Asymptotic behavior of solutions to a higher-order KdV-type equation with critical nonlinearity

We consider the Cauchy problem of the higher-order KdV-type equation: \[ \partial_t u + \frac{1}{\mathfrak{m}} |\partial_x|^{\mathfrak{m}-1} \partial_x u = \partial_x (u^{\mathfrak{m}}) \] where $\mathfrak{m} \ge 4$. The nonlinearity is critical in the sense of long-time behavior. Using the method of testing by wave packets, we prove that there exists a unique global solution of the Cauchy problem satisfying the same time decay estimate as that of linear solutions. Moreover, we divide the long-time behavior of the solution into three distinct regions.


Introduction
We consider the Cauchy problem for the higher-order Korteweg-de Vries (KdV) type equation where u is a real valued function, |∂ x | = (−∂ 2 x ) 1 2 , and m ∈ Z ≥3 . This equation describes the propagation of nonlinear waves in a dispersive medium. In particular, (1.1) with m = 3, 5 are called the modified KdV (mKdV) and the modified Kawahara equations, respectively.
If u is a solution to (1.1), then the total mass, the momentum, and the energy are conserved: Local-in-time well-posedness of the Cauchy problem for (1.1) follows from the same argument as in [19] (see also [17,18,4] and references therein). Owing to the conserved quantities, the local-in-time solution can be extended to the global-intime one if the values of the initial data are small. Thus, it is of interest to obtain the asymptotic behavior of solutions to (1.1).
Sidi et al. [27] studied the long-time behavior of solutions to the generalized KdV equations for α ∈ R, α ≥ 1, and p ∈ Z ≥2 . More precisely, they proved that when the initial data values are small, the global-in-time solution scatters to a linear solution if α ≥ 1 and p > α+ √ α 2 +4α 2 + 1. Kenig et al. [18] improved the results in [27], that is the scattering for small initial data values holds true if α ≥ 1 and p > max(α+1, α 2 +3). Because there exists a blow up solution in some cases when initial data values are large ( [21,23]), the assumption of small initial data values is essential.
The asymptotic behavior for (1.2) with α = 3 has been studied by several researchers (see [1,7,8,9,28,22,13,5,3,2] and references therein). In particular, p = 3 is critical in the sense of long-time behavior. In other words, while the solutions scatter for p > 3, asymptotic behavior of the solution differs from that of the linear solutions when p = 3. Moreover, Hayashi and Naumkin [10,12] showed the criticality of the quartic derivative fourth-order nonlinear Schrödinger equation (see also [11,14]), which is related to (1.1) with m = 4.
From these results, we expect that the nonlinearity of (1.1) is critical in the sense of long-time behavior. However, there is a gap between the exponent of nonlinearity in previous results and that to be critical in general. In this paper, we study the asymptotic behavior of solutions to (1.1) for m ≥ 4. Even though we used u m in (1.1) in our study, the same asymptotic behavior is obtained for (1.1) with short-range perturbations (see Remark 1.3).
To explain the critical phenomenon, we roughly derive the asymptotic behavior of linear solutions for Schwartz initial data u 0 ∈ S(R). Let U(t) denote the linear propagator, i.e., U(t) := e − 1 m t|∂x| m−1 ∂x . We note that, for t > 0, the linear solution is written as follows: where Q 0 is defined by Q 0 (y) := 1 2π R e i(yξ− 1 m |ξ| m−1 ξ) dξ, that is, Q 0 satisfies the ordinary differential equation Sidi et al. [27] proved the following estimate for Q 0 : where y := (1 + |y| 2 ) We expect that, for v > 0, solutions initially localized spatially near zero and at frequencies near ±ξ v , where ξ v := v 1 m−1 , travel along the ray Γ v := {x = vt}. Hence, the linear solution U(t)u 0 (x) decays rapidly as t − 1 m x → −∞ and oscillates as t − 1 m x → +∞. In particular, the stationary phase method shows that, as t − 1 m x → +∞, there exists a constant c 0 such that where the phase function is given by Moreover, in the self-similar region |t − 1 m x| 1, This observation implies that if u 0 L 2 + xu 0 L 2 ≤ ε then we have for k = 0, 1, . . . , m − 2. We expect that solutions to (1.1) satisfy the same pointwise estimates as linear solutions above when the initial data values are small. Then, Because the integral is not bounded by sup t≥1 u(t) L 2 , we only expect that the solution behaves like a linear solution up to t ∼ exp(ε −m+1 ). Especially, the asymptotic behavior of the solution differs from that of linear solutions.
1.1. Main result. To state our results precisely, we introduce notation and function spaces. We denote the set of positive and negative real numbers by R + and R − , respectively. We denote the usual Sobolev space by H s (R). For s ∈ R, we denote the weighted Sobolev spaces by Σ s (R) equipped with the norm f Σ s := f H s + xf L 2 .
Theorem 1.1. Let m ≥ 4. Assume that the initial datum u 0 at time 0 satisfies Then, there exists a unique global solution u to (1.
In the decaying region ε.
In the self-similar region X 0 (t) := {x ∈ R : t − 1 m |x| t (m−1)ρ }, there exists a solution Q = Q(y) to the nonlinear ordinary differential equation In the oscillatory region X + (t) : where the error, err x , satisfied the estimates ε.
In the corresponding frequency region X + (t) : ε.
As (1.1) has time reversal symmetry given by u(t, x) → u(−t, −x), we get the corresponding asymptotic behavior as t → −∞.
Theorem 1.1 presents the leading asymptotic term not only in L ∞ (R), but also in L 2 (R). In addition, as with linear solutions, we divide the long-time behavior of the solution to (1.1) into three distinct regions. Moreover, Theorem (1.1) says that there is a difference between u and linear solutions in X 0 (t), while the leading term of u in X + (t) behaves like a linear solution.
The regularity assumption u 0 ∈ H 2m m−1 (R) needs to show the existence of a localin-time solution u with U(−t)u ∈ C([−T, T ]; Σ 0 (R)) (see Remark 1.5 below). In other words, regularity is no longer required in the proof of the global existence and asymptotic behavior.
is a solution to (1.1) with the initial datum u(0) = R u 0 (x)dxδ 0 , where δ 0 denotes the Dirac delta measure concentrated at the origin. Moreover, by (5.2) below, we can roughly state that then the self-similar solution vanishes, and the solution u to (1.1) decays faster than t − 1 m . Accordingly, the nonlinearity of (1.1) is not critical in the long-time behavior in this case. Remark 1.3. Theorem (1.1) is also true for short-range perturbations of the form where there exists a real number p > m such that F ∈ C 3 (R) and 2m . For completeness, we briefly outline these modifications in Appendix B.

