by Lemma 2.3 we have the estimate ‖aj,1(x,D)φ‖L2 x ≤ C‖φ‖L2 for all t ≥ 1. The sym- bol aj,2(x, ξ) satisfies the estimate supx,ξ∈R |∂jx∂lξaj,2(x, ξ)| ≤ C for j, l = 0, 1, then by Lemma 2.1 we have ‖aj,2(x,D)φ‖L2 x ≤ C‖φ‖L2 . By the Cauchy-Schwarz inequal- ity we find that | ∫ R e ixξaj,3(x, ξ)φ̂(ξ)dξ| ≤ |χ(x)|‖χ(ξ)‖L2‖φ̂‖L2 = C|χ(x)|‖φ‖L2 . Then we obtain ‖aj,3(x,D)φ‖L2 x = ‖ ∫ R eixξaj,3(x, ξ)φ̂(ξ)dξ‖L2 ≤ C‖χ(x)‖L2‖φ‖L2 ≤ C‖φ‖L2 . Therefore the pseudodifferential operator aj(x,D) = ∑3 l=1 aj,l(x,D) is L2-bounded. Hence the estimate follows ‖V∗Kjφ‖L2 ≤ ‖a∗j (ξ,D)F−1DtBMηφ‖L2 ≤ C‖BMηφ‖L2 ≤ C‖|Λ′′|1/2ηφ‖L2 . The proof is complete. � 5. A priori estimates 5.1. Uniform norm. In the next lemma we estimate the large time behavior of FU(−t)∂x(|u|2u). We denote ξ̂ = ξt1/3, and define the norm ‖φ‖Y = ‖φ‖L∞ + t−1/6‖∂ξφ‖L2 . Lemma 5.1. The asymptotic equality tFU(−t)∂x(|u|2u) = |ξ̂|6〈ξ̂〉−6 iξ Λ′′(ξ) |ϕ̂|2ϕ̂+O(|ξ̂|〈ξ̂〉−−1−ν‖ϕ̂‖3Y ) holds for all t ≥ 1 and ξ ∈ R, where ϕ̂(t) = FU(−t)u(t), ν > 0 is small. 20 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 Proof. In view of (2.1), it follows that tFU(−t)∂x(|u|2u) = V∗(t)A1(t)|ψ|2ψ and tFU(−t)∂x(|u|2u) = iξV∗(t)|ψ|2ψ, where ψ = Vϕ̂. Then by Lemma 4.1 with α = 1 2 + ν, β = 2ν, ν ∈ (0, 11 32 ], we obtain tFU(−t)∂x(|u|2u) = iξV∗(t)|ψ|2ψ = iξA∗|ψ|2ψ +O ( t ν 3− 1 2 ξ̂‖{η} 1 2 +ν〈η̂〉−2ν〈η〉 34 +4ν∂η|ψ|2ψ‖L2 ) +O ( t ν 3− 1 2 ξ̂‖{η}ν− 1 2 〈η̂〉−2ν〈η〉 34 +4ν |ψ|2ψ‖L2 ) when |ξ| < t−1/3, and tFU(−t)∂x(|u|2u) = V∗A1|ψ|2ψ = A∗A1|ψ|2ψ +O ( t ν 3− 1 6 〈ξ̂〉−2ν‖{η} 1 2 +ν〈η̂〉−2ν〈η〉 34 +4ν∂ηA1|ψ|2ψ‖L2 ) +O ( t ν 3− 1 6 〈ξ̂〉−2ν‖{η}ν− 1 2 〈η̂〉−2ν〈η〉 34 +4νA1|ψ|2ψ‖L2 ) when |ξ| > t−1/3. Now we consider the remainder terms. Using that {η} ≤ t−1/3{η̂}〈η̂〉, we have ‖{η} 1 2 +ν〈η̂〉−2ν〈η〉 34 +4ν∂η|ψ|2ψ‖L2 ≤ C‖{η} 1 2 +ν〈η̂〉−2ν〈η〉 34 +4ν |ψ|2∂ηψ‖L2 ≤ C‖|Λ′′|−1/2{η} 1 2 +ν〈η̂〉−2ν〈η〉 34 +4νψ2‖L∞‖|Λ′′|1/2∂ηψ‖L2 ≤ Ct 1 6− ν 3 ‖φ‖Y ‖ψ‖2L∞ and denoting ψ1 = V(iξ)ϕ̂, we have ‖{η} 1 2 +ν〈η̂〉−2ν〈η〉 34 +4ν∂ηA1|ψ|2ψ‖L2 ≤ C‖{η} 1 2 +ν〈η̂〉−2ν〈η〉 34 +4νψ2∂ηψ1‖L2 + C‖{η} 1 2 +ν〈η̂〉−2ν〈η〉 34 +4νψψ1∂ηψ‖L2 ≤ C‖|Λ′′|−1/2{η} 1 2 +ν〈η̂〉−2ν〈η〉 34 +4νηψ2‖L∞‖|Λ′′|1/2η−1∂ηψ1‖L2 + C‖|Λ′′|−1/2{η} 1 2 +ν〈η̂〉−2ν〈η〉 34 +4νψψ1‖L∞‖|Λ′′|1/2∂ηψ‖L2 ≤ Ct 1 6− ν 3 ‖φ‖Y (‖ηψ2‖L∞ + ‖ψψ1‖L∞), since by Corollary 3.6, 3.7, 3.8 we have ‖|Λ′′|1/2η−j∂ηψj‖L2 ≤ C(‖∂ξφ‖L2 + t1/6|φ(0)|) = Ct1/6‖φ‖Y for j = 0, 1. Next using Lemma 3.9, we have ‖ψ2‖L∞ ≤ Ct1/3‖φ‖2Y , ‖ηψ2‖L∞ ≤ C‖φ‖2Y and ‖ψψ1‖L∞ ≤ C‖φ‖2Y . Also ‖{η}ν− 1 2 〈η̂〉−2ν〈η〉 34 +4ν |ψ|2ψ‖L2 + ‖{η}ν− 1 2 〈η̂〉−2ν〈η〉 34 +4νA1|ψ|2ψ‖L2 ≤ Ct 1 2− ν 3 ‖φ‖3Y . Therefore, tFU(−t)∂x(|u|2u) = iξA∗|ψ|2ψ +O ( {ξ̂}〈ξ̂〉−2ν‖φ‖3Y ) for |ξ| < t−1/3, and tFU(−t)∂x(|u|2u) = A∗A1|ψ|2ψ +O ( {ξ̂}〈ξ̂〉−2ν‖φ‖3Y ) EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 21 for |ξ| > t−1/3. Next by Lemma 3.9 we have ψj(t, ξ) = t1/2|ξ|(iξ)j√ i〈tξ3〉Λ ′′(ξ) |ξ| ϕ̂(ξ) +O ( t 1 6− j 3 〈tξ3〉 j 3− 1 2 ‖ϕ̂‖Y ) for j = 