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Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval

  • Xiangqing Zhao EMAIL logo , Chengqiang Wang and Jifeng Bao
Published/Copyright: December 16, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

We have established the existence and uniqueness of the local solution for

(0.1) ∂ t u + ∂ x 5 u − u ∂ x u = 0 , 0 < x < 1 , t > 0 , u ( x , 0 ) = φ ( x ) , 0 < x < 1 , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x u ( 0 , t ) = h 4 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) , t > 0 ,

in the study of Zhao and Zhang [Non-homogeneous boundary value problem of the fifth-order KdV equations posed on a bounded interval, J. Math. Anal. Appl. 470 (2019), 251–278]. A question arises naturally: Can the local solution be extended to a global one? This article will address this question. First, through a series of logical deductions, a global a priori estimate is established, and then the local solution is naturally extended to a global solution.

Keywords: fifth-order KdV equation; initial-boundary value problem; a priori estimate; global well-posedness
MSC 2010: 35G16; 35Q53

1 Introduction

The Korteweg-de Vries (KdV) equation was originally derived to describe shallow water waves [1]:

∂ t u + α ∂ x 3 u + ∂ x ( u 2 ) = 0 .

The KdV equation has been extensively applied to describe the propagation of long, weakly nonlinear, dispersive waves in one spatial dimension and has led to many important developments in the field of nonlinear wave theory and integrable systems.

However, in certain specific scenarios, the third-order dispersion effect falls short in accurately capturing real physical phenomena (for instance, situations such as the angle between the propagation direction and the magneto-acoustic wave in a cold collision-free plasma, along with the influence of an external magnetic field reaching critical values [2], as well as shallow water conditions near the critical point of surface tension [3]), which demand the consideration of higher-order dispersion effects. Consequently, the fifth-order KdV equation

∂ t u + α ∂ x 5 u + β ∂ x 5 u + ∂ x ( u 2 ) = 0 ,

gains significance in such contexts.

The fifth-order KdV equation is almost as prominently studied in the mathematical community as the KdV equation itself. In terms of initial value problems, the pace of well-posedness research closely follows that of the KdV equation. In 2005, Cui and Tao [4] employed the oscillation integral technique to establish the Kato smoothing effect. They subsequently demonstrated that the initial value problem is locally well-posed in H s for s > 1 4 . Then, in 2007, Wang et al. [5] expanded upon these findings, proving that the initial value problem can be locally solved in H s ( R ) for s > − 7 5 , and global solutions exist for s > − 1 2 using the “I method,” which relies on conservative laws. Subsequently, there has been significant progress in further enhancing the global well-posedness of this problem, ultimately leading to the sharp results as documented in [6–9].

However, the study of initial-boundary value problems for the fifth-order KdV equation:

(1.1) ∂ t u + ∂ x 5 u − u ∂ x u = 0 , 0 < x < 1 , t > 0 , u ( x , 0 ) = φ ( x ) , 0 < x < 1 , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x u ( 0 , t ) = h 4 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) , t > 0 ,

presents a different picture, progressing much more slowly compared to the initial-boundary value problem for the KdV equation. The significant progress mainly occurred after 2014, for instance, as referenced in [10–13], a series of studies were conducted ranging from smoothness estimates, sharp trace regularity, up to local well-posedness. For the convenience of presenting the results, we first introduce several notations.

Notations: We express the boundary values in vector:

h → ( t ) ≡ ( h 1 ( t ) , h 2 ( t ) , h 3 ( t ) , h 4 ( t ) , h 5 ( t ) ) .

Furthermore, expressing the initial and boundary values as

( ϕ ( x ) , h → ( t ) ) ≡ ( ϕ ( x ) ; h 1 ( t ) , h 2 ( t ) , h 3 ( t ) , h 4 ( t ) , h 5 ( t ) ) .

Correspondingly, the function space to which the initial and boundary values belong is denoted as:

X T s ≡ H 0 s ( 0 , 1 ) × H 0 s + 2 5 ( 0 , T ) × H 0 s + 2 5 ( 0 , T ) × H 0 s + 1 5 ( 0 , T ) × H 0 s + 1 5 ( 0 , T ) × H 0 s 5 ( 0 , T ) .

Additionally, the function space to which the solution belongs is denoted as:

Y T s ≡ L 2 ( 0 , T ; H 2 + s ( 0 , 1 ) ) ∩ C ( [ 0 , T ] ; H s ( 0 , 1 ) ) .