1.2.
Outline of the proof. We give an outline of the proof of Theorem 1.1. Let L denote the linear operator of (1.1): To obtain pointwise estimates of solutions, we use the "vector field" However, J does not behave well with respect to the nonlinearity, so as in [7,8,5] we instead work with Λ := ∂ −1 x (mt∂ t + x∂ x + 1). Here, S := mt∂ t + x∂ x + 1 is the generator of the scaling transformation for (1.1): for any λ > 0. Moreover, S is related to L and J as follows: We introduce the norm with respect to the spatial variable as follows: We note that 2m . Well-posedness in X follows from a similar argument as in [19].
We give a proof of this well-posedness result in Appendix A.
For initial data u 0 with u 0 Σ m−1 2m ≪ 1, we can find an existence time T > 1 and a unique solution u ∈ C([−T, T ]; X) to (1.1). Moreover, Proposition 1.4 says that the existence of a global solution u ∈ C(R; X) follows from the decay estimate (1.4).
Because (1.1) is time reversal invariant, it suffices to consider the case t ≥ 0. We then make the bootstrap assumption that u satisfies the linear pointwise estimates: In §2, under this assumption, for ε > 0 sufficiently small, we have the energy estimate where C is a constant independent of D and T . To complete the proof of global existence, we need to close the bootstrap estimate (1.10).
In §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 Γ v . We also observe that (1.10) holds true at t = 1.
To close the bootstrap argument, we use the method of testing by wave packets as in [5,6,15,25]. 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 §4 is to construct a wave packet Ψ v (t, x) to the corresponding linear equation and observe its properties. Here, to show that Ψ v (t, x) is a good approximate solution to the linear solution, we use the fact that m is a natural number. Because Ψ v (t, x) is essentially frequency localized near ξ v = v 1 m−1 (see Lemma 4.1), the linear operator L acts on Ψ v as ∂ t + i(−i∂ x ) m . Hence, we can avoid applying the nonlocal operator |∂ x | directly in the calculation of LΨ v (see (4.9)).
To observe decay of u along the ray Γ v , we use the function In §4, we also prove that γ is a reasonable approximation of u. We then reduce closing the bootstrap estimate (1.10) to proving global bounds for γ.
In §5, we show that γ is the leading asymptotic term of u in X + (t). Moreover, we prove existence of a solution Q to (1.5).
1.3. Notation. We summarize the notation used throughout this paper. We set N 0 := N ∪ {0}. We denote the space of all rapidly decaying functions on R by S(R). We 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 considered 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 σ(ξ) = 1, if |ξ| ≤ 1, 0, if |ξ| ≥ 2. For For any N, N 1 , N 2 ∈ 2 Z with N 1 < N 2 , we define We denote the characteristic function of an interval I by 1 I . For N ∈ 2 Z , we define

Energy estimates
We show energy estimates for solutions u to (1.1) under (1.10).
and (1.10). Then, where the implicit constant is independent of D, T , and ε.
To treat the fractional derivatives, we use the Kato-Ponce commutator estimate (see [16,19]): Proof of Lemma 2.1. Because the desired bound for 0 ≤ t ≤ 1 follows from Proposition 1.4, we consider the case t ≥ 1. Integration by parts yields By Lemma 2.2 and (1.10), we have which imply that for solutions u to (1.1) From (Dε) m−1 ≪ ε, Gronwall's inequality with the above estimates implies We define the auxiliary space where the implicit constant is independent of D, T , and ε Proof. By (1.10), we see that From (1.9) and Lemma 2.1, we therefore have We use a self-similar change of variables by defining A direct calculation shows Hence, we have By integrating this with respect to t, we have , we obtain the desired bound.

Decay estimates
In this section, we prove a number of estimates for u without assuming (1.10). We divide u into two parts on which J acts hyperbolically and elliptically. For simplicity, we use the following notation: For t ≥ 1 and N > t − 1 m , we define the hyperbolic and elliptic parts of u + N as follows: and κ := 2 2m+3 . This large constant κ is needed to show (3.6) in Lemma 3.3 below. Moreover, we define The functions u hyp,+ N and u ell,+ N are essentially localized at frequency N in the following sense: For any a ≥ 0, b ∈ R, and c ≥ 0, These estimates are consequences of the following lemma (see, for example, [24,25]): Lemma 3.1. For 2 ≤ p ≤ ∞, any a, b, c ∈ R with a ≥ 0 and a + c ≥ 0, and any R > 0, we have Moreover, we may replace σ R on the left hand side by σ >R if a + c > b + 1 and by σ <R if a + c ≥ 0 and b = 0.
3.1. Frequency localized estimates. Since by factorizing out the term x − tξ m−1 , we define These operators are useful in our analysis. We begin with the following preliminary observation.
Let a be a real number and let g be a (C-valued) smooth function supported in R + or R − . For any integer k > 0, the following equations hold: where C a,± k,l and D a,± k,l are real constants depending on a, k, and l. In particular, Proof. For k = 1, integration by parts yields Similarly, we have Hence, the equations hold when k = 1 with C a,± and D a,± 2,1 = ±a. Next, we assume that these equalities hold up to k − 1. Integration by parts yields Hence, by setting we obtain the equations for the real part. Similarly, by setting we obtain the equations for the imaginary part. From the recurrence relations with We show the frequency localized estimates.
Proof. Set f := J + u hyp,+ N . We apply Lemma 3.2 to obtain Similarly, we get Hence, we have On the other hand, because 2) and tN m > 1 yield Combining this with (3.7), we obtain (3.5).
For the elliptic bound, we decompose u ell,+ N into three parts We observe that the equation holds for any smooth function f and odd m. Similarly, for even m, holds for any smooth function f . In what follows, we only consider the case when m is even, because the case when m is odd can be similarly handled. Since Taking f = σ ≥κtN m−1 u ell,+ N in (3.9), we have which shows that

3.2.
Decay estimates in L 2 (R) and L ∞ (R). First, by summing up the frequency localized estimates, we show the L 2 -estimates.
For the elliptic bound, we note that We use (3.1), (3.3), and (3.6) to obtain Second, we show the pointwise decay estimates.
Because u hyp,+ (t, x) is a finite sum of u hyp,+ N (t, x)'s, this yields the desired hyperbolic bound (3.14).
Next, we show the elliptic bound. First, we consider the low frequency part. For Second, we consider the high frequency part. For |x| ≤ t 1 m , the Gagliardo-Nirenberg inequality and (3. Thus, by (3.6), we have For |x| > t Then, the Gagliardo-Nirenberg inequality and Lemma 3.1 lead to Hence, we have Remark 3.6. For t ≥ 1, the estimate holds true. Indeed, the Gagliardo-Nirenberg inequality, (3.2), and (3.5) yield Accordingly, by combining this estimate with (3.15) and Remark 2.4, we obtain (1.4) at t = 1.
For v ≥ t − m−1 m , we define the nearest dyadic number to ξ v by N v ∈ 2 Z . Then, Integration by parts with (4.2) yields for any a, l ≥ 0, which implies and any a, c ≥ 0. Next, we show that Ψ v is a good approximate solution for the linear equation. We begin with the following preliminary observation. Set S j := {k = (k 1 , . . . , k j ) ∈ N 0 : 0 ≤ k 1 ≤ · · · ≤ k j ≤ j, k 1 + · · · + k j = j} for j ∈ N.
is a constant depending on j ∈ N, k ∈ S j . In particular, Proof. A direct calculation shows Hence, we have (0,1,2) = 3.
We assume that (4.7) holds up to j − 1. Because , which shows (4.7). In particular, the following recurrence equations hold true:

Lemma 4.2 and ∂
where and X is a nonnegative continuous function supported in [− 1 2 , 1 2 ]. Therefore, we obtain the following: Because χ has the same localization as χ, the first term on the right hand side of (4.9) is essentially localized at frequency ξ v . For the sake of completeness, we give a proof here, although the proof is similar to that of Lemma 4.1.
and any a, c ≥ 0, we have Proof. We write The same calculation as in the proof of Lemma 4.1 yields where Since χ 0 (·, α) ∈ S(R) for α ≥ 1, we have , integration by parts yields which implies the desired bound.

4.2.
Testing by wave packets. Let C 2 > 0 be the constant appearing in (4.2) with k = 2 and l = 0, that is, For t ≥ 1, we define Here, C 1 is the constant appearing in (4.3). The large constant C * is needed to show the pointwise estimate (4.13) in the frequency space. We observe that the output γ(t, v) defined by (1.11) is a "good" approximation of u for v ∈ Ω(t).  ∂ Moreover, in the frequency space, we have where R ξ is a function satisfying Proof. First, we show that holds true. By a change of variables using z = λ(x − vt), L.H.S. of (4.14) = t . Setting for v ∈ Ω(t) and |z| ≤ 1 2 . Then, we have L.H.S. of (4.14) t .
Second, we show that u in the definition of γ is replaced with u hyp,+ up to error terms; In fact, Proposition 3.5 and (4.6) imply In addition, by (4.14) and (3.12), we have Hence, from λv , we obtain (4.15). Third, we observe that the equation holds true. We note that Here, (3.14) yields By (4.14) and (3.10), we have These estimates and (4.15) show (4.17).
For the L 2 -estimate in the frequency space, we change variables using v = ξ v + λζ(1 − θ). Because which concludes the L 2 -estimate in (4.13)

Proof of the main theorem
We show the following estimate forγ.
where the implicit constant is independent of D, T , and ε.

Proof. A direct calculation yieldṡ
The bootstrap assumption (1.10) yields Moreover, from (4.9), (4.6), Lemma 4.3, and Proposition 3.5, we have Here, from (3.11), we obtain In addition, we use (3.11) and (4.14) to obtain When t − 1 m |x| 1, owing to (4.12), we only need to show that where the implicit constant is independent of D, T , and ε.
For v ≥ C * , where C * is defined by (4.11), the Gagliardo-Nirenberg inequality, Proposition 1.4 and Lemma 4.1 lead to The fundamental theorem of calculus and Proposition 5.1 yield Then, Bernstein's inequality, (4.6), Proposition 3.5, and Lemma 2.3 yield The fundamental theorem of calculus and Proposition 5.1 lead to which implies |γ(t, v)| ε for t ∈ [t 0 , T ]. Accordingly, we conclude that (1.4) holds for any t ∈ [1, T ].
By (1.4) and the first estimate in (5.1), we see that Because Lemma 2.3 implies that by taking the limit as t → ∞, we have that Q is a solution to (1.5). Moreover, (5.1) and the mass conservation law yield Therefore, u(t, x) is a solution to (1.1) with u(0) = R u 0 (x)dxδ 0 , where δ 0 denotes the Dirac delta measure concentrated at the origin.
Finally, we prove the asymptotic behavior of the global solution. The estimates in the self-similar region X 0 (t) follow from (5.1). Moreover, the estimates in the decaying region X − (t) are consequences of Lemma 2.3, (3.12), and (3.15). Hence, we only need to show the estimates in the oscillatory region X + (t). Proposition 5.1 yields ε.
Here, we set and extend W to R by defining Then, from (1.4) and (4.14), we see that Appendix A. Well-posedness In this appendix, we show the local-in-time well-posedness of the Cauchy problem of (1.6) as well as (1.1). Here, we assume p ≥ m ≥ 3.
To show well-posedness for (1.1), we can apply the Fourier restriction norm method. In fact, Grünrock [4] proved well-posedness for (1.1) in H s (R) for odd values of m ≥ 5 and s > − 1 2 . On the other hand, because F may not be a polynomial with respect to u, some regularity is needed to be well-posed for (1.6).
We need to introduce some notation. Let 1 ≤ p, q ≤ ∞ and T > 0. Define with T = t to indicate the case when T = ∞. The maximal function estimate and the local smoothing estimate are the main tools in the proof (see Theorems 2.5 and 4.1 in [18] respectively).
We note that if u is a solution to (1. ( u(t) H s + Λu(t) L 2 ) u 0 Σ s .
Moreover, the flow map u 0 ∈ Σ s (R) → u ∈ Z s T is locally Lipschitz continuous. Proof. The fractional Leibnitz and chain rules (see Appendix in [19]) and Lemma A.3 yield (A. 6) YT . In addition, Hölder's inequality and Lemma A.3 imply We define the operator K u0 (u) by A similar calculation as above yields we find that the mapping K u0 is a contraction on the ball B(2C u 0 Σ s ) := {u ∈ Z s T : u Z s T ≤ 2C u 0 Σ s }. Accordingly, there exists a unique solution u ∈ B(2C u 0 Σ s ) to (1.6).

Appendix B. Short-range perturbations
In this appendix, we outline some modifications to Theorem 1.1 in the case of short-range perturbations (1.6). The main differences appear in the energy estimate.
Let p be a real number with p > m. The existence of a local-in-time solution u with U(−t)u(t) ∈ Σ We need some modifications in the energy estimate for Λu when p ∈ (m, m + 1 2 ), because the second and third terms on the right hand side of (A.5) do not have enough decay. Set a := max( 2m+1−2p 2m , 0). Because (1.10) yields Hence, Gronwall's inequality shows u(t) X ε t a+ε . The remaining arguments in §5 are unchanged as long as ρ = 1 m ( 1 2m − ε) is replaced with ρ := 1 m ( 1 2m − a − ε).