0, 1. Then we obtain iξA∗|ψ|2ψ = iξA∗ ∣∣∣ t1/2|ξ|√ 〈tξ3〉Λ ′′(ξ) |ξ| ϕ̂(ξ) ∣∣∣2 t1/2|ξ|√ 〈tξ3〉 iΛ ′′(ξ) |ξ| ϕ̂(ξ) +O ( tξ3〈tξ3〉−2‖ϕ̂‖3Y ) = t2ξ6 〈tξ3〉2 iξ Λ′′(ξ) |ϕ̂|2ϕ̂+O ( |ξ̂|〈ξ̂〉−1−ν‖ϕ̂‖3Y ) and A∗A1|ψ|2ψ = A∗ ( 2ψψ1ψ + ψ2ψ1 ) = A∗ ( 2 ∣∣ t1/2|ξ|√ 〈tξ3〉Λ ′′(ξ) |ξ| ∣∣2 t1/2|ξ|iξ√ i〈tξ3〉Λ ′′(ξ) |ξ| + ( t1/2|ξ|√ 〈tξ3〉Λ ′′(ξ) |ξ| )2 t1/2|ξ|iξ√ i〈tξ3〉Λ ′′(ξ) |ξ| ) |ϕ̂|2ϕ̂+O ( t1/3ξ〈tξ3〉−4/3‖ϕ̂‖3Y ) = t2ξ6 〈tξ3〉2 iξ Λ′′(ξ) |ϕ̂|2ϕ̂+O ( |ξ̂|〈ξ̂〉−1−ν‖ϕ̂‖3Y ) . The proof is complete. � We next prove a priori estimate in the L∞-norm for ϕ̂. Lemma 5.2. Assume that supt∈[1,T ] ‖ϕ̂‖Y ≤ Cε. Then there exists an ε such that the estimate supt∈[1,T ] ‖ϕ̂‖L∞ < Cε holds for any T > 1. Proof. On the contrary we assume that there exists a first time T > 0 such that supt∈[1,T ] ‖ϕ̂‖L∞ = Cε. We use (2.1) for a new dependent variable ϕ̂ = FU(−t)u(t). In view of Lemma 5.1, we obtain ∂tϕ̂ = FU(−t)∂x(|u|2u) = |ξ̂|6iξ 〈ξ̂〉6tΛ′′(ξ) |ϕ̂|2ϕ̂+O ( ε3t−1|ξ̂|〈ξ̂〉−1−ν ) . (5.1) For the case of |ξ| < t−1/3, we integrate in time directly |ϕ̂(t, ξ)| ≤ |ϕ̂(1, ξ)|+ Cε3|ξ| ∫ t 1 τ−2/3dτ ≤ ε+ Cε3|ξ|t1/3 ≤ ε+ Cε3. For the case of |ξ| ≥ t−1/3 multiplying equation (5.1) by ϕ̂ and taking the real part of the result we obtain ∂t|ϕ̂(t, ξ)| = O ( ε3t−1|ξ̂|〈ξ̂〉−1−ν). Integration in time yields |ϕ̂(t, ξ)| ≤ |ϕ̂(|ξ|−3, ξ)|+ Cε3 ∫ t |ξ|−3 |ξτ1/3|〈ξτ1/3〉−1−ν dτ τ ≤ ε+ Cε3 ∫ |ξ|t1/3 1 〈y〉−1−νdy ≤ ε+ Cε3. Therefore ‖ϕ̂‖L∞ < Cε. The proof is complete. � 22 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 5.2. L2-norms of Pa and Pb. There are many papers devoted to Kato-Ponce type estimates of the commutators of the form [〈i∂x〉α, u]v (see, e.g. [17, 7, 31, 35]). Below we will use the following estimates (see [41]). Lemma 5.3. The estimate ‖[|∂x|, u]w‖L2 ≤ C‖∂xu‖L∞‖w‖L2 is true. Denote K1 = ∂4 x Λ(−i∂x) , K2φ = |∂x|3∂x Λ(−i∂x)φ. Lemma 5.4. The estimate ‖[Kj , u]v‖L2 ≤ C‖∂xu‖L∞‖v‖H1 holds for j = 1, 2. Next we consider a-priori estimates of solutions in the norm ‖∂−1 x Pbu(t)‖L2 + ‖Pau(t)‖L2 + ‖u(t)‖H1 uniformly in time. Lemma 5.5. Let supt∈[1,T ] ‖ϕ̂‖Y ≤ Cε. Then the estimate sup t∈[1,T ] t−γ(‖∂−1 x Pbu(t)‖L2 + ‖Pau(t)‖L2 + ‖u(t)‖H1) < Cε holds for all T > 1, where γ > 0 is small. Proof. We apply operators Pa = 4t∂t + ∂xx − a∂a and Pb = 3t∂t + ∂xx + b∂b to equation (1.1). Using the commutators [L,Pa] = 4L and [L,Pb] = 3L, we obtain LPau = (Pa + 4)Lu = ∂x(Pa + 3)(|u|2u) = 2∂x|u|2Pau+ ∂xu 2Pau+ 3∂x(|u|2u) and L∂−1 x Pbu = (∂−1 x Pb + 3∂−1 x )Lu = (Pb + 2)(|u|2u) = 2|u|2Pbu+ u2Pbu. Then we have d dt ‖∂−1 x Pbu‖2L2 = 4 Re(∂−1 x Pbu, |u|2Pbu) + 2 Re(∂−1 x Pbu, u2Pbu) ≤ C‖|u|ux‖L∞‖∂−1 x Pbu‖2L2 and d dt ‖Pau‖L2 ≤ C‖|u|ux‖L∞(‖Pau‖L2 + ‖u‖L2). In the same manner we obtain d dt‖u‖H1 ≤ C‖|u|ux‖L∞‖u‖H1 . We denote ∂kxu = vk+wk, vk = DtBM(V1(iξ)kϕ̂+V2(iξ)kϕ̂+V4(iξ)kϕ̂), and wk = DtBM(V3(iξ)kϕ̂+ V5(iξ)kϕ̂). Via Lemmas 3.4, 3.9 we have ||η| 12−jVkξjϕ̂| ≤ Cε, for j ≥ 