Long-time behavior of solutions to a fourth-order nonlinear Schrödinger equation with critical nonlinearity

We consider the long-time behavior of solutions to a fourth-order nonlinear Schrödinger (NLS) equation with a derivative nonlinearity. By using the method of testing by wave packets, we construct an approximate solution and show that the solution for the fourth-order NLS has the same decay estimate for linear solutions. We prove that the self-similar solution is the leading part of the asymptotic behavior.


Introduction
We consider the Cauchy problem for a fourth-order nonlinear Schrödinger (NLS) equation where u = u(t, x) : [0, ∞) × R → C is an unknown function and u 0 is a given function. Here, F satisfies the following assumptions: A-1. F ∈ C 1 (C; C) ∩ C 2 (C\{0}; C) 1 with F(0) = F (0) = 0 and F(αu) = α 4 F(u) for α ≥ 0 and u ∈ C, where F denotes any of F u := ∂ F ∂u and F u := ∂ F ∂u . Moreover, We use the assumption (A-1) to show the local-in-time well-posedness of (1.1). More precisely, we can prove the local well-posedness of (1.1) with the quartic homogeneity replaced by |F ( j) (u)| |u| 4− j (1.2) for j = 0, 1, 2 and u = 0. However, we only consider the quartic homogeneous nonlinearity in this paper for simplicity. See also Remark 1.
To obtain the global existence (and asymptotic behavior), we employ the quartic homogeneity and (A-2). Indeed, we use these assumptions in energy estimates in Sect. 2. A typical example of F is given by F(u) = a|u| 3 u + bu 4 (1. 3) for a ∈ R and b ∈ C. We note that the first term |u| 3 u in (1.3) can be generalized as follows: for a real-valued cubic homogeneous function g ∈ C 1 (C; R)∩C 2 (C\{0}; R), u 0 g(v)dv satisfies assumptions (A-1) and (A-2), where we calculate this integral as if u is a real-variable. For example, when g(u) = |u| 3 = u Here, we mention some properties of solutions to (1.1). If u is a solution to (1.1), we have the following conservation law: (1.4) Note that (1.1) is invariant under the scaling transformation u(t, x) → λu(λ 4 t, λx) (1.5) for any λ > 0. Hence, the scaling critical Sobolev regularity is s c := − 1 2 . Asymptotic behavior of the fourth-order NLS and its related equations have been studied by several researchers. See [1,2,[5][6][7][8][9][10][11][12]14,15,19] and references therein. In particular, Ben-Artzi, Koch, and Saut [2] showed the dispersive estimates for the fourth-order Schrödinger equations. From the dispersive estimates, we can expect that a quartic nonlinearity with a derivative is critical in the sense of the asymptotic behavior of solutions to (1.1). This is a reason why we assume quartic nonlinearity in (A-1).
Hayashi and Naumkin [6,7] derived the asymptotic behavior of the solution to the fourth-order NLS equation with the gauge invariant nonlinearity: (1. 6) They proved that the asymptotic behavior of (1.6) is the same as that of the linear solution and the self-similar solution to (1.6) when λ ∈ C, ρ > 3 and λ = i, ρ = 3, respectively. They employed the factorization technique for the evolution operator of the fourth-order Schrödinger equation. For (1.1) with F(u) = u 4 , namely Hirayama and the first author [12] showed the small data global well-posedness and the scattering in the scaling critical Sobolev spaceḢ − 1 2 (R). They used the Fourier restriction norm method adapted to the spaces V p of functions of bounded p-variation and their pre-duals U p .
To state the main result, we denote H s,r (R) the weighted Sobolev space equipped with the norm u H s,r := x r i∂ x s u L 2 x for s, r ∈ R and we set H s (R) := H s,0 (R). Define the phase function (1.7) Here, a 1 3 = 3 √ a denotes the unique real cubic root of a ∈ R.
Theorem 1. Assume that the initial datum u 0 at time 0 satisfies

