In this paper, an algebraic decay rate for the radius of spatial analyticity of solutions to the Kawahara equation is investigated. With given analytic initial data having a fixed radius of analyticity σ₀, we derive an algebraic decay rate σ(t) ~ |t|^−1/2 for the uniform radius of spatial analyticity of solutions to the Kawahara equation. This improves a recent result due to Ahn et al.’s study, where they demonstrated a decay rate of order |t|⁻¹. Our strategy mainly relies on an approximate conservation law in a modified Gevrey space and bilinear estimate in Bourgain space.

1. Introduction

We consider an initial value problem (IVP) for the Kawahara equation:

()

where

, and u : = u(x, t) is a real-valued function of space and time. This equation is also called the fifth-order KdV-type equation, and it has been derived to model gravity-capillary waves on a shallow layer and magnetosound propagation in plasmas [1, 2].

The low regularity well-posedness of the IVP (1.1) has been investigated by Zhang et al. [3]. Following the ideas of the authors in [4, 5], first, the authors showed that the local well-posedness is established for the initial data with s ≥ −7/4. Then, using these results and conservation laws, they also proved that the IVP is globally well-posed for the initial data with s = 0. Recently, Zhang et al. [6] investigated the IVP of the higher order nonlinear dispersive equation for the initial data in the Sobolev space . For further survey, we refer the readers to [7, 8] and references therein for additional information about the well-posedness of (1) in the Sobolev space.

The principal goal of this paper is to improve the decay rate for the radius of spatial analyticity of the solution u(x, t) to the IVP (1) obtained by authors in [9], given real analytic initial data ψ(x) with a uniform radius of spatial analyticity σ₀ > 0, so there is a holomorphic extension to a complex strip:

()

For any

, the class of analytic function space we used is called a modified Gevrey space introduced by authors in [10], denoted by

, which equipped with the following norm:

()

where 〈⋅〉² = 1 + |⋅|², and

is the spatial Fourier transform of f(x). This space is obtained from the Gevrey space

by replacing the exponential weight e^σ|ξ| with the hyperbolic weight cosh(σ|ξ|). Here, for every

, G^σ,s is introduced by Foias and Temam [11], which is defined by

()

where

()

It is clear that, for σ = 0, both the Gevrey and modified Gevrey spaces coincide with the standard Sobolev space

via the following norm:

()

We note that the modified Gevrey space satisfy the following embedding:

()

According to the Paley–Wiener theorem [12], any function f ∈ G^σ,s with σ > 0 can be extended to an analytic function in the strip . This is the reason why one can take initial data in the Gevrey space G^σ,s.

As a consequence of

()

the norms

and

are equivalent, which in turn implies that, for functions in H^σ,s, the statement of the Paley–Weiner theorem holds true.

Recently, Ahn et al. [9] proved that (1) is locally well-posed on the analytic Gevrey space

for s > −7/4, with existence time δ > 0. The authors used bilinear estimates in combination with the conservation of mass functional

()

to prove an almost conservation law in the analytic space G^σ,0. Moreover, they used repeatedly the local well-posedness and the almost conservation law on the interval, [0, δ], [δ, 2δ], ⋯, and investigated the algebraic decay rate σ(t) ≥ c|t|⁻¹, |t|⟶∞. This method is introduced by Selberg and Tesfahun [13] in the context of the 1D Dirac–Klein–Gordon equations.

The method of Gevrey approximate conservation law results decay rate of order t^−1/ρ for some 0 < ρ ≤ 1 on the radius of spatial analyticity of solutions to a number of nonlinear dispersive and wave equations (see, e.g., [6, 14–31]). 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/(2ρ) for some 0 < ρ ≤ 1 (see, e.g., [10, 16, 32–34]). Such a decay rate is obtained in the view of the inequality.

()

which follows from an interpolation of

In the paper [35], Zhang et al. investigated the Cauchy problem for the higher order nonlinear dispersive equation with initial data in Gevrey space G^σ,s. First, the authors used Tao’s [k, Z]− multiplier method to establish the basic estimate on dyadic blocks. Also, using the Fourier restriction norm method, they established the bilinear estimate and approximate conservation law. Then, using the contraction mapping principle, iteration technique, and the bilinear estimate, they proved the local well-posedness for the initial data u₀ ∈ G^σ,s with s ≥ −11/4. Finally, based on the local well-posedness and approximate conservation law, the authors obtained that the analytic radius dores not decay faster than t^−4/11 as time t goes to infinity.