0, k = 1, 2, 4, therefore denoting µ̃ = µ(xt−1) we obtain ‖µ̃ 1 2−jvj‖L∞ = ∑ k=1,2,4 ‖µ̃ 1 2−jDtMBVkξjϕ̂‖L∞ ≤ Ct−1/2 ∑ k=1,2,4 ‖|η| 12−jVkξjϕ̂‖L∞ ≤ Cεt−1/2 for j ≥ 0. Also via Lemma 3.9 we have ‖µ̃1/2w‖L∞ = ∑ k=3,5 ‖µ̃1/2DtMBVkϕ̂‖L∞ ≤ Ct−1/2 ∑ k=3,5 ‖|η|1/2Vkϕ̂‖L∞ ≤ Cεt−1/2. Similarly ‖w1‖L∞ ≤ Cεt−2/3. Hence ‖|u|∂xu‖L∞ = ‖µ̃1/2|v|µ̃−1/2v1‖L∞ + ‖µ̃1/2|w|µ̃−1/2v1‖L∞ + ‖(|v|+ |w|)w1‖L∞ ≤ Cε2t−1. EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 23 Therefore d dt (‖∂−1 x Pbu‖L2 + ‖Pau‖L2 + ‖u‖H1) ≤ Cε2t−1(‖∂−1 x Pbu‖L2 + ‖Pau‖L2 + ‖u‖H1). Integration in time of the above inequality yields ‖∂−1 x Pbu‖L2 +‖Pau‖L2 +‖u‖H1 ≤ ε+ Cε3tγ . The proof is complete. � 5.3. L2-norms of Ia and Ib. In this subsection we prove L2-estimates for Iau(t) and ∂−1 x Ibu(t). We define the norm ‖u‖XT = sup t∈[1,T ] ( ‖ϕ̂‖L∞ + t−γ‖∂xJ u(t)‖L2 + t−γ‖∂−1 x Ibu(t)‖L2 + t−γ‖∂−1 x Pbu(t)‖L2 + t−γ‖u(t)‖H1 ) . Lemma 5.6. Let ‖u‖XT ≤ Cε. Then the estimate sup t∈[1,T ] t−γ(‖Iau(t)‖L2 + ‖∂−1 x Ibu(t)‖L2) < Cε holds for any T > 1. Proof. Arguing by the contradiction we assume that there exists a first time T > 1 such that supt∈[1,T ] t −γ(‖Iau(t)‖L2 +‖∂−1 x Ibu(t)‖L2) = Cε. We apply Ia and ∂−1 x Ib to (1.1). Via the commutator relations [L, Ia] = [L, Ib] = 0 and by the definitions Ia = ∂b + it 1 3 |∂x| 3 and Ib = ∂b + it 1 4∂ 4 x we find LIau = Ia∂x(|u|2u) = ∂a∂x(|u|2u) + it 3 |∂x|3∂x(|u|2u) = 2∂x(|u|2Iau) + ∂x(u2Iau) + it 3 Na (5.2) and L∂−1 x Ibu = Ib(|u|2u) = ∂b(|u|2u) + it 4 ∂4 x(|u|2u) = 2|u|2Ibu+ u2Ibu+ it 4 Nb, (5.3) where Na = |∂x|3∂x(|u|2u)− 2∂x|u|2|∂x|3u+ ∂xu 2|∂x|3u = −2(|∂x|(|u|2uxxx)− |u|2|∂x|uxxx)− (|∂x|(u2uxxx)− u2|∂x|uxxx) − 2(∂x|u|2)|∂x|3u+ 2uux|∂x|3u − 6|∂x|(uuxuxx + |ux|2ux + (∂x|u|2)uxx)− 2u2|∂x|uxxx and Nb = ∂4 x(|u|2u)− 2|u|2∂4 xu+ u2∂4 xu = 8uux∂3 xu+ 8(∂x|u|2)∂3 xu+ 12u2 xuxx + 24|ux|2uxx + 12u|uxx|2 + 6uu2 xx + 2u2∂4 xu. We write ∂kxu = vk+wk, vk = DtBM(V1(iξ)kϕ̂+V2(iξ)kϕ̂+V4(iξ)kϕ̂), and wk = DtBM(V3(iξ)kϕ̂ + V5(iξ)kϕ̂), v|k| = DtBM(V1|ξ|kϕ̂ + V2|ξ|kϕ̂ + V4|ξ|kϕ̂), w|k| = 24 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 DtBM(V3|ξ|kϕ̂ + V5|ξ|kϕ̂). Also ρk = DtBMV2(iξ)kϕ̂, ωk = DtBM(V1(iξ)kϕ̂ + V4(iξ)kϕ̂), i.e. vk = ρk+ωk. We denote f ≈ g if f−g = OL2(εtγ−1), i.e. ‖f−g‖L2 ≤ Cεtγ−1. Substituting uxxx = v3 + w3, we obtain t(|∂x|(|u|2uxxx)− |u|2|∂x|uxxx) ≈ t|∂x|(|u|2(iµ̃)2ρ1)− t|u|2|∂x|(iµ̃)2ρ1 . By Lemma 5.3 we find ‖|∂x|(|u|2w3)− |u|2|∂x|w3‖L2 + ‖|∂x|(|u|2ω3)− |u|2|∂x|ω3‖L2 ≤ ‖∂x|u|2‖L∞(‖w3‖L2 + ‖ω3‖L2) ≤ Cε3tγ−1, by Lemma 3.3, 3.5 we have ‖w3‖L2 ≤ ∑ k=3,5 ‖|Λ′′|1/2Vkξ2(ξϕ̂)‖L2 ≤ Ct−1‖∂ξξϕ̂‖L2 ≤ Cεtγ−1, by Lemmas 3.1 and 3.4 we have ‖ω3‖L2 ≤ ∑ k=1,4 ‖|Λ′′|1/2Vkξ2(ξϕ̂)‖L2 ≤ Ct−1‖∂ξξϕ̂‖L2 ≤ Cεtγ−1, and by Lemma 3.2 we find ‖ρ3 − (iµ̃)2ρ1‖L2 ≤ 2∑ j=1 ‖|Λ′′|1/2η2−j [iη,V2]ξj−1(ξϕ̂)‖L2 ≤ Ct−1‖∂ξξϕ̂‖L2 ≤ Cεtγ−1. Next we represent t|∂x|(|u|2(iµ̃)2ρ1) ≈ t |∂x| ∂x ((∂x|u|2)(iµ̃)2ρ1 + |u|2∂x((iµ̃)2ρ1)) ≈ t |∂x| ∂x (uux(iµ̃)3ρ+ uxu(iµ̃)3ρ+ |u|2(iµ̃)4ρ) ≈ t |∂x| ∂x (−µ̃4u|ρ|2 + µ̃4uρ2 + µ̃4|u|2ρ) ≈ t |∂x| ∂x (−µ̃4(ω + ρ)|ρ|2 + µ̃4(ω + ρ)ρ2 + µ̃4(ωρ+ ρω + |ρ|2)ρ) ≈ t |∂x| ∂x (µ̃4|ρ|2ρ+ 2µ̃4ωρ2). Next we represent |u|2|∂x|(iµ̃)2ρ1 ≈ (iµ̃)2|u|2|∂x|ρ1 ≈ (iµ̃)3|µ̃||u|2ρ, since taking p = 2 + γ, q = 2p γ we obtain ‖t|u|2[|∂x|, µ̃2]ρ1‖L2 ≤ Ct‖u‖2L2p‖[|∂x|, µ̃2]ρ1‖Lq ≤ Ct‖u‖2L2p‖∂xµ̃2‖L∞‖ρ1‖Lq ≤ Cε3t− 2 3 (1− 1 2p )− 1 2 (1− 2 q )‖ρ1‖Lq ≤ Cε3tγ−1 and denoting ρ|1|1 = DtBMV2|ξ|(iξ)ϕ̂, and similarly ω|1|1, w|1|1 we obtain µ̃2|u|2|∂x|ρ1 = µ̃2|u|2|∂x|ux − |u|2[|∂x|, µ̃2](ω1 + w1) − [|∂x|, |u|2]µ̃2(ω1 + w1)− |∂x| ∂x ((∂x|u|2)µ̃2(ω1 + w1)) EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 25 − |∂x| ∂x (|u|2(∂xµ̃ 2)(ω1 + w1))− |∂x| ∂x (|u|2µ̃2(∂xω1 + ∂xw1)) ≈ µ̃2|u|2|∂x|ux ≈ µ̃2|u|2(ω|1|1 + ρ|1|1 + w|1|1) ≈ µ̃2|u|2ρ|1|1 ≈ iµ̃3|µ̃||u|2ρ ≈ iµ̃3|µ̃|(ωρ+ ωρ+ |ρ|2)ρ. Thus we obtain t(|∂x|(|u|2uxxx)− |u|2|∂x|uxxx) ≈ t |∂x| ∂x (µ̃4|ρ|2ρ+ 2µ̃4ωρ2) + itµ̃3|µ̃|(ωρ+ ωρ+ |ρ|2)ρ. In the same manner we obtain t(|∂x|(u2uxxx)− u2|∂x|uxxx) ≈ t|∂x|(u2(iµ̃)2ρ1)− tu2|∂x|(iµ̃)2ρ1 ≈ −t |∂x| ∂x (µ̃4|ρ|2ρ)− it|µ̃|3µ̃(2ω|ρ|2 + |ρ|2ρ) and −2t(∂x|u|2)|∂x|3u+ 2tuux|∂x|3u ≈ −2t(∂x|u|2)ρ|3| + 2tuuxρ|3| ≈ 4it|µ̃|3µ̃ω|ρ|2 − 2it|µ̃|3µ̃ωρ2 + 2it|µ̃|3µ̃|ρ|2ρ. Next we obtain 6t|∂x|(uuxuxx + |ux|2ux + (∂x|u|2)uxx) ≈ 12t |∂x| ∂x (µ̃4ωρ2), and writing w|1|3 = DtBMV3|ξ|(iξ)3ϕ̂, we have tu2|∂x|uxxx ≈ tu2w|1|3+it|µ̃|µ̃3|ρ|2ρ+ 2it|µ̃|µ̃3|ρ|2ω. Therefore tNa ≈ −t |∂x| ∂x µ̃4(|ρ|2ρ+ 16µ̃4ωρ2)− it|µ̃|µ̃3(|ρ|2ρ+ 4ωρ2)− 2tu2w|1|3. Next using the factorization formula (2.1), we represent FU(−t) ( t |∂x| ∂x (µ̃4|ρ|2ρ) + it|µ̃|µ̃3|ρ|2ρ ) ≈ −itFU(−t)(|∂x|(µ̃3|ρ|2ρ)− |µ̃|µ̃3|ρ|2ρ) ≈ −i[|ξ|,V∗](η3|V2ϕ̂|2V2ϕ̂) ≈ 0, since by Corollary 3.6 ‖|Λ′′|−1/2η2∂ηV2ϕ̂‖L2 ≤ ‖|Λ′′|1/2η∂ηV2ξ −1(ξϕ̂)‖L2 ≤ Cεt−1‖∂ξ(ξϕ̂)‖L2 ≤ Cεtγ−1 and by Lemma 4.2 ‖[|ξ|,V∗](η3|V2ϕ̂|2V2ϕ̂)‖L2 ≤ Ct−1‖|Λ′′|−1/2η3|V2ϕ̂|2∂ηV2ϕ̂‖L2 + Ct−1‖|Λ′′|− 1 2 η2|V2ϕ̂|2V2ϕ̂‖L2 ≤ Cε3tγ−1. Thus we obtain tNa ≈ −16t |∂x| ∂x (µ̃4ωρ2)− 4it|µ̃|3µ̃ωρ2 − 2tu2w|1|3 ≈ 4t(|∂x|∂x(µ̃2ωρ2)− i|µ̃|µ̃3ωρ2)− 2tu2w|1|3. 26 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 In the same manner we transform uux∂3 xu≈ −µ̃4(ρ+ω)|ρ|2, (∂x|u|2)∂3 xu ≈ −µ̃4ω|ρ|2+ µ̃4ωρ2, u2 xuxx ≈ µ̃4ρ|ρ|2, |ux|2uxx ≈ −µ̃4ρ|ρ|2, u|uxx|2 ≈ µ̃4(ρ + ω)|ρ|2, uu2 xx ≈ µ̃4(ρ+ ω)ρ2, u2∂4 xu ≈ µ̃4ρ|ρ|2 + 2µ̃4ω|ρ|2 + u2w4. Therefore tNb ≈ 14tµ̃4ωρ2 + 2tu2w4 ≈ −14 3 t(∂2 x(µ̃2ωρ2)− (iµ̃)2(µ̃2ωρ2)) + 2tu2w4. Now we need to transform the terms 2tu2w|1|3 and 2tu2w4. We have ∂t(V∗Mφ) = 1 2tV ∗Mφ+iΛV∗Mφ−iξV∗Λ′Mφ+V∗M∂tφ and ∂t(MVlφ) = MV∂tΨlφ+ 1 2tMVlφ− MVliΛφ + MΛ′Vliξφ + MVl∂tφ. We denote K1 = ∂4 x Λ(−i∂x) , and K2 = |∂x|3∂x Λ(−i∂x) and the symbols K1(ξ) = ξ4 Λ(ξ) and K2(ξ) = |ξ|3iξ Λ(ξ) . Then by a direct calculation we obtain ∂t(V∗(Vϕ̂)2VlKjϕ̂) = ∂t(V∗M(MVϕ̂)2MVlKjϕ̂) = 1 2t V∗(Vϕ̂)2VlKjϕ̂+ iΛV∗(Vϕ̂)2VlKjϕ̂− iξV∗Λ′(Vϕ̂)2VlKjϕ̂ + 2V∗(Vϕ̂)(VlKjϕ̂)( 1 2t Vϕ̂− ViΛϕ̂+ Λ′Viξϕ̂+ V∂tϕ̂) + V∗(Vϕ̂)2(V∂tΨlKjϕ̂+ 1 2t VlKjϕ̂− VliΛKjϕ̂+ Λ′VliξKjϕ̂+ VlKj∂tϕ̂). Hence ∂t(V∗(Vϕ̂)2VlKjϕ̂) = 2 t V∗(Vϕ̂)2VlKjϕ̂+ V∗(Vϕ̂)2V∂tΨlϕ̂ + 2V∗(Vϕ̂)(VlKjϕ̂)(V∂tϕ̂) + V∗(Vϕ̂)2VlKj∂tϕ̂ − iξV∗Λ′(Vϕ̂)2VlKjϕ̂+ 2V∗Λ′(Vϕ̂)(Viξϕ̂)VlKjϕ̂+ V∗Λ′(Vϕ̂)2VliξKjϕ̂ + iΛV∗(Vϕ̂)2VlKjϕ̂− 2V∗(Vϕ̂)(ViΛϕ̂)(VlKjϕ̂)− V∗(Vϕ̂)2VliΛKjϕ̂. Next we use the relations iξV∗Λ′(Vϕ̂)2VlKjϕ̂ = 2V∗Λ′(Vϕ̂)(Viξϕ̂)VlKjϕ̂+ V∗Λ′(Vϕ̂)2VlKjiξϕ̂ + 1 t V∗(Vϕ̂)2VlKjϕ̂+ V∗Λ′(Vϕ̂)2 1 tΛ′′ V(∂ηΨl)Kjϕ̂ ≈ 2V∗Λ′(Vϕ̂)(Viξϕ̂)VlKjϕ̂+ V∗Λ′(Vϕ̂)2VlKjiξϕ̂ and iΛV∗(Vϕ̂)2VlKjϕ̂ = iΛ iξ iξV∗(Vϕ̂)2VlKjϕ̂ = 2 iΛ iξ V∗(Vϕ̂)(Viξϕ̂)VlKjϕ̂+ iΛ iξ V∗(Vϕ̂)2VlKjiξϕ̂ + iΛ iξ V∗(Vϕ̂)2 1 tΛ′′ V(∂ηΨl)Kjϕ̂ ≈ iΛ iξ V∗(Vϕ̂)2VlKjiξϕ̂. EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 27 Therefore ∂t(V∗(Vϕ̂)2VlKjϕ̂) ≈ 2 t V∗(Vϕ̂)2VlKjϕ̂+ V∗(Vϕ̂)2V∂tΨlϕ̂+ 2V∗(Vϕ̂)(VlKjϕ̂)(V∂tϕ̂) + V∗(Vϕ̂)2VlKj∂tϕ̂− 2V∗(Vϕ̂)(ViΛϕ̂)(VlKjϕ̂) + iΛ iξ V∗(Vϕ̂)2VlKjiξϕ̂+ iV∗(Vϕ̂)2VlΛKjϕ̂. We have 2 tV ∗(Vϕ̂)2VlKjϕ̂+ V∗(Vϕ̂)2V∂tΨlϕ̂ ≈ 0 and 2V∗(Vϕ̂)(VlKjϕ̂)(V∂tϕ̂)− 2V∗(Vϕ̂)(ViΛϕ̂)(VlKjϕ̂) ≈ 0. Also we represent∑ l=3,5 V∗(Vϕ̂)2VlKj∂tϕ̂ = t−1 ∑ l=3,5 V∗(Vϕ̂)2VlKjiξV∗(|Vϕ̂|2Vϕ̂) = t−1V∗(Vϕ̂)2VKjiξV∗(|Vϕ̂|2Vϕ̂)− t−1 ∑ l=1,2,4 V∗(Vϕ̂)2VlKjiξV∗(|Vϕ̂|2Vϕ̂). Then by Lemma 5.4, U(t)Ft−1V∗(Vϕ̂)2VKjiξV∗(|Vϕ̂|2Vϕ̂) = u2Kj∂x(|u|2u) = [Kj , u2]∂x(|u|2u) +Kju2∂x(|u|2u) ≈ Kju2∂x(|u|2u) ≈ 0, since t‖[Kj , u2]∂x(|u|2u)‖L2 ≤ Ct‖∂xu2‖L∞‖∂x(|u|2u)‖H1 ≤ Cε5tγ−1, and t‖Kju2∂x(|u|2u)‖L2 ≤ Ct‖(∂xu2)∂x(|u|2u)‖L2 + Ct‖u2∂2 x(|u|2u)‖L2 ≤ Ct‖∂xu2‖2L∞‖u‖L2 + Ct‖u‖4L∞‖ω2‖L2 + Ct‖|µ̃|2|u|4|µ̃|−2ρ2‖L2 + Ct‖u‖4L∞‖w2‖L2 ≤ Cε5tγ−1, and by Lemmas 3.1, 3.2 and 3.4, for l = 1, 2, 4, j = 1, 2, we have ‖V∗(Vϕ̂)2VlKjiξV∗(|Vϕ̂|2Vϕ̂)‖L2 ≤ C‖|η|1/2Vϕ̂‖2L∞‖|Λ′′|1/2η−1VlKjiξV∗(|Vϕ̂|2Vϕ̂)‖L2 ≤ Cε2‖|Λ′′|1/2ξV∗(|Vϕ̂|2Vϕ̂)‖L2 ≤ Cε2‖∂xu2‖L∞‖u‖L2 ≤ Cε5tγ−1. Therefore using Λ iξV ∗(Vϕ̂)2VliξKjϕ̂− V∗(Vϕ̂)2Vl Λ iξ iξKjϕ̂ ≈ 0, we obtain ∂t ( ∑ l=3,5 V∗(Vϕ̂)2VlKjϕ̂ ) ≈ iΛ iξ ∑ l=3,5 V∗(Vϕ̂)2VlKjiξϕ̂+ i ∑ l=3,5 V∗(Vϕ̂)2VlΛKjϕ̂ 28 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 ≈ 2i ∑ l=3,5 V∗(Vϕ̂)2VlΛKjϕ̂ = { 2i ∑ l=3,5 V∗(Vϕ̂)2Vlξ4ϕ̂, j = 1, 2i ∑ l=3,5 V∗(Vϕ̂)2Vl|ξ|iξ3ϕ̂, j = 2, if we choose K1 = ξ4 Λ(ξ) and K2 = |ξ|3iξ Λ(ξ) . Since L(tu2wKj ) = U(t)F−1∂t( ∑ l=3,5 V∗(Vϕ̂)2VlKjϕ̂), we have L(tu2wK1 ) ≈ 2itu2w4 and L(tu2wK2 ) ≈ −2tu2w|1|3. Next we transform the terms t∂2 x(µ̃2ωρ2)− t(iµ̃)2(µ̃2ωρ2) and t|∂x|∂x(µ̃2ωρ2)− it|µ̃|µ̃3ωρ2. Using the factorization formula (2.1), we write FU(−t)∂x(|u|2u) = iξt−1V∗(|Vϕ̂|2Vϕ̂). Hence FU(−t)(t∂2 x(µ̃2ωρ2)− t(iµ̃)2(µ̃2ωρ2)) ≈ [(iξ)2,V∗]η2 ( (V2ϕ̂)2V1ϕ̂ ) , FU(−t)(t|∂x|∂x(µ̃2ωρ2)− it|µ̃|µ̃3ωρ2) ≈ [|ξ|(iξ),V∗]η2 ( (V2ϕ̂)2V1ϕ̂ ) . Using ∂t(V∗KMφ) = 1 2tV ∗ KMφ+iΛV∗KMφ−iξV∗KΛ′Mφ+V∗KM∂tφ and ∂t(MVkφ) = MV∂tΨkφ + 1 2tMVkφ − MVkiΛφ + MΛ′Vkiξφ + MVk∂tφ, as above by a direct computation we obtain ∂t(V∗K(V2ϕ̂)2V1ϕ̂) = ∂t(V∗KM(MV2ϕ̂)2MV1ϕ̂) = 1 2t V∗K(V2ϕ̂)2V1ϕ̂+ iΛV∗K(V2ϕ̂)2V1ϕ̂− iξV∗KΛ′(V2ϕ̂)2V1ϕ̂ + 2V∗K ( (V2ϕ̂)V1ϕ̂ )( V∂tΨ2ϕ̂+ 1 2t