Let F satisfy (A-1) and (A-2). Then, there exists a unique global solution u to
for t ≥ 1 and k = 0, 1, 2. Moreover, we have the following asymptotic behavior as t → +∞. Set ρ := 1 4 ( 1 8 − ε). In the self-similar region X self (t) := {x ∈ R : t − 1 4 |x| t 3ρ }, there exists a solution Q = Q(y) to the nonlinear ordinary differential equation In the oscillatory region X osc (t) := {x ∈ R : t − 1 4 |x| t 3ρ }, there exists a unique complex-valued function W satisfying W L ∞ ∩L 2 ε such that where the error satisfies the estimates ε.
In Theorem 1, we divide R into two regions R = X self (t) ∪ X osc (t). Note that, in the results on KdV equations in [3,17,18], the asymptotic behavior is classified into three regions: self-similar, oscillatory, and decaying. This difference comes from the asymptotic behavior of the linear solutions. Indeed, the corresponding linear equation to (1.1) is invariant under the spatial inversion. Namely, if u satisfies (1.15), then u defined by also satisfies the same equation. Hence, the asymptotic behaviors for x > 0 and x < 0 are the same. On the other hand, the linear KdV (Airy) equation is not invariant under the spatial inversion (1.16). More precisely, the transformation (1.16) changes the sign of the coefficient of ∂ 3 x . Indeed, the solution to (1.17) (the Airy function) is oscillating for x > 0 and decaying for x < 0.
As mentioned above, by using the factorization technique for the fourth-order NLS equation, Hayashi and Naumkin [6] studied the asymptotic behavior of (1.1) with F(u) = |u| 3 u for small initial data in H 1,1 (R). More precisely, they proved the existence of a global solution u with e i 1 4 t∂ 4 when u 0 H 1,1 ≤ ε 1. In this paper, we employ the method of testing by wave packets as in [13]. Since we use (1.9) instead of (1.18) (as a bootstrap assumption), our assumption u 0 ∈ H 1 (R) ∩ H 0,1 (R) is better than u 0 ∈ H 1,1 (R) in [6]. See also Remark 2.

Remark 1.
We can obtain the same result as in Theorem 1 for short-range perturbations of the form where G ∈ C 2 (C; C), G u is real-valued, and there exists p 0 > 4 such that Since we can apply the same argument as in Appendix A in [3] and Appendix B in [18], we omit the details here.  [18] with the fourth-order dispersion.