In this paper, we follow the idea from the authors in [35] and use an approximate conservation law in the modified Gevrey space to improve the required lower bound.

As a consequence of the embedding (8) for σ^′ = 0, and the existing global well-posedness theory in H^s for s ≥ −7/4 (see [6]), we deduce that the IVP (1) has a unique, smooth solution for all time, given initial data for any σ₀ > 0 and .

We now state our main result on the algebraic decay rate.

Theorem 1. Let for some σ₀ > 0 and for . Then, the global C^∞ solution u of the IVP (1) satisfies

()

with the radius of analyticity σ≔σ(T) satisfying the decay rate

()

where c > 0 is a constant depending only on

The organization of this paper is as follows: In Section 2, we introduce some function spaces. We also present bilinear estimates and energy estimates in the restriction of the Bourgain space. In Section 3, we prove the local well-posedness result using energy inequalities and contraction mapping principles. In Section 4, we prove an approximate conservation law by making use of the bilinear estimate in the Bourgain space. Finally, in Section 5, we prove our main result Theorem 1 by iterating the local result on the approximate conservation law.

1.1. Notations

We use C to denote various space- and time-independent constants whose exact values may vary from line to line. For two positive quantities a and b, we use a≲b to mean a ≤ Cb, while a ~ b means a≲b and b≲a. Moreover, we use m± to mean m ± ϵ for a sufficiently small constant ϵ > 0 and any real number m.

2. Preliminary Results

In this section, we introduced the function spaces, bilinear estimates, and energy estimates necessary for our proofs.

2.1. Function Spaces

For given real numbers s and b, the Bourgain space

corresponding to (1) is the completion of the Schwartz function space on

with respect to the following norm:

()

where p(ξ) = −βξ⁵ + αξ³ and

denote the space-time Fourier transform of f defined by

()

For a positive time δ > 0, the restriction to time slab

of the Bourgain space, denoted

, is a Banach space via the norm

()

For any s₁ ≤ s₂ and/or b₁ ≤ b₂ in

, we note that the definition of X^s,b ensures a continuous embedding (see, e.g., [5, 36])

()

For any real numbers σ > 0, s, and b, the Gevrey–Bourgain space

is defined via the norm

()

where D = −i∂_x is a Fourier multiplier with symbol |ξ|. This space coincides with the Bourgain space X^s,b when σ = 0. Furthermore, its restriction to time slab

is denoted by

and is defined in a similar way as mentioned above.

2.2. Bilinear Estimates and Energy Estimates

This section is dedicated to present bilinear estimates and energy estimates, which will use in proving the local existence result and approximate conservation law.

Now, we consider the IVP for the linear Kawahara equation:

()

Then, its solution is given by Duhamels’ formula:

()

It is noted that W(t)ψ(x) represents the solution to the homogeneous IVP for the Kawahara equation with initial data ψ, i.e.,

()

Then, for 0 < δ ≤ 1, b > 1/2 and

, the following

energy estimates follows from [3] (see also [5, 6, 9, 37]).

()

Lemma 1. Let δ > 0 and 1/2 < b ≤ b^′ < 1. Let |D| be the Fourier multiplier with symbol |ξ|. Then, we have

()

Proof 1. Jia and Huo [[38], Theorem 3.5] proved that

()

where s > −1/4, b > 1/2, and b^′ ∈ [b, b + ϵ] with sufficiently small ϵ > 0.

An application of Plancherel’s theorem yields

()

Since all estimates of X^s,b can apply to its restriction, the desired estimate follows from (26) and (25).

Lemma 2. Let s > −7/4, δ > 0 and 1/2 < b ≤ b^′ < 1. Then,

()

Proof 2. Zhang and Huang ([5], Lemma 5.1) (see also [[37], Lemma 3.2] and [[9], Lemma 3.1]) proved that

()

for s > −7/4 and 1/2 < b ≤ b^′ < 1. On the other hand, we have by Plancherel’s theorem,

()

As estimates of X^s,b can hold for its restrictions, (29) in combination with (28) yields the estimate what we claimed.

