Asymptotic behavior in time of solution to the nonlinear Schr"odinger equation with higher order anisotropic dispersion

We consider the asymptotic behavior in time of solutions to the nonlinear Schr"odinger equation with fourth order anisotropic dispersion (4NLS) which describes the propagation of ultrashort laser pulses in a medium with anomalous time-dispersion in the presence of fourth-order time-dispersion. We prove existence of a solution to (4NLS) which scatters to a solution of the linearized equation of (4NLS).


Introduction
We consider the asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion: where u : R × R d → C is an unknown function, α, β, γ, λ are real constants and p > 1. Equation (1.1) arises in nonlinear optics to model the propagation of ultrashort laser pulses in a medium with anomalous time-dispersion in the presence of fourth-order time-dispersion (see [39,12,4] and the references therein). It also arises in models of propagation in fiber arrays (see [1,11]). To simplify (1.1), we introduce a new unknown function v(t, x) = e −i( αβ 2 16γ 2 + 5β 4 256γ 3 )t+i β Then the equation (1.1) can be rewritten as Therefore if α, β and γ satisfy (α + 3β 2 8γ )α > 0 and αγ < 0, then by using a suitable scaling transform, we can rewrite (1.1) into the Schrödinger equation with fourth order anisotropic dispersion: In this paper, we study the asymptotic behavior in time of solutions to (1.2).
The Cauchy problem for the homogeneous fourth order nonlinear Schrödinger type equation has been studied by many authors, most of the results holding also when lower dispersive terms are added. By using the Strichartz estimates in [3] one shows that the Cauchy problem is locally well-posed in the energy space H 2 (R d ) for the energy subcritical case (i.e., 1 < p < 1+8/(d−4) when d ≥ 5 and 1 < p < ∞ when d ≤ 4) and in L 2 (R d ) for the mass subcritical case (1 < p < 1 + 8/d). We also refer to Bouchel [6] who studies the Cauchy problem and furthermore gives non-existence, existence and qualitative properties results of solitary wave solutions for (1.1). See also [12] for results on the Cauchy problem for slightly more general situations. There are several results concerning the scattering and blow-up of solutions for (1.3). For the defocusing case λ < 0, the global well-posedness and scattering for (1.3) with the energy-critical nonlinearity (i.e., (1.3) with d 5, and p = 1 + 8/(d − 4)) was studied by Pausader [32] for radially symmetric initial data by combining the concentration-compactness argument by Kenig-Merle [25] and Morawetz-type estimate. Later, Miao, Xu and Zhao [28] proved a similar result for (1.3) in the energy-critical and higher dimensional case d 9 without radial assumption on initial data. In [33], Pausader has shown the global well-posedness and scattering of (1.3) with cubic nonlinearity for the case 5 d 8. Pausader and Xia [34] proved the global well-posedness and scattering for (1.3) with mass super-critical nonlinearity (i.e., (1.3) with p > 1 + 8/d) for low dimensions 1 d 4 by using a virial-type estimate instead of the Morawetz-type estimates.
For the focusing case λ > 0, Pausader [31] and Miao, Xu and Zhao [27] independently showed the global well-posedness and scattering for (1.3) with the energy-critical nonlinearity for radially symmetric initial data withḢ 2 and energy norms below that of the ground state. When the initial data is sufficiently small, Hayashi, Mendez-Navarro and Naumkin [14] proved the global existence and the scattering for (1.3) with d = 1 and p > 5 by using the factorization technique developed by the authors [16]. In [14] they also shown the small data global existence and the decay estimates for (1.3) with d = 1 and p > 4 under the assumption that the initial data is odd. In the subsequent paper [15], they proved that when d = 1, p = 5 and λ < 0, a solution to (1.3) has dissipative structure and gains additional logarithmic decay. Aoki, Hayashi and Naumkin [2] showed the global existence and scattering of (1.3) with d = 1, 2 and p > 1 + 4/d. We refer also to the series of paper by Hayashi and Naumkin [18,19,21] and work by Hirayama and Okamoto [22] for interesting phenomena on the long time behavior of solutions to (1.3) with a derivative nonlinearity.
Recently, a blow-up result is proved by Boulenger and Lenzmann [7] for (1.3) with the mass critical and super critical focusing nonlinearity in the radial case which solves a long standing conjecture suggested by several numerical studies (see [11] for instance). Notice that most of their results hold also when lower dispersive term µ∆u is added. See also the work by Bonheure, Casteras, Gou and Jeanjean [5] for the extension of the blow up results by [7].
We now return to the inhomogeneous case. Since the point-wise decay of the solution to the linear fourth order Schrödinger equation where {W (t)} t∈R is a unitary group generated by the operator (1/2)i∆ − (1/4)i∂ 4 x 1 , and ψ + is a given function. Our main results in this paper are as follows: Theorem 1.1. Let d = 2 and let 2 < p < 3. Then for any ψ + ∈ H 0,s (R 2 ) with (p + 1 + 2ε)/2 < s < p and sufficiently small number ε > 0, there exists a unique global solution u ∈ C(R; L 2 Theorem 1.2. Let d = 3 and let 9/5 < p < 7/3. Then for any ψ + ∈ H 0,s (R 3 ) with p − 1/6 < s < p, there exists a unique global solution u ∈ C(R; L 2 To prove Theorems 1.1 and 1.2, we employ the argument by Ozawa [30], Hayashi and Naumkin [16,17]. Let us introduce a modified asymptotic profile where µ = (µ 1 , µ ⊥ ) is given by (1.7) Notice that the right hand side of (1.6) with S + ≡ 0 is the leading term of the solution to the linear fourth order equation (1.4). We first construct a solution u to the final state problem   To prove this, we first rewrite (1.8) as the integral equation For the derivation of (1.9), see Section 4 below. Next, we apply the contraction mapping principle to the integral equation (1.9) in a suitable function space. In this step the asymptotic formula (Proposition 2.1) and the Strichartz estimate (Lemma 2.3) for the linear equation (1.4) play an important role. Finally, we show that the solutions of (1.8) converge to W (t)ψ + in L 2 as t → ∞. We introduce several notations and function spaces which are used throughout this paper. For ψ ∈ S ′ (R d ),ψ(ξ) = F[ψ](ξ) denote the Fourier transform of ψ. Let ξ = |ξ| 2 + 1. The differential operators |∇| s = (−∆) s/2 and ∇ s = (1 − ∆) s/2 denote the Riesz potential and Bessel potential of order −s, respectively. We define ∂ x 1 s = F −1 ξ 1 s F −1 . For 1 q, r ∞, L q (t, ∞; L r x (R d )) is defined as follows: We will use the Sobolev spaces and their homogeneous versioṅ The weighted Sobolev spaces is defined by We denote various constants by C and so forth. They may differ from line to line, when this does not cause any confusion. The plan of the present paper is as follows. In Section 2, we prove several linear estimates for the fourth order Schrödinger type equation (1.4). In Section 3, we give several estimates for the asymptotic profile 1.6. In Section 4, we prove Theorems 1.1 and 1.2 by applying the contraction mapping principle to the integral equation (1.9). Finally in Section 5, we give several additional remarks.