Outline of proof
We give here an outline of the proof. Denote by L the linear operator of (1.1): To obtain pointwise estimates for solutions, we use the vector field x . Since J has the third derivative, it is difficult to apply J directly for the energy estimates. We then use the generator of the scaling transformation (1.5) given by S := 4t∂ t + x∂ x + 1. (1.21) Moreover, by (1.19)-(1.21), we have As in [3,17,18], we also use the operator Roughly speaking, since the operator acts as the first-order derivative for the nonlinearity, we use instead of J . We introduce the norm with respect to the spatial variable We note that By a standard fixed point argument, we have the local well-posedness in X of (1.1).
The proof is a slight modification of that in Appendix in [18]. We then make the bootstrap assumption that u satisfies the linear pointwise estimates: there exists a large constant D such that for t ∈ [1, T ] and k = 0, 1, 2. Note that we take ε > 0 small enough so that ε ≤ D −2 .
In Sect. 2, by using (1.26), for ε > 0 sufficiently small, we prove the a priori bound: where C T is a constant depending only on T . Namely, C T is independent of D and ε. Then, by the local well-posedness with (1.27), the global existence follows from closing the bootstrap estimate (1.26). In Sect. 3, we prove decay estimates in L ∞ (R) and L 2 (R) that allow us to reduce closing the bootstrap argument to considering the behavior of u along the ray v := {x = vt}. We also observe that (1.26) holds true at t = 1. Since u is complex-valued, we have to pay attention to the sign of frequencies. We thus need to slightly modify the argument in [18]. See, for example, (3.11) and the proof of Lemma 4.
To close the bootstrap argument, we use the method of testing by wave packets as in [3,4,13,18]. Here, a wave packet is an approximate solution localized in both space and frequency on the scale of the uncertainty principle. Our main task in Sect. 4 is to construct a wave packet v (t, x) to the corresponding linear equation and observe its properties.
To observe decay of u along the ray v , we use the function In Sect. 4, we prove that γ is a reasonable approximation of u. We then reduce closing the bootstrap estimate (1.26) to proving global bounds for γ . In Sect. 5, by solving an ordinary differential equation with respect to γ , we show the global existence of u. Moreover, we prove that the leading part of the asymptotic behavior is given by the self-similar solution where Q is a solution to (1.10).
Let C ∞ 0 (R) be the space of all smooth and compactly supported functions. We denote the space of all smooth and rapidly decaying functions on R by S(R). Define the Fourier transform of f by F[ f ] or f .
In estimates, we use C to denote a positive constant that can change from line to line. If C is absolute or depends only on parameters that are fixed, then we often write X Y , which means X ≤ CY . When an implicit constant depends on a parameter a, we sometimes write X a Y . We define X Y to mean Let σ be a smooth even function with 0 ≤ σ ≤ 1 and For any R, R 1 , Moreover, we define the corresponding Fourier multipliers as usual: We denote the characteristic function of an interval I by 1 I . For N ∈ 2 Z , we define We also set σ ± = σ 1 R ± and σ ± ≤R := σ ≤R 1 R ± , etc.

Energy estimates
In this section, we prove some a priori estimates of a solution u to (1.1) satisfying (1.26). First, we use an energy estimate to obtain the bound for u(t) X .

Lemma 1. Assume that F satisfies (A-1) and (A-2). Let u be a solution to
and (1.26). Then, we have where X is defined in (1.23) and the implicit constant is independent of D, T , and ε.

Remark 3.
To obtain (2.6) in the proof of Lemma 1, we only use (1.2) (instead of the quartic homogeneity). However, (2.7) is a consequence of (A-1), and we rely on (A-1) in the calculation in (2.8).
Second, we prove a priori bound for J u(t) L 2 x . We define the auxiliary space where J is defined in (1.20).

Lemma 2. Assume that F satisfies (A-1) and (A-2). Let u be a solution to
where the implicit constant is independent of D, T , and ε.

Decay estimates
In this section, we prove decay estimates for u without the bootstrap assumption (1.26). In Sect. 3.1, we decompose u into a part on which J acts hyperbolically and a part on which it acts elliptically. Since u is complex-valued, the decomposition is (a bit) different from the previous papers [3,17,18]. In Sect. 3.2, by using the decomposition in Sect. 3.1, we prove some decay estimates for u.

Hyperbolic and elliptic parts of u
We write u N := P N u. Let N (t) ∈ 2 Z be the smallest dyadic integer satisfying where σ is a derivative of σ . Hence, it follows from (2.9) and (3.1) that We decompose u N into positive and negative frequencies: For t ≥ 1 and N ≥ t − 1 4 , we define the hyperbolic and elliptic parts of u ± N as follows: The largeness of κ uses in the proof of (3.13) in Lemma 4. While the explicit value of κ is not important (e.g., we can choose κ with κ ≥ 2 10 ), we fix κ as in (3.4) for simplicity. Next, we define Hence, u hyp,± (t, x) is a finite sum of u The functions u hyp N and u ell N are essentially frequency localized near N . This is a consequence of the following lemma. See Lemma 3.1 in [16] and Lemma 4.1 [17] for the proof.
In addition, for any 0 < r < R, we have Lemma 3 yields that for any a ≥ 0, b ∈ R, and c ≥ 0, (3.10) Factorizing the symbol x − tξ 3 of J , we define These operators are useful in our analysis. Note that J − and J + are elliptic on positive and negative frequencies, respectively.