V2ϕ̂− V2iΛϕ̂+ Λ′V2iξϕ̂+ V2∂tϕ̂ ) + V∗K ( (V2ϕ̂)2 )( V∂tΨ1ϕ̂+ 1 2t V1ϕ̂− V1iΛϕ̂+ Λ′V1iξϕ̂+ V1∂tϕ̂ ) . Hence ∂t ( V∗K(V2ϕ̂)2V1ϕ̂ ) = 2 t V∗K(V2ϕ̂)2V1ϕ̂+ 2V∗K(V2ϕ̂)(V∂tΨ2ϕ̂)V1ϕ̂+ V∗K(V2ϕ̂)2(V∂tΨ1ϕ̂) + 2V∗K(V2ϕ̂)V1ϕ̂(V2∂tϕ̂) + V∗K(V2ϕ̂)2V1∂tϕ̂− iξV∗KΛ′(V2ϕ̂)2V1ϕ̂ + 2V∗KΛ′(V2ϕ̂)(V2iξϕ̂)V1ϕ̂+ V∗KΛ′(V2ϕ̂)2V1iξϕ̂+ iΛV∗K(V2ϕ̂)2V1ϕ̂ − 2V∗K(V2ϕ̂)(V2iΛϕ̂)V1ϕ̂− V∗K(V2ϕ̂)2V1iΛϕ̂. Then we use the identity iξV∗KΛ′(V2ϕ̂)2V1ϕ̂ = V∗A1KΛ′(V2ϕ̂)2V1ϕ̂ = 2V∗KΛ′(V2iξϕ̂)(V2ϕ̂)V1ϕ̂+ V∗KΛ′(V2ϕ̂)2V1iξϕ̂+ V∗A0(KΛ′)(V2ϕ̂)2V1ϕ̂ + 2V∗KΛ′(V2ϕ̂)V1ϕ̂(VA0Ψ2ϕ̂) + V∗KΛ′A0(V2ϕ̂)2VA0Ψ1ϕ̂ ≈ 2V∗KΛ′(V2iξϕ̂)(V2ϕ̂)V1ϕ̂. Also we use 2 t V∗K(V2ϕ̂)2V1ϕ̂+ 2V∗K(V2ϕ̂)(V2∂tΨkϕ̂)V1ϕ̂+ V∗K(V2ϕ̂)2(V1∂tΨkϕ̂) + 2V∗K(V2ϕ̂)V1ϕ̂(V2∂tϕ̂) + V∗K(V2ϕ̂)2V1∂tϕ̂+ V∗KΛ′(V2ϕ̂)2V1iξϕ̂ EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 29 − V∗K(V2ϕ̂)2V1iΛϕ̂ ≈ 0. Hence we obtain ∂t ( V∗K(V2ϕ̂)2V1ϕ̂ ) ≈ iΛV∗K(V2ϕ̂)2V1ϕ̂− 2V∗K(V2ϕ̂)(V2iΛϕ̂)V1ϕ̂ ≈ iΛV∗K(V2ϕ̂)2V1ϕ̂− 2iV∗KΛ(V2ϕ̂)2V1ϕ̂. On the domain |ξ| ≤ 8|η| we represent iΛV∗Kχ( ξ 4η )(V2ϕ̂)2V1ϕ̂ = 2 Λ ξ V∗Kχ(V2iξϕ̂)(V2ϕ̂)V1ϕ̂+ Λ ξ V∗Kχ(V2ϕ̂)2V1iξϕ̂+ Λ ξ V∗(A0χK)(V2ϕ̂)2V1ϕ̂ ≈ 2i Λ ξ V∗Kχη(V2ϕ̂)2V1ϕ̂. Thus we obtain ∂t(V∗K(V2ϕ̂)2V1ϕ̂) ≈ iV∗KZ(ξ, η)(V2ϕ̂)2V1ϕ̂, where Z(ξ, η) = (Λ(ξ)− 2Λ(η)) ( 1− χ( ξ 4η ) ) + 2 (Λ(ξ) ξ − Λ(η) η ) ηχ( ξ 4η ). Next we choose K3(ξ, η) = ((iξ)2−(iη)2)η2 Z(ξ,η) and K4(ξ, η) = (|ξ|(iξ)−|η|(iη))η2 Z(ξ,η) , then we have [(iξ)2,V∗]η2((V2ϕ̂)2V1ϕ̂) ≈ ∂t(V∗K3 (V2ϕ̂)2V1ϕ̂) and [|ξ|(iξ),V∗]η2((V2ϕ̂)2V1ϕ̂) ≈ ∂t(V∗K4 (V2ϕ̂)2V1ϕ̂). By Lemma 4.3 we have the estimate ‖V∗Kjφ‖L2 ≤ C‖|Λ′′|1/2ηφ‖L2 for j = 3, 4. Then equations (5.2) and (5.3) yield L ( Iau− it 3 u2wK2 − 4i 3 U(t)F−1V∗K4 (V2ϕ̂)2V1ϕ̂ ) ≈ 2∂x(|u|2Iau) + ∂x(u2Iau), L ( ∂−1 x Ibu− t 4 u2wK1 + 7i 6 U(t)F−1V∗K3 (V2ϕ̂)2V1ϕ̂ ) ≈ 2|u|2Ibu+ u2Ibu. We denote Φa = Iau − it 3 u 2wK2 − 4i 3 U(t)F−1V∗K4 (V2ϕ̂)2V1ϕ̂ and Φb = ∂−1 x Ibu − t 4u 2wK1 + 7i 6 U(t)F−1V∗K3 (V2ϕ̂)2V1ϕ̂, then we obtain LΦa ≈ 2∂x|u|2Φa+∂xu 2Φa and LΦb ≈ 2|u|2∂xΦb + u2∂xΦb. Integration in time of these equation yields ‖Φa‖L2 + ‖Φb‖L2 ≤ ε+Cε3tγ . Consequently, we obtain ‖Iau‖L2 + ‖∂−1 x Ibu‖L2 ≤ ε+Cε3tγ . This is the desired contradiction. The proof is complete. � 6. Proof of Theorem 1.1 By Lemmas 5.2, 5.5 and 5.6 we see that a priori estimate ‖u‖XT ≤ Cε holds for all T > 0. Therefore the existence of global solutions of the Cauchy problem (1.1) follows by a standard continuation argument. Now we turn to the proof of the asymptotic formulas for the solutions u of the Cauchy problem (1.1). We need to compute the check the asymptotic behavior of the function ϕ̂(t, ξ). As in the proof of Lemma 5.2 we obtain on the domain |ξ| ≥ tγ− 1 3 ∂tϕ̂ = |ξ̂|6〈ξ̂〉−6 iξ tΛ′′(ξ) |ϕ̂|2ϕ̂+O ( t−1|ξt1/3|〈ξt1/3〉−2‖ϕ̂‖3Y ) . Hence for a new dependent variable z(t, ξ) = ϕ̂(t, ξ)Ψ(t, ξ), where Ψ(t, ξ) = exp ( − iξ |Λ′′(ξ)| ∫ t 1 |ϕ̂(τ, ξ)|2 |ξτ 1/3|6 τ〈ξτ 1 3 〉6 dτ ) , 30 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 