Linear Estimates
In this section, we derive several linear estimates for the fourth order Schrödinger type equation (2.1) Proposition 2.1. Let u be a solution to (2.1). Then we have To calculate the oscillatory integral effectively, we show the following elementary lemma.
Proof of Lemma 2.2. Lemma 2.2 follows from the combination of the identity and integration by parts.
Proof of Proposition 2.1. Let u be a solution to (2.1). Then we have By the Fresnel integral formula for j = 2. · · · , d, we have Therefore, we find We split u into the following two pieces: To evaluate L, we split L into We rewrite L 1 as follows: Let We rewrite L 1 as follows: In addition, changing the variable 1 ) and using the Fresnel integral formula, we obtain For L 1,2 , using Lemma 2.2, we have Since Furthermore, we easily see that Combining above three inequalities, we have We split the right hand side of above inequality into the following two pieces: Using the inequalities where max(0, 1/4 − 1/(2p)) < β < 1/2. By (2.4), (2.5) and (2.7), we have Hence the Hölder and Sobolev inequalities yield for 2 p ∞ and 1/(4p) < β < 1/2, where s 1 > 1/2. Combining the above inequality and the Minkowski inequality, we have where s = s 1 + s ⊥ and s ⊥ = (d − 1)(1/2 − 1/p) for 2 p < ∞ and s ⊥ > (d − 1)/2 for p = ∞.
Next we evaluate L 2 . We write Using Lemma 2.2, we have Form (2.6), we have The same argument as that in (2.7) yields Hence for 2 p ∞ and 1/4 + 1/(4p) < β < 1/2. Combining the above inequality and the Minkowski inequality, we have Finally let us evaluate R. R can be rewritten as where {W 4LS (t)} t∈R is a unitary group generated by the linear operator (i/2)∂ 2 By using the decay estimate (see [3,35] for instance), Combining the above inequality and the Minkowski inequality, we have and (q j , r j , d) = (2, ∞, 2).