Decay estimates in L 2 and L ∞
First, we show the following frequency localized estimates.
Next, we prove (3.13). We decompose u ell,± N into three parts for any smooth function g.
We consider the estimate of the third part on the right-hand side of (3.16). By the Cauchy-Schwarz inequality, (3.10), (3.16), and (3.4), we have Hence, it follows from taking g = u ell,±,H Next, we consider the estimate of the first part on the right-hand side of (3.16). By the Cauchy-Schwarz inequality, (3.9), and (3.4), we have R t xu ell,±,L Hence, it follows from taking g = u ell,±,L N in (3.17) and (3.2) that Finally, we consider the estimate of the second part on the right-hand side of (3.16). It follows from Hence By summing up the frequency localized estimates, we obtain the L 2 -estimates.

Corollary 1. For t ≥ 1, we have
The proof is the same as that in Corollary 3.4 in [18]. We thus omit the details here. Moreover, by a repetition of the proof of Proposition 3.5 in [18], we have the pointwise decay estimates.

Testing by wave packets
In this section, we prove some properties of wave packets. In Sect. 4.1, we construct wave packets corresponding to the fourth-order Schrödinger equation. Moreover, we show that the wave packet is a good approximate solution to the linear equation. In Sect. 4.2, we prove the output (1.28) is a good approximation of u.

Construction of wave packets
where χ is a smooth function with and φ is defined by (1.7). The spatial support for v < 0. In particular, the sign of x is the same as that of v. We show that v (t, x) is essentially localized at frequency in the following sense (see Lemma 4.1 in [18], for example): where for any k, ∈ N 0 . Moreover, there exists a constant C 1 > 0 such that for any |α| ≥ 1, For |v| ≥ t − 3 4 , we define the nearest dyadic number to |ξ v | by N v ∈ 2 Z . Then, we have Moreover, let ± be the sign of v: Lemma 5 yields the following bound.

Approximation of u
In this subsection, by using wave packets constructed in Sect. 4.1, we prove the output γ (t, v) defined in (1.28) is a "good" approximation of u.
The main goal in this subsection is to prove the following proposition: Proposition 3. For t ≥ 1 and k = 0, 1, 2, we have the bound where γ and φ are defined in (1.28) and (1.7), respectively, and R k is a function satisfying t k+1 (4.23) Moreover, in the frequency space, we have where R ξ is a function satisfying Before the proof of Proposition 3, we provide two preliminary lemmas.  Setting v = t 1 2 v 1 3 z + vt, we note that for v ∈ (t) and |z| ≤ 1 2 . Then, we have L.H.S. of (4.25) t , which shows (4.25).
The second lemma says that we can replace (iξ v ) k u in (1.28) with ∂ k x u hyp,± .
Lemma 8. For t ≥ 1 and k = 0, 1, 2, we have where ± is as in (4.8) and R k is a function satisfying (4.23).

Proof of the main theorem
In this section, we prove Theorem 1. In Sect. 5.1, we derive an ordinary differential equation with respect to γ . In Sect. 5.2, we prove the global existence of the solution to (1.1). In Sect. 5.3, we show the asymptotic behavior of the global solution.

ODE with respect to γ
In this subsection, we prove the following proposition: F satisfies (A-1) and (A-2). Let u be a solution to (1.1) satisfying (1.26). Then, we have We use err to denote error terms that satisfy the estimates ε.
For the proof of Proposition 4, we use the following lemmas.

Asymptotic behavior
In this subsection, we present the proof of the asymptotic behavior of the global solution to (1.1).
Proposition 4 yields that there exists a unique function W defined on R\{0} such that for t ≥ 1, where (t 3 4 |v|) We extend W to R by defining Then, by (5.10), we have W L ∞ ξv ≤ ε.  ε.