In this work, consideration is given to the initial value problem associated with the periodic fifth-order KdV–BBM equation. It is shown that the uniform radius of spatial analyticity σ(t) of solution at time t is bounded from below by ct^−2/3 (for some c > 0), given initial data η₀ that is analytic on the circle and has a uniform radius of spatial analyticity σ₀. The proof of our main theorems is based on a contraction mapping argument, a method of approximate conservation law in a modified Gevrey spaces, Hölder’s inequality, Sobolev algebra, Cauchy–Schwartz inequality, and Sobolev embedding.

1. Introduction

We consider the initial value problem (IVP) for the fifth order Korteweg-de Vries (KdV)–Benjamin–Bona–Mahony (BBM) type equation on the circle

()

where the parameters γ₁, γ₂, δ₁, δ₂, and γ are constants that satisfy certain constraints; see [1–3] for more details.

The IVPs for some dispersive equations have plenty of significant issues under the periodic setting compared to nonperiodic IVPs. The problem that makes a difference from the nonperiodic IVP is the presence of nontrivial resonance. In what follows, the fifth-order KdV–BBM equation (1) describes the unidirectional propagation of water waves, and this was introduced by Bona et al. [1] by using the second-order approximation in the two-way model, the so-called abcd-system derived in [4, 5].

If the parameter γ = 7/48, then sufficiently smooth solutions for the IVP (1) enjoys the energy conservation

()

Local well-posedness of the IVP (1) with data in the Sobolev spaces for s ≥ 1 was established by Carvajal et al. [2]. Moreover, for γ₁, δ₁ > 0 and γ = 7/48, the authors used the conserved energy quantity to prove global well-posedness of (1) for data in , s ≥ 2. Furthermore, the authors used the method of splitting argument to improve the global well-posedness result in for 1 ≤ s < 2. In addition to this, the authors also showed that the IVP (1) is ill-posed in the sense that the flow-map that takes initial data to solution can not be continuous for given data in . In line with this, the decay rate for (1) on the real line has been studied in [3, 6, 7].

As far as the author knows, the decay rate for (1) with initial data η₀(x) that is analytic on the circle having a uniform radius of spatial analyticity σ₀ > 0 has not yet been studied. Thus, in this work, we are interested in studying the property of spatial analyticity for the solution η(x, t) to the IVP (1), given initial data η₀(x) that are real analytic in a strip of width 2σ₀ around the x-axis of the complex plane. The question is then whether this property persists for all later times t, but with a possibly smaller and shrinking radius of analyticity σ(t) > 0, i.e., it is the solution u(x, t) of (1) analytic in S_σ(t for all t? For short times, it is shown that the radius of analyticity remains at least as large as the initial radius, i.e., one can take σ(t) = σ₀. For large times on the other hand, we use the idea introduced in [8] (see also [9]) to show that σ(t) can decay no faster than t^−2/3 as t⟶∞. For this purpose, we need the modified Gevrey space

, for σ > 0 and

which is endowed with the norm

()

where 〈k〉² = 1 + k² and

denotes the spatial Fourier transform of f given by

()

In fact, the space

was introduced by Dufera et al. [10] from the Gevrey space

by replacing the exponential weight e^σ|k| with the hyperbolic weight cos h(σ|k|). Here, the Gevrey space

is a Hilbert space of periodic functions endowed with the norm

()

It is clear that G^0,s = H^0,s = H^s, where denotes the L²-based Sobolev space of order s.

Since 1/2e^σ|k| ≤ cos h(σ|k|) ≤ e^σ|k|, the

and

-norms are equivalent in the sense that

()

The interest in the space is due to (6), and the Paley–Wiener Theorem [11], which states that for σ > 0 and s ≥ 0, a 2π-periodic function f belongs to if and only if f is the restriction to the real line of a function F, which is holomorphic in the strip and satisfies the bound .

Information about the domain of analyticity of a solution to a partial differential equation (PDE) can be used to gain a quantitative understanding of the structure of the equation, and to obtain insight into underlying physical processes. The idea of obtaining spatial analyticity was introduced by Kato and Masuda [12]. There are various studies concerning the properties of spatial analyticity of solutions for a large class of nonlinear partial differential equations. For instance, parabolic equations [13], Navier–Stoke equation [14], cubic Szegő equation [15], periodic generalized KdV equation [16,17], generalized Euler equation [18], second-order analytic nonlinear differential equations [19], semilinear Klein–Gordon equations [20] and the references therein.