Then, the inequalitiy
holds.

Nonlinear Estimates
In this section, we derive several estimates for the asymptotic profile (1.6).
Proposition 3.1. Assume d = 2, 3 and let S + be given by (1.6). Then if 1 < s < 2 and s < p < 3, then we have for t 1, ψ + e iS + (t,ξ) If 2 s < p < 3, then we have for t 1, In particular, if we further assume s > d/2, then we have where P is a polynomial without constant term.
To prove Proposition 3.1, we employ the Leibniz and chain rules for the fractional derivatives.
We first consider the case 1 < s < p < 2. Since we obtain By Lemma 3.2, we have

Lemma 3.2, Lemma 3.3 (ii) and the interpolation inequality yield
where σ satisfies (s − 1)/(p − 1) < σ < 1. Lemma 3.3 and (3.2) imply The interpolation inequality yields is a pseudo-differential operator of order zero, by the L p boundedness of pseudo-differential operator (see [36,Chapter VI] for instance) and the interpolation inequality, we obtain Next we consider the case 1 < s < 2 p. As in the previous case, we obtain (3.1). Lemma 3.2, Lemma 3.3 (i) and the interpolation inequality yield

Lemma 3.3 and (3.7) imply
Finally let us consider the case 2 s < p < 3. Since we obtain Hence by Lemma 3.2, we have
Next, we show that the solution to (1.8) with finite X T norm is unique. Let u 1 and u 2 be two solutions satisfying u 1 X T < ∞ and u 2 X T < ∞. We put t 1 = inf{t ∈ [T, ∞); u 1 (s) = u 2 (s) for any s ∈ [t, ∞)} and ρ = max{ u 1 X T , u 2 X T }. If t 1 = T , then u 1 (t) = u 2 (t) on [T, ∞) which is desired result. If T < t 1 , as in (4.14) by the Strichartz inequality (Lemma 2.3), we have . This contradicts the assumption of t 1 . Hence u 1 (t) = u 2 (t) on [T, ∞).
From (1.8), we obtain Since u(T ) ∈ L 2 x (R 3 ), combining the argument by [37] with the Strichartz estimate (Lemma 2.3) and L 2 conservation law for (4.15), we can prove that (4.15) has a unique global solution in C(R; L 2 ). Therefore the solution u of (1.8) can be extended to all times.
To show the existence of u satisfying (4.12), we shall prove that the map Φ given by (4.13) is a contraction on for some T 3, where (q, r) = 2 p − 2 + 2ε , 2 3 − p − 2ε .

Final comments
Finally, we give several comments.
1. As mentioned in the introduction, we restricted our study of (1.1) to the case where α, β and γ satisfy (α + 3β 2 8γ )α > 0 and αγ < 0. If those conditions are violated, one should replace (1.2) by the "non-elliptic" NLS It is very likely that similar scattering results hold true in this case since the linear estimates should be essentially the same.
2. We were concerned in this paper with scattering issues for equation (1.1). On a different regime, one might look for a possible blow-up of solutions in the focusing case (λ < 0 in (1.2)). It is conjectured in [12] that a finite time blow-up should arise when p ≥ 1 + 8/(2d − 1). Proving this fact is an interesting open question.