3. Local Well-Posedness Result

In this section, we prove existence of the local result in H^σ,s for σ > 0 and s > −7/4.

Theorem 2. Let s > −7/4 and σ > 0. Then, for any given Ψ ∈ H^σ,s, there exists a time δ > 0 depending on and a unique solution u of (1) on the time interval (−δ, δ) such that

()

with existence time

()

for some constant a > 2 depending on s. Moreover, for some 1/2 < b < 1, the local solution u satisfies the following estimate:

()

In proving this theorem, the objective is to perform energy estimates, Sobolev embedding, and contraction mapping principle.

For δ > 0, s > −7/4, and 1/2 < b < 1, taking the

norm to both sides of (20) gives

()

Now, by applying cosh(σ|D|) to (20), we can rewrite (33) as

()

Indeed, we claim the following nonlinear estimate.

Lemma 3. Let δ > 0, σ ≥ 0, s > −7/4 and 1/2 < b ≤ b^′ < 1. Then, for any , we have

()

Proof 3. From the left-hand side of (35), we have the following estimate (see [39], Lemma 2.11], [38], Lemma 2.15])

()

We have |ξ| ≤ |ξ₁| + |ξ₂| by the triangle inequality. As a consequence of it, we then have

()

Setting U_σ≔cosh(σ|D|)u. Then, by using Plancherel’s theorem and (37), the right-hand side of (36) is estimated as

()

where η denotes the conditions ξ − ξ₁ = ξ₂ and τ − τ₁ = τ₂.

Denoting , and then by Plancherel’s theorem, Lemma 2, and Sobolev embedding, we have

()

It follows from (34) and (35) that the energy inequality for the IVP (1) is

()

Now, by combining (40) with the contraction mapping principle in the space , we obtain a unique solution for (1), with existence time

()

The other key ingredient that we will use to derive the lower bound estimate for solutions to (1) is an approximate conservation law.

4. Approximate Conservation Law

To derive an approximate conservation law, we first apply cosh(σ|D|) to both sides of (1) and then set U_σ(x, t)≔cosh(σ|D|)u(x, t) to obtain

()

where u is the local solution to (1) and

()

We define a modified mass functional for U_σ as

()

Since cosh(σ|D|) = 1, for σ = 0, we have U₀(x, t) = u(x, t). Then, we obtain the mass conservation for the solution u of (1), i.e.,

()

But this fails to hold for σ > 0. In what follows, for δ as in Theorem 2, we will nevertheless derive the approximate conservation law

()

In taking the limit σ⟶0, we recover the conservation for all t ∈ [0, δ].

Now, we need to prove (46). To do this, differentiating

with respect to time t, and then using (42) (43) and integration by parts¹ gives

()

Consequently, integrating in time over the interval (0, s) for s ≤ δ gives

()

where

()

The integral (49) satisfies the following a priori estimate.

Lemma 4. Let δ > 0. Then, for all and 0 ≤ s ≤ δ, we have

()

Now, to finish the proof of (46), we need to use (50). By (32), we have the bound

()

Also, we have

()

Therefore,

()

Plugging (53) into (51) yields

()

Finally, inserting (54) into (50) and then using the result in (48) gives the desired estimate (46).

4.1. Proof of (50)

We have by Plancherel’s theorem,

()

where dμ(ξ) is the measure

()

In particular, this measure imposes the condition ξ = ξ₁ + ξ₂. Again by triangle inequality, we have

()

Now, we recall the following estimate from [[16], Lemma 3]

()

In addition to estimate (58), we need to recall the following estimate:

Lemma 5 ([35], Lemma 2.3]). Assume that and δ > 0. Then, for any time interval I ⊂ [−δ, δ], we have the estimate

()

where χ_I is the characteristic function of the interval I, and C > 0 is a constant depending only on b.

Putting

and using Plancherel’s theorem, Lemma 5, (58), Hölder inequality, Lemma 1 and one dimensional Sobolev embedding, we obtain

()

5. Proof of Theorem 1

We closely follow the argument in [35] to derive the claimed decay rate on the lower bound for the radius of analyticity.