we find ∂tz(t, ξ) = O ( t−1|ξt1/3|〈ξt1/3〉−2‖ϕ̂‖3Y ) on the domain |ξ| ≥ tγ− 1 3 . Integra- tion in time yields |z(t, ξ)− z(s, ξ)| ≤ Cε3|ξ| ∫ s t dτ τ2/3(1 + ξ2τ2/3) ≤ Cε3 ∫ ∞ t3γ dz z 4 3 ≤ Cε3t−γ for any s > t > 0 in the domain |ξ|t1/3 ≥ tγ . Then there exists a unique final state z+ ∈ L∞ such that ‖z(t)− z+‖ L∞(|ξ|≥tγ− 1 3 ) ≤ Cε3t−γ for all t > 0. We write∫ t 1 |ϕ̂(τ, ξ)|2 dττ = ∫ t 1 |z(τ, ξ)|2 dττ = |z+|2 log t + Φ(t). For the remainder term Φ(t), we have Φ(s) − Φ(t) = ∫ s t ( |z(τ)|2 − |z(s)|2 ) dτ τ + ( |z(s)|2 − |z+|2 ) log s t . Then we obtain ‖Φ(s)−Φ(t)‖ L∞(|ξ|≥tγ− 1 3 ) ≤ Cε3t−γ for any s > t > 0. Hence there exists a unique real-valued function Φ+ ∈ L∞ such that ‖Φ(t)−Φ+‖ L∞(|ξ|≥tγ− 1 3 ) ≤ Cε3t−γ for all t > 0. Therefore,∥∥Ψ(t, ξ)− exp ( iξ |Λ′′(ξ)| (|z+|2 log t+ Φ+) )∥∥ L∞(|ξ|≥tγ− 1 3 ) ≤ Ct−γ for all t > 0. Thus for large time, ‖ϕ̂(t, ξ)− z+Ψ(t, ξ)‖ L∞(|ξ|≥tγ− 1 3 ) ≤ Ct−γ and∥∥z+Ψ(t, ξ)−W+ exp ( iξ |Λ′′(ξ)| |z+|2 log t )∥∥ L∞(|ξ|≥tγ− 1 3 ) ≤ Ct−γ with W+ = z+ exp( iξ |Λ′′(ξ)|Φ+). Thus, ϕ̂(t, ξ) = W+ exp ( 3iξ |Λ′′(ξ)| |W+|2 log t ) +O(t−δ) on the domain |ξ| ≥ tγ− 1 3 with some δ > 0. Using the factorization formulas we have u = DtBMVϕ̂. Then by Lemma 3.9 we have u(t, x) = DtBMA0ϕ̂(t) +O ( t−1/3〈t1/3µ( x t )〉−3/4 ) = M√ itΛ′′(µ(xt )) W+(µ( x t )) exp ( iµ(xt ) |Λ′′(µ(xt ))| |W+(µ( x t ))|2 log t ) +O ( t−1/3〈t1/3µ( x t )〉−3/4 ) +O(t− 1 3−δ) on the domain |x| ≥ t 1 3 +2γ . Next we consider the self-similar region |x| ≤ t 1 3 +2γ . We represent u(t) = U(t)F−1ϕ̂ = √ 1 2π ∫ |ξ|≥2t− 1 3 +γ eixξ−itΛ(ξ)ϕ̂(t, ξ)dξ + √ 1 2π ∫ |ξ|≤2t− 1 3 +γ eixξ ( e−itΛ(ξ) − e− it3 |ξ| 3 ) ϕ̂(t, ξ)dξ + √ 1 2π ∫ |ξ|≤2t− 1 3 +γ eixξ− it 3 |ξ| 3 ϕ̂(t, ξ)dξ = I1 + I2 + I3. We show that the first term I1 is a remainder in the region |x| ≤ t 1 3 +2γ . Integrating by parts we have I1 = C eixξ−itΛ(ξ)ϕ̂(t, ξ) i(x− tΛ′(ξ)) ∣∣∣ |ξ|=2t− 1 3 +γ − C ∫ |ξ|≥2t− 1 3 +γ eixξ−itΛ(ξ)ϕ̂(t, ξ) tΛ′′(ξ)dξ (x− tΛ′(ξ))2 − C ∫ |ξ|≥2t− 1 3 +γ eixξ−itΛ(ξ) ϕ̂ξ(t, ξ)dξ i(x− tΛ′(ξ)) . EJDE-2020/77 HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION 31 Hence |I1| ≤ Ct− 1 3−2γ‖ϕ̂‖L∞ + C‖ϕ̂‖L∞ ∫ |ξ|≥2t− 1 3 +γ tΛ′′(ξ) (x− tΛ′(ξ))2 dξ + Ct−1‖ϕ̂ξ‖L2 (∫ |ξ|≥2t− 1 3 +γ dξ ξ4 )1/2 ≤ Ct− 1 3−2γ‖ϕ̂‖L∞ + Ct− 1 2− 3 2γ‖ϕ̂ξ‖L2 ≤ Cεt− 1 3− 3 2γ . The second term I2 is also a remainder since |I2| ≤ Ct‖ϕ̂‖L∞ ∫ |ξ|≤2t− 1 3 +γ |ξ|4dξ ≤ Cεt− 2 3 +5γ ≤ Cεt− 1 3−γ , if γ ≤ 1 18 . Changing ξt1/3 = ξ̂, y = xt−1/3, we obtain for the last summand I3 = t−1/3 √ 1 2π ∫ |ξ̂|≤2tγ eiyξ̂e− i 3 |ξ̂| 3 ϕ̂(t, ξ̂t−1/3)dξ̂ = t−1/3 √ 1 2π ∫ |ξ̂|≤2tγ eiS̃(y,ξ̂)ϕ̂(t, ξ̂t−1/3)dξ̂ = t−1/3F (t, y), where S̃(y, ξ̂) = yξ̂ − 1 3 |ξ̂| 3, Φ(t, ξ̂) = ϕ̂(t, ξ̂t−1/3), and F (t, y) = √ 1 2π ∫ |ξ̂|≤2tγ eiS̃(y,ξ̂)Φ(t, ξ̂)dξ̂. Let us show that F (t, y) is a Cauchy sequence on the domain |y| ≤ tγ . We can write ∫ |ξ̂|≤2tγ1 eiS̃(y,ξ̂)Φ(t1, ξ̂)dξ̂ − ∫ |ξ̂|≤2tγ1 eiS̃(y,ξ̂)Φ(t2, ξ̂)dξ̂ = ∫ t2 t1 ∫ |ξ̂|≤2tγ1 eiS̃(y,ξ̂)∂tΦ(t, ξ̂)dξ̂dt. Using the identity ∂tΦ(t, ξ̂) = ∂t ( ϕ̂(t, ξ̂t−1/3) ) = 1 3t (3t∂tϕ̂−ξ∂ξϕ̂) = 1 3 ξ̂t −4/3(ξ−1P̂b− bξ−1Îb)ϕ̂, and applying Lemmas 5.5, 5.6, by the Cauchy-Schwarz inequality, we ob- tain ‖∂tΦ(t, ξ̂)‖L1 ξ̂ (|ξ̂|≤2tγ) ≤ Ct 3 2γ− 4 3 ‖(ξ−1P̂b − 2bξ−1Îb)ϕ̂‖L2 ξ̂ (|ξ̂|≤2tγ) ≤ Ct 3 2γ− 7 6 ‖(ξ−1P̂b − 2bξ−1Îb)ϕ̂‖L2 ξ ≤ Ct 3 2γ− 7 6 (‖∂−1 x Pbu‖L2 + ‖∂−1 x Ibu‖L2) ≤ Cεt 5 2γ− 7 6 . Integrating by parts as above we find∫ 2tγ1≤|ξ̂|≤2tγ2 eiS̃(y,ξ̂)Φ(t2, ξ̂)dξ̂ = ∫ 2tγ1≤|ξ̂|≤2tγ2 ei(yξ̂− 1 3 |ξ̂| 3)ϕ̂(t2, ξ̂t −1/3 2 )dξ̂ = eiS̃(y,ξ̂) ϕ̂(t2, ξ̂t −1/3 2 ) i(y − |ξ̂|ξ̂) ∣∣∣|ξ̂|=2tγ2 |ξ̂|=2tγ1 − ∫ 2tγ1≤|ξ̂|≤2tγ2 eiS̃(y,ξ̂)ϕ̂(t2, ξ̂t −1/3 2 )∂ξ̂ 1 i(y − |ξ̂|ξ̂) dξ̂ 32 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 − ∫ 2tγ1≤|ξ̂|≤2tγ2 eiS̃(y,ξ̂) t −1/3 2 ϕ̂ξ(t2, ξ̂t −1/3 2 ) i(y − |ξ̂|ξ̂) dξ̂ on the domain |y| ≤ tγ1 for t2 > t1. Hence∣∣ ∫ 2tγ1≤|ξ̂|≤2tγ2 eiS̃(y,ξ̂)Φ(t2, ξ̂)dξ̂ ∣∣ ≤ Ct−2γ 1 ‖ϕ̂‖L∞ + C‖ϕ̂‖L∞ ∫ 2tγ1≤|ξ̂|≤2tγ2 ξ̂dξ̂ (x− |ξ̂|ξ̂)2 + Ct −1/3 2 (∫ 2tγ1≤|ξ̂|≤2tγ2 |ϕ̂(t2, ξ̂t −1/3 2 )|2dξ̂ )1/2(∫ 2tγ1≤|ξ̂|≤2tγ2 dξ̂ ξ̂4 )1/2 ≤ Ct−2γ 1 ‖ϕ̂‖L∞ + Ct − 1 6− 3 2γ 1 ‖ϕ̂ξ‖L2 ≤ Cεt− 3 2γ 1 . Then we obtain |F (t1, y)− F (t2, y)| ≤ ∣∣ ∫ |ξ̂|≤2tγ1 eiS̃(y,ξ̂)Φ(t1, ξ̂)dξ̂ − ∫ |ξ̂ ∣∣≤2tγ1 eiS̃(y,ξ̂)Φ(t2, ξ̂)dξ̂| + | ∫ 2tγ1≤|ξ̂|≤2tγ2 eiS̃(y,ξ̂)Φ(t2, ξ̂)dξ̂| ≤ Cεt− 3 2γ 1 + ∫ t2 t1 ‖∂tΦ(t, ξ̂)‖L1 ξ̂ (|ξ̂|≤2tγ)dt ≤ Cεt− 3 2γ 1 + Cε ∫ t2 t1 t 5 2γ− 7 6 dt ≤ Cεt−δ1 for any t2 ≥ t1 with some small δ > 0. Hence there exists a limit fm(y) = limt→∞ F (t, y) such that |F (t, y) − fm(y)| ≤ Cεt−δ for |y| ≤ tγ . Thus in the self- similar region |x| ≤ t 1 3 +γ we have the asymptotic formula u(t, x) = t− 1 3 fm(xt−1/3)+ O(t− 1 3−δ). Taking the limit t→∞ in equation (1.1) we can see that t−1/3fm(xt−1/3) is the self-similar solution of equation (1.2). This completes the proof of the as- ymptotic part of Theorem 1.1. Acknowledgments. 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Douay; Third-order dispersion for generating optical rogue solitons, Phys. Lett. A, 374 (2010), pp. 691–695. [50] M. Tsutsumi; On global solutions of the generalized Korteweg - de Vries equation, Publ. Res. Inst. Math. Sci., 7 (1972), pp. 329-344. Pavel I. Naumkin Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, México Email address: pavelni@matmor.unam.mx Isahi Sánchez-Suárez Universidad Politécnica de Uruapan, CP 60210 Uruapan Michoacán, México Email address: isahi ss@hotmail.com 1. Introduction 2. Preliminaries 2.1. Factorization techniques 2.2. Boundedness of pseudodifferential operators 3. Estimates for the operator V 3.1. Estimates for commutators 3.2. Asymptotic behavior 4. Estimates for the operator V 4.1. Asymptotic behavior 4.2. Estimates for commutators 5. A priori estimates 5.1. Uniform norm 5.2. L2-norms of Pa and Pb 5.3. L2-norms of Ia and Ib 6. Proof of Theorem ?? Acknowledgments. References