Thus, local well-posedness can be formulated as follows:

Theorem A

(Local well-posedness [11]) Let T > 0 , s ∈ [ 0 , 5 ] (with s ≠ 2 j − 1 2 , j = 1 , 2 , 3 , 4 , 5 ) be given. For any s-compatible ( φ , h → ) ∈ X T s , there exists T * ∈ ( 0 , T ] such that the initial-boundary value problem (1.1) admits a unique solution u ∈ Y T * s . Moreover, the solution depends Lipschtiz continuously on ( φ , h → ) in the corresponding space.

The well-posedness result presented in Theorem A is local in the sense that the time interval ( 0 , T * ) on which the solution u exists depends on the size r of the initial-boundary data ( ϕ , h → ) in the space X T s (we will rewrite it as Proposition 3.1). In general, the larger the size r , the smaller the length T * of the time interval ( 0 , T * ) . If T * can be chosen to be T no matter how large the size r is, the initial-boundary value problem will be said to be globally well-posed.

The issue of concern in this article is whether the local solutions established in [5] can be extended to global solutions.

It follows from the standard extension argument that to show the initial-boundary value problem is globally well-posed it suffices to establish global a priori estimate for solutions of the initial-boundary value problem: If u ∈ C ( [ 0 , T ] ; H 5 ( 0 , 1 ) ) solves the initial-boundary value problem, then for certain s ∈ R

sup 0 ≤ t ≤ T ‖ u ( ⋅ , t ) ‖ H s ( 0 , 1 ) ≤ β s ( ‖ ( φ , h → ) ‖ X T s ) ,

where β s : R + → R + is a non-increasing continuous function depending only on s.

From local well-posedness and a priori estimates, we immediately obtain global well-posedness:

Theorem 1.1

(Global well-posedness) Let T > 0 , s ∈ [ 0 , 5 ] (with s ≠ 2 j − 1 2 , j = 1 , 2 , 3 , 4 , 5 ). For any ( φ , h → ) ∈ X T s , the initial-boundary value problem (1.1) admits a unique solution u ∈ Y T s . Moreover, the solution depends Lipschtiz continuously on initial-boundary vaules ( φ , h → ) .

The following sections are arranged as follows:

  1. Section 2 is the core of this article, primarily devoted to establishing a priori estimates. We combine the smoothing effect, trace regularity, and nonlinear estimates, and employ the Gronwall’s theorem to establish the a priori estimates.

  2. Section 3 is dedicated to establishing global well-posedness. By combining local well-posedness with global a priori estimates, we extend the local solution to a global solution.

  3. In Section 4, we provide an overview of unresolved issues pertaining to the global well-posedness of the fifth-order KdV equation.

2 Global a priori estimate

We recall some existing estimates including smoothing effect and nonlinear estimate which will be used in the proof of global a priori estimate.

Lemma 2.1

(Smoothing effect) Let T > 0 be given. For any compatible ϕ ∈ H s ( 0 , 1 ) , h → ∈ ℋ s ( R + ) , f ∈ L 1 ( 0 , T ; H s ( 0 , 1 ) ) , the initial-boundary value problem

∂ t u − ∂ x 5 u = f ( x , t ) , 0 < x < 1 , t > 0 , u ( x , 0 ) = ϕ ( x ) , 0 < x < 1 , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x u ( 0 , t ) = h 4 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) , t > 0 ,

admits a solution u ∈ C ( [ 0 , T ] ; H s ( 0 , 1 ) ) ∩ L 2 ( 0 , T ; H 2 + s ( 0 , 1 ) ) satisfying

‖ u ‖ Y T s + ∑ k = 0 4 sup x ∈ R ‖ ∂ x k u ‖ H s + 2 − k 5 ( 0 , T ) ≤ C ( ‖ ( ϕ , h → ) ‖ X T s + ‖ f ‖ L 1 ( 0 , T ; H s ( 0 , 1 ) ) ) ,

which implies

‖ u ‖ Y T s ≤ C ( ‖ ( ϕ , h → ) ‖ X T s + ‖ f ‖ L 1 ( 0 , T ; H s ( 0 , 1 ) ) )

and

∑ k = 0 4 sup x ∈ R ‖ ∂ x k u ‖ H s + 2 − k 5 ( 0 , T ) ≤ C ( ‖ ( ϕ , h → ) ‖ X T s + ‖ f ‖ L 1 ( 0 , T ; H s ( 0 , 1 ) ) ) .

Proof

See [11].□

Lemma 2.2

(Nonlinear estimate) For s ≥ 0 , there is a C > 0 such that for any T > 0 and u , v ∈ Y T s ,

∫ 0 T ‖ u ∂ x v ‖ H s ( 0 , 1 ) d τ ≤ C T 1 2 + T 1 4 ‖ u ‖ Y T s ‖ v ‖ Y T s .