Now, we fix σ₀ > 0 and

. Because of the invariance of the Kawahara equation under the reflection (x, t)⟶(−x, −t), we may restrict ourself to a positive time. We want to prove that the solution u of (1) satisfies

()

where

()

for a constant c > 0 depending only on

and σ₀.

By Theorem 1 there is a maximal time δ ∈ (0, ∞] such that

()

If δ = ∞ then σ(t) = σ₀, so we are done. If not, we remain to prove

()

Now, we first consider the case s = 0 by taking the advantage of the almost conservation law in . By the embedding property (8), the general case, , will essentially reduce to s = 0 which is shown in the next subsection.

5.1. Case: s = 0

We fix T > δ. Setting

()

We claim to prove for sufficiently small σ > 0 that,

()

using Theorem 1 and (46) repeatedly with the time step

()

where c₀ > 0 and a > 2 are in Theorem 1 with s = 0.

If we do this, becomes finite for all t ∈ [0, T], which in turn imply that u ∈ C([0, T]; H^σ,0), and thus the proof of (34) is completed for s = 0.

Now, it remains to prove (66). We choose so that T ∈ [nδ, (n + 1)δ].

Using induction for k ∈ {1, …, n + 1}, we will demonstrate that

()

with the smallest conditions on σ such that

()

and

()

Since σ ≤ σ₀, using the fact

, for k = 1, we can deduce from (46) that

()

This implies that (68) holds true for k = 1.

To obtain (69) for k = 1,

()

is required, but this follows from (71) because σ ≤ σ₀ and

We will now assume that (68) and (69) hold true for some k ∈ {1, …, n}. Then, using (46), (68), and (69), we obtain

()

Combining (74) with the induction hypothesis (68) for k yields

()

This proves (68) for k + 1, which also implies (69) whenever

. But the latter follows from (69), since

()

holds because of the inequality nδ ≤ T < (n + 1)δ. We have thus proved (69) under the smallest assumptions (70) and (71) on σ. Since δ ≤ T, condition (71) must fail for σ = σ₀. That is,

()

otherwise we would be able to continue the solution in

beyond the time T, which contradicts the maximality of δ. Thus, there must be some σ ∈ (0, σ₀) in which the equality in (71) holds and using (67), we obtain

()

Thus,

()

where

()

This completes the proof of our goal Theorem 1 for s = 0.

5.2. General Case:

For any

, we use the embedding (8) with s^′ = 0 and σ^′ = σ₀/2 to obtain

()

From the local well-posedness result, there is a

such that

()

and

()

where κ > 0 is a constant depending on

and σ₀. It follows again from the embedding (8) that

()

and

()

Conflicts of Interest

The author declares no conflicts of interest.

Funding

The author received no financial support for this manuscript.

Endnotes

¹Since we may assume that U_σ(x, t) and all its spatial derivatives vanish as |x|⟶∞, integration by parts is justified [27, 35].

Open Research

Data Availability Statement

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

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New Lower Bounds of Spatial Analyticity Radius for the Kawahara Equation

Abstract

1. Introduction

1.1. Notations

2. Preliminary Results

2.1. Function Spaces

2.2. Bilinear Estimates and Energy Estimates

3. Local Well-Posedness Result

4. Approximate Conservation Law

4.1. Proof of (50)

5. Proof of Theorem 1

5.1. Case: s = 0

5.2. General Case:

Conflicts of Interest

Funding

Endnotes

Open Research

Data Availability Statement

References

Citing Literature

References

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New Lower Bounds of Spatial Analyticity Radius for the Kawahara Equation

Abstract

1. Introduction

1.1. Notations

2. Preliminary Results

2.1. Function Spaces

2.2. Bilinear Estimates and Energy Estimates

3. Local Well-Posedness Result

4. Approximate Conservation Law

4.1. Proof of (50)

5. Proof of Theorem 1

5.1. Case: s = 0

5.2. General Case:

Conflicts of Interest

Funding

Endnotes

Open Research

Data Availability Statement

References

Citing Literature

References

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