Rapid progress has been made lately in obtaining an algebraic decay rate of the radius, i.e., σ(t) ~ t^−α for some α ≥ 1, to various nonlinear dispersive PDEs. For example, Himonas et al. in [21] studied the persistence of spatial analyticity for periodic solutions of the dispersion-generalized (g) KdV equation for β ≥ 2. For a class of analytic initial data with a uniform radius of analyticity σ₀ > 0, the authors obtained an asymptotic lower bound σ(t) ≥ ct^−p on the uniform radius of analyticity σ(t) at time t, as t⟶∞, where p = max{1, 4/β}. Moreover, Tesfahun [22] showed that the uniform radius of spatial analyticity σ(t) of solutions u(t) at time t to the KdV equation cannot decay faster than t^−4/3 as t⟶∞ given initial data that is analytic with fixed radius σ₀ > 0. Furthermore, in a paper [23], Himonas and Petronilho studied the IVP of the Benjamin–Bona–Mahony (BBM) equation with initial data that are analytic on the torus and have a uniform radius of analyticity σ₀ > 0. In their work, they examined the evolution of the radius of spatial analyticity σ(t) of the solution u(t) at any future time t. The authors showed that the size of the radius of spatial analyticity persists for some time and after that, it evolves in such a way that its size at any time t is bounded below by ct⁻¹ for some c > 0. A similar lower bound of spatial analyticity was obtained in [24, 25] for the Schrödinger equation with cubic nonlinearity on both the real line and the circle. The method used in these papers was first introduced by Selberg and Tesfahun [8] in the context of the 1d Dirac–Klein–Gordon equations, which is based on approximate conservation laws in a Gevrey space and Bourgan’s Fourier restriction method.

In an attempt to improve the decay rate obtained so far, the use of approximate conservation law in the modified Gevrey space can yield a decay rate of order t^−1/(2α) for some 0 < α ≤ 1. For instance, in the paper [10], Dufera et al. obtained an asymptotic lower bound

on the uniform radius of analyticity σ(t) of the solution u(·, t) of the Beam equation as t⟶∞. Moreover, Esfahani and Tesfahun [26] showed persistence of spatial analyticity of solutions for the sixth-order Boussinesq equation with cubic nonlinearity, and they obtained a decay rate of order t^−1/2 as t⟶∞. The similar lower bound of spatial analyticity has been obtained in [6, 9, 27] for the fourth-order Schrödinger equation with cubic nonlinearity, fifth-order KdV–BBM equation and Kawahara equation on the real line. Furthermore, in the paper [28], Getachew shows that the radius of spatial analyticity σ(t) of the solution u(t) for the BBM equation on the circle does not decay faster than c|t|^−2/3 (for some constant c > 0) as |t|⟶∞. Such a decay rate is obtained in the view of the inequality

()

which follows from an interpolation of

We remark that the modified Gevrey spaces satisfy the embeddings

()

for any

, 0 ≤ σ < σ^′ and some constant C > 0. In particular, for σ = 0, and all

, σ^′ > 0, we have

()

As a consequence of property (9) and the existing well-posedness theory in (see [2]), we conclude that the IVP (1) with initial data η₀ in for all σ₀ > 0 and s ≥ 1 have a unique and global in time solution provide that γ₁, δ₁ > 0 and γ = 7/48.

The main result in this paper is as follows.

Theorem 1. Assume γ₁, δ₁ > 0. Let s ≥ 1, γ = 7/48, and for σ₀ > 0. Then, the global solution η of (1) satisfies

()

with asymptotic lower bound

()

where c > 0 is a constant depending on the initial data norm

Notation: throughout this paper, we use C to denote generic constant that may vary at each occurrence. For any positive quantities a and b, we use a≲b to indicate an estimate of the form a ≤ Cb. Moreover, we write a ~ b if a≲b and b≲a. Furthermore, we denote by a+ the quantity a + ε for any ε > 0 and by D^α the differential operator given by .

2. Local Well-Posedness in H^σ,s

In this section, we prove the local well-posedness of the IVP (1) in the space H^σ,s for s ≥ 1 and σ > 0, using multilinear estimates in combination with a contraction mapping argument. In fact, we outline the argument in [3] that enables the authors to obtain the local well-posedness result for the IVP associated to the fifth oder KdV–BBM model on the real line in G^σ,s with s ≥ 1, σ > 0.

Taking the spatial Fourier transform of (1), we obtain

()

where

()

with

()

Defining the Fourier multipliers

()

In view of (12), the IVP (1) can be rewritten as in the following form:

()

where

()

Then, due to the Duhamel’s integral formula the integral equation for (16) and (17) is given by

()

Lemma 1 (see [23], Lemma 2.1, [2], Proposition 2.1.)Let with σ ≥ 0, and s ≥ 0. Then, we have

()

where ω(D) is the Fourier multiplier operator defined by

with

()

Lemma 2. Let τ(D) be the operator given in (6), σ > 0, and s ≥ 0. Then, we have

()

Proof 1. Since γ₁, δ₁ > 0, there exists a constant C > 0 such that τ(k) ≤ Cω(k) for . Using this fact, the definition of G^σ,s-norm and the estimate in Lemma 1, we obtain

()

Lemma 3. Let ψ(D) be the operator given in (6), σ > 0, and s > 1/2. Then, we have

()

Proof 2. Note that

()

for all

. Let

()

Now, using the definition of G^σ,s-norm, (24) and (25), triangle inequality, Parseval’s identity, the convolution (see [29], pp. 381),

()

From Lemma 1 and Sobolev algebra, we obtain

()

Lemma 4. Let ψ(D) be the operator given in (6), σ > 0, and s ≥ 1. Then, we have

()

Proof 3. Using Lemma 1 with s − 1 ≥ 0, we have

()

Combining the inequalities found in Lemmas 2–4, we obtain the following nonlinear estimate.

Lemma 5. Let F(η) be given as in (8). Then, for any s ≥ 1 and σ > 0, we have

()

Now, applying the contraction mapping argument to the integral equation (9) and using Lemma 5 yields the following local result.

Theorem 2. Let η₀ ∈ H^σ,s for s ≥ 1 and σ > 0. Then, there exists a unique solution

()

of the IVP (1) with existence time

()

Moreover,

()

Here, we use the notation

()

3. Approximate Conservation Law

Let us fix γ₁, δ₁ > 0 and γ = 7/48. By applying the operator cosh(σ|k|) to both sides of equation (1), the function v(x, t) = cos h(σ|k|)η(x, t) satisfies the equation

()

where

()

with

()

Define a modified energy functional associated with the function v by

()

For σ = 0, we have v = η. Since E(t) = E(0), we have the conservation

()

However, this fails to hold for σ > 0. Nevertheless, we will prove an approximate conservation law by establishing growth estimate. To do this, we differentiate the modified energy functional, and then using (35) and (36) and integration by parts, and obtain

()

Consequently, integration in time gives

()

Lemma 6. For all v ∈ H², we have

()

Proof 4. Using (36) and (37), Plancherel’s Theorem and Cauchy–Schwartz inequality, we get

()

Then, estimate (42) reduces to proving the following estimates:

()

where α ∈ (0, 1] is any arbitrary constant.

To prove the estimates (44)–(46), we recall the following estimate from ([10], Lemma 3).

Lemma 7. Let p > 1 be any integer and such that |ξ₁| ≤ |ξ₂| ≤ ⋯≤|ξ_p| and . Let . Then,

()

From Lemma 7, one can deduce that

()

Then, for any α ∈ [0, 1], the interpolation between (47) and (48) yields

()

Proof of (44): by taking the Fourier transform, we write

()

where M(σ|ξ|) be given as in Lemma 7.

By symmetry, we may assume |k₁| ≤ |k₂|. Now, let

()

Then, for any choice of α ∈ (0, 1], applying (26) and (49) gives

()

Therefore, using Parseval’s identity, Hölder’s inequality, and Sobolev embedding, we obtain

()

Proof of (45): by taking the Fourier transform, we write

()

where M(σ|ξ|) be as in (49). Then, using (49) for α = 3/4, we get

()

By Parseval’s identity, Hölder’ inequality, and Sobolev embedding, we obtain

()

Proof of (46): similarly, by taking the Fourier transform, we write

()

where M(σ|k|) be given as in (49).

By symmetry, we may assume |k₁| ≤ |k₂| ≤ |k₃|. Then, for any choice of α ∈ (0, 1], by using (26) and (49), we obtain

()

Therefore, using Parseval’s identity, Hölder’s inequality, and Sobolev embedding, we obtain

()

Theorem 3. (Approximate conservation law). Let η₀ ∈ H^σ,2 for all σ > 0, and η be the local-in-time solution to the IVP (1) that is constructed in Theorem 2. Then,

()

Proof 5. From (15), we have the bound

()

where T is as in (32).

On the other hand, we have

()

Then, plugging (61) and (62) into (42) and combining it with (41) yields the desired estimate (60).

4. Proof of Theorem 1

We closely follow the argument presented in [24, 30]. First, we consider the case s = 2. Then, the general case, s ≥ 1, will essentially reduce to s = 2 as shown in the next subsection.

4.1. The Case: s = 2

Suppose that

for some σ₀ > 0. From the local theory there is a unique solution

()

of (1) constructed in Theorem 2 with existence time T as in (32).

Since

()

we have

()

Now, following the argument in [24, 30], we can construct a solution on [0, T₀] for arbitrarily large time T₀ by applying the approximate conservation law in Theorem 3 so as to repeat the above local result on successive short time intervals of size T to reach T₀ by adjusting the strip width parameter σ according to the size of T₀. Doing so, we establish the bound

()

for σ satisfying

()

Thus, E_σ(t) < ∞ for t ∈ [0, T₀], which in turn implies

()

4.2. The General Case: s ≥ 1

For any

, we apply the Gevrey embedding inequality in (8) to get

()

From the local theory, there is a

such that

()

and

()

where a_∗ > 0 is a constant depending on

and σ₀. Applying the Gevrey embedding inequality in (8), we now conclude that

()

and

()

This completes the proof of Theorem 1.

Conflicts of Interest

The author declares no conflicts of interest.

Funding

The author received no financial support for this manuscript.

Open Research

Data Availability Statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

References

1 Bona J. L., Carvajal X., Panthee M., and Scialom M., Higher-Order Hamiltonian Model for Unidirectional Water Waves, Journal of Nonlinear Science. (2018) 28, no. 2, 543–577, https://doi.org/10.1007/s00332-017-9417-y, 2-s2.0-85030667716.
10.1007/s00332-017-9417-y
Web of Science® Google Scholar
2 Carvajal X., Panthee M., and Pastran R., On the Well-Posedness, Ill-Posedness and Norm-Inflation for a Higher Order Water Wave Model on a Periodic Domain, Nonlinear Analysis. (2020) 192, https://doi.org/10.1016/j.na.2019.111713.
10.1016/j.na.2019.111713
Web of Science® Google Scholar
3 Carvajal X. and Panthee M., On Propagation of Regularities and Evolution of Radius of Analyticity in the Solution of the Fifth-Order KdV–BBM Model, Zeitschrift für Angewandte Mathematik und Physik. (2022) 73, no. 2, https://doi.org/10.1007/s00033-022-01704-0.
10.1007/s00033-022-01704-0
Web of Science® Google Scholar
4 Bona J. L., Chen M., and Saut J.-C., Boussinesq Equations and Other Systems for Small- Amplitude Long Waves in Nonlinear Dispersive Media I. Derivation and Linear Theory, Journal of Nonlinear Science. (2002) 12, no. 4, 283–318, https://doi.org/10.1007/s00332-002-0466-4, 2-s2.0-0011913755.
10.1007/s00332-002-0466-4
Web of Science® Google Scholar
5 Bona J. L., Chen M., and Saut J.-C., Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media II, The Nonlinear Theory, Nonlinearity. (2004) 17, no. 3, 925–952, https://doi.org/10.1088/0951-7715/17/3/010, 2-s2.0-2442660478.
10.1088/0951-7715/17/3/010
Web of Science® Google Scholar
6 Mebrate S., Dufera T., Tesfahun A., Dufera T. T., and Tesfahun A., Improved Lower Bound for the Radius of Analyticity of Solutions to the Fifth Order, KdV-BBM Type Equation, Bulletin of the Iranian Mathematical Society. (2024) 50, no. 4, 62, https://doi.org/10.1007/s41980-024-00882-z.
10.1007/s41980-024-00882-z
Web of Science® Google Scholar
7 Belayneh B., Tegegn E., and Tesfahun A., Lower Bound on the Radius of Analyticity of Solution for Fifth Order KdV-BBM Equation, Nonlinear Differential Equations and Applications. (2022) 29, no. 1, https://doi.org/10.1007/s00030-021-00738-z.
10.1007/s00030-021-00738-z
PubMed Web of Science® Google Scholar
8 Selberg S. and Tesfahun A., On the Radius of Spatial Analyticity for the 1d Dirac-Klein-Gordon Equations, Journal of Differential Equations. (2015) 259, no. 9, 4732–4744, https://doi.org/10.1016/j.jde.2015.06.007, 2-s2.0-84938213779.
10.1016/j.jde.2015.06.007
Web of Science® Google Scholar
9 Getachew T., Belayneh B., and Tesfahun A., Propagation of Radius of Analyticity for Solutions to a Fourth Order Nonlinear Schrödinger Equation, Mathematical Methods in the Applied Sciences. (2024) 1–11.
Web of Science® Google Scholar
10 Dufera T. T., Mebrate S., and Tesfahun A., On the Persistence of Spatial Analyticity for the Beam Equation, Journal of Mathematical Analysis and Applications. (2022) 509, no. 2, https://doi.org/10.1016/j.jmaa.2022.126001.
10.1016/j.jmaa.2022.126001
Web of Science® Google Scholar
11 Katznelson Y., An Introduction to Harmonic Analysis, 1976, Dover Publications, Inc.
Google Scholar
12 Kato T T. and Masuda K., Nonlinear Evolution Equations and Analyticity I, Annales de l′Institut Henri Poincaré C, Analyse non Linéaire. (1986) 3, 455–467.
10.1016/s0294-1449(16)30377-8
Google Scholar
13 Ferrari A. B. and Titi E. S., Gevrey Regularity for Nonlinear Analytic Parabolic Equations, Communications in Partial Differential Equations. (1998) 23, no. 1, 424–448, https://doi.org/10.1080/03605309808821336.
10.1080/03605309808821336
Web of Science® Google Scholar
14 Foias C. and Temam R., Gevrey Class Regularity for the Solutions of the Navier-Stokes Equations, Journal of Functional Analysis. (1989) 87, no. 2, 359–369, https://doi.org/10.1016/0022-1236(89)90015-3, 2-s2.0-33746636159.
10.1016/0022-1236(89)90015-3
Web of Science® Google Scholar
15 Gérard P., Guo Y., and Titi E. S., On the Radius of Analyticity of Solutions to the Cubic Szegő Equation, Annales de l′Institut Henri Poincaré C, Analyse non linéaire. (2015) 32, no. 1, 97–108, https://doi.org/10.1016/j.anihpc.2013.11.001, 2-s2.0-84923212024.
10.1016/j.anihpc.2013.11.001
Web of Science® Google Scholar
16 Hannah H., Himonas A. A., and Petronilho G., Gevrey Regularity of the Periodic gKdV Equation, Journal of Differential Equations. (2011) 250, no. 5, 2581–2600, https://doi.org/10.1016/j.jde.2010.12.020, 2-s2.0-78751590930.
10.1016/j.jde.2010.12.020
Web of Science® Google Scholar
17 Himonas A. A. and Petronilho G., Analytic Well-Posedness of Periodic gKdV, Journal of Differential Equations. (2012) 253, no. 11, 3101–3112, https://doi.org/10.1016/j.jde.2012.08.024, 2-s2.0-84866184393.
10.1016/j.jde.2012.08.024
Web of Science® Google Scholar
18 Levermore C. D. and Oliver M., Analyticity of Solutions for a Generalized Euler Equation, Journal of Differential Equations. (1997) 133, no. 2, 321–339, https://doi.org/10.1006/jdeq.1996.3200, 2-s2.0-0031579130.
10.1006/jdeq.1996.3200
Web of Science® Google Scholar
19 Oliver M. and Titi E. S., On the Domain of Analyticity for Solutions of Second Order Analytic Nonlinear Differential Equations, Journal of Differential Equations. (2001) 174, no. 1, 55–74, https://doi.org/10.1006/jdeq.2000.3927, 2-s2.0-0035920061.
10.1006/jdeq.2000.3927
Web of Science® Google Scholar
20 Panizzi S., On the Domain of Analyticity of Solutions to Semilinear Klein-Gordon Equations, Nonlinear Analysis: Theory, Methods and Applications. (2012) 75, no. 5, 2841–2850, https://doi.org/10.1016/j.na.2011.11.031, 2-s2.0-84855549507.
10.1016/j.na.2011.11.031
Web of Science® Google Scholar
21 Himonas A. A., Kalisch H., and Selberg S., On Persistence of Spatial Analyticity for the Dispersion-Generalized Periodic Kdv Equation, Nonlinear Analysis: Real World Applications. (2017) 38, 35–48, https://doi.org/10.1016/j.nonrwa.2017.04.003, 2-s2.0-85019119046.
10.1016/j.nonrwa.2017.04.003
Web of Science® Google Scholar
22 Tesfahun A., Asymptotic Lower Bound for the Radius of Spatial Analyticity to Solutions of KdV Equation, Communications in Contemporary Mathematics. (2019) 21, no. 08, https://doi.org/10.1142/s021919971850061x, 2-s2.0-85053851639.
10.1142/S021919971850061X
Web of Science® Google Scholar
23 Himonas A. A. and Petronilho G., Evolution of the Radius of Spatial Analyticity for the Periodic BBM Equation, Proceedings of the American Mathematical Society. (2020) 148, no. 7, 2953–2967, https://doi.org/10.1090/proc/14942.
10.1090/proc/14942
Web of Science® Google Scholar
24 Tesfahun A., On the Radius of Spatial Analyticity for Cubic Nonlinear Schrödinger Equations, Journal of Differential Equations. (2017) 263, no. 11, 7496–7512, https://doi.org/10.1016/j.jde.2017.08.009, 2-s2.0-85032198862.
10.1016/j.jde.2017.08.009
Web of Science® Google Scholar
25 Tesfahun A., Remark on the Persistence of Spatial Analyticity for Cubic Nonlinear Schrödinger Equation on the Circle, NoDEA Nonlinear Differential Equations Appl.(2019) 26, no. 2, 1–13.
10.1007/s00030-019-0558-6
Web of Science® Google Scholar
26 Esfahani A. and Tesfahun A., Well-Posedness and Analyticity of Solutions for the Sixth-Order Boussinesq Equation, Communications in Contemporary Mathematics. (2024) 27, no. 02, https://doi.org/10.1142/s0219199724500056.
10.1142/s0219199724500056
Web of Science® Google Scholar
27 Getachew T., New Lower Bounds of Spatial Analyticity Radius for the Kawahara Equation, International Journal of Differential Equations. (2025) 2025, no. 1, https://doi.org/10.1155/ijde/2947966.
10.1155/ijde/2947966
Web of Science® Google Scholar
28 Getachew T., Lower Bounds of Spatial Analyticity Radius for Benjamin-Bona-Mahony Equation on the Circle, Journal of Mathematical Modeling. (2024) 13, 201–208.
Google Scholar
29 Kwak C., Well-Posedness Issues on the Periodic Modified Kawahara Equation, Annales de l′Institut Henri Poincaré C, Analyse non Linéaire. (2020) 37, no. 2, 373–416, https://doi.org/10.1016/j.anihpc.2019.09.002, 2-s2.0-85073570598.
10.1016/j.anihpc.2019.09.002
Web of Science® Google Scholar
30 Selberg S. and da Silva D. O., Lower Bounds on the Radius of Spatial Analyticity for the KdV Equation, Annales Henri Poincaré. (2016) .
Google Scholar

All articles

Asymptotic Lower Bound on the Spatial Analyticity Radius for Solutions of the Periodic Fifth Order KdV–BBM Equation

Abstract

1. Introduction

2. Local Well-Posedness in H^σ,s

3. Approximate Conservation Law

4. Proof of Theorem 1

4.1. The Case: s = 2

4.2. The General Case: s ≥ 1

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Asymptotic Lower Bound on the Spatial Analyticity Radius for Solutions of the Periodic Fifth Order KdV–BBM Equation

Abstract

1. Introduction

2. Local Well-Posedness in Hσ,s

3. Approximate Conservation Law

4. Proof of Theorem 1

4.1. The Case: s = 2

4.2. The General Case: s ≥ 1

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

References

Related

Information

2. Local Well-Posedness in H^σ,s