Proof

See [14].□

Now we state the global a priori estimate precisely:

Proposition 2.3

(Global a priori estimate) Let T > 0 , s ∈ [ 0 , 5 ] (with s ≠ 2 j − 1 2 , j = 1 , 2 , 3 , 4 , 5 ) be given. Let u ∈ C ( [ 0 , T ] ; H 5 ( 0 , 1 ) ) solves the initial-boundary value problem

(2.1) ∂ t u + ∂ x 5 u − u ∂ x u = 0 , 0 < x < 1 , t > 0 , u ( x , 0 ) = φ ( x ) , 0 < x < 1 , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x u ( 0 , t ) = h 4 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) , t > 0 ,

then

sup 0 ≤ t ≤ T ‖ u ( ⋅ , t ) ‖ H s ( 0 , 1 ) ≤ β s ( ‖ ( φ , h → ) ‖ X T s ) ,

where β s : R + → R + is a non-increasing continuous function.

Proof

The proof is divided into three steps:

Step 1. s = 0. The initial-boundary value problem (2.1) can be decomposed into

(2.2) ∂ t v − ∂ x 5 v = 0 , 0 < x < 1 , t > 0 , v ( x , 0 ) = 0 , 0 < x < 1 , v ( 0 , t ) = h 1 ( t ) , v ( 1 , t ) = h 2 ( t ) , ∂ x v ( 1 , t ) = h 3 ( t ) , ∂ x v ( 0 , t ) = h 4 ( t ) , ∂ x 2 v ( 1 , t ) = h 5 ( t ) , t > 0

and

(2.3) ∂ t w − ∂ x 5 w = w ∂ x w + ∂ x ( w v ) + v ∂ x v , 0 < x < 1 , t > 0 , w ( x , 0 ) = ϕ ( x ) , 0 < x < 1 , w ( 0 , t ) = 0 , w ( 1 , t ) = 0 , ∂ x w ( 1 , t ) = 0 , ∂ x w ( 0 , t ) = 0 , ∂ x 2 w ( 1 , t ) = 0 , t > 0 .

According to Lemma 2.1, for (2.2), we have the following estimate:

‖ v ‖ Y T 0 + ∑ k = 0 4 sup x ∈ R ‖ ∂ x k v ‖ H 2 − k 5 ( 0 , T ) ≤ ‖ ( 0 , h → ) ‖ X T 0 ,

which implies that

(2.4) ‖ v ‖ L 2 ( 0 , T ; H 2 ( 0 , 1 ) ) ≤ ‖ ( 0 , h → ) ‖ X T 0 .

Multiplying both sides of the equation in (2.3) by w and then integrating with respect to x over the interval (0, 1), and after performing integration by parts, we obtain:

d d t ∫ 0 1 w 2 d x ≤ C ∫ 0 1 ∣ ∂ x v ∣ w 2 d x + C ∫ 0 1 ∣ v ∂ x v w ∣ d x .

Observe that

∫ 0 1 ∣ ∂ x v ∣ w 2 d x ≤ sup x ∈ ( 0 , 1 ) ∣ ∂ x v ∣ ‖ w ‖ L 2 ( 0 , 1 ) 2 ≤ C ε ‖ v ‖ H 3 + ε 2 ( 0 , 1 ) ‖ w ‖ L 2 ( 0 , 1 ) 2

and

∫ 0 1 ∣ v ∂ x v w ∣ d x ≤ sup x ∈ ( 0 , 1 ) ∣ ∂ x v ∣ ‖ v ‖ L 2 ( 0 , 1 ) ‖ w ‖ L 2 ( 0 , 1 ) ≤ C ε ‖ v ‖ H 3 + ε 2 ( 0 , 1 ) 2 ‖ w ‖ L 2 ( 0 , 1 ) ,

where ε is any fixed positive constant, we deduce that

d d t ‖ w ‖ L 2 ( 0 , 1 ) 2 = d d t ∫ 0 1 w 2 d x ≤ C ε ‖ v ‖ H 3 + ε 2 ( 0 , 1 ) ‖ w ‖ L 2 ( 0 , 1 ) 2 + C ε ‖ v ‖ H 3 + ε 2 ( 0 , 1 ) 2 ‖ w ‖ L 2 ( 0 , 1 )

for any t ≥ 0 . Thus, we have

d d t ‖ w ‖ L 2 ( 0 , 1 ) = d d t ∫ 0 1 w 2 d x ≤ C ε ‖ v ‖ H 3 + ε 2 ( 0 , 1 ) ‖ w ‖ L 2 ( 0 , 1 ) + C ε ‖ v ‖ H 3 + ε 2 ( 0 , 1 ) 2 .

Applying the Gronwall inequality, Hölder’s inequality, and combining with (2.4), we obtain:

‖ w ( ⋅ , t ) ‖ L 2 ( 0 , 1 ) = e C ε ∫ 0 T ‖ v ( ⋅ , t ) ‖ H 3 + ε 2 ( 0 , 1 ) d t ‖ w ( ⋅ , 0 ) ‖ L 2 ( 0 , 1 ) + C ε ∫ 0 T ‖ v ( ⋅ , t ) ‖ H 3 + ε 2 ( 0 , 1 ) d t 2 ≤ e C ε T ‖ v ‖ L 2 ( 0 , T ; H 3 + ε 2 ( 0 , 1 ) ) ‖ ϕ ‖ L 2 ( 0 , 1 ) + C ε ‖ v ‖ L 2 ( 0 , T ; H 3 + ε 2 ( 0 , 1 ) ) 2 ≤ e C ε T ‖ v ‖ L 2 ( 0 , T ; H 2 ( 0 , 1 ) ) ( ‖ ϕ ‖ L 2 ( 0 , 1 ) + C ε ‖ v ‖ L 2 ( 0 , T ; H 2 ( 0 , 1 ) ) 2 ) ≤ C ‖ ( ϕ , h → ) ‖ X T 0 .

Step 2. s = 5 . For a smooth solution u of (2.2), v = ∂ t u solves

(2.5) ∂ t v − ∂ x 5 v = ∂ x ( u v ) , 0 < x < 1 , t > 0 , v ( x , 0 ) = ϕ * ( x ) , 0 < x < 1 , v ( 1 , t ) = h 1 ′ ( t ) , v ( 0 , t ) = h 2 ′ ( t ) , ∂ x v ( 1 , t ) = h 3 ′ ( t ) , ∂ x v ( 0 , t ) = h 4 ′ ( t ) , ∂ x 2 v ( 1 , t ) = h 5 ′ ( t ) , t > 0 ,

where

ϕ * ( x ) = ϕ ′ ϕ + ϕ ( 5 ) .

By Lemmas 2.1 and 2.2, there exists a constant C > 0 such that for any T ′ ≤ T ,

‖ v ‖ Y T ′ 0 ≤ ‖ ( ϕ * , h ′ → ) ‖ X T 0 + C ( T ′ 1 2 + T ′ 1 4 ) ‖ u ‖ Y T 0 ‖ v ‖ Y T 0 .

Choose T ′ < T such that C ( T ′ 1 2 + T ′ 1 4 ) ‖ u ‖ Y T 0 < 1 2 , with such a choice,

‖ v ‖ Y T ′ 0 ≤ 2 ‖ ( ϕ * , h ′ → ) ‖ X T 0 .

Note that T ′ only depends on ‖ u ‖ Y T 0 , and therefore depends only on ‖ ( ϕ , h → ) ‖ X T 0 , by the estimate proved in step 1. By a standard extension argument, we have

‖ v ‖ Y T 0 ≤ C 1 ‖ ( ϕ , h → ) ‖ X T 5 ,

where C 1 depends only on T and ‖ ( ϕ , h → ) ‖ X T 0 . The estimate (2.1) with s = 5 then follows from

v = ∂ t u = ∂ x 5 u + u ∂ x u .

Step 3. 0 < s < 5 . Through nonlinear interpolation, we demonstrate the validity of equation (2.1) for 0 < s < 5 . We omit the proof in this context, for a comprehensive understanding, we direct readers to the referenced source [14,15].□

3 Proof of the main theorem

Proposition 3.1

(Local well-posedness) Let T > 0 , s ∈ [ 0 , 5 ] (with s ≠ 2 j − 1 2 , j = 1 , 2 , 3 , 4 , 5 ) be given. For any r > 0 , there exists a T * ∈ ( 0 , T ] depending only on r such that for any ( φ , h → ) ∈ X T s with

‖ ( φ , h → ) ‖ X T s ≤ r ,

the initial-boundary value problem

(3.1) ∂ t u + ∂ x 5 u − u ∂ x u = 0 , 0 < x < 1 , t > 0 , u ( x , 0 ) = φ ( x ) , 0 < x < 1 , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x u ( 0 , t ) = h 4 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) , t > 0

admits a unique solution u ∈ Y T * s . Moreover, the solution depends Lipschtiz continuously on ( φ , h → ) in the corresponding spaces.

Finally, we give the proof of Theorem 1.1.

Proof

By Proposition 3.1, for any s -compatible ( φ , h → ) ∈ X T s , there exists T 1 ∈ ( 0 , T ] such that the initial-boundary value problem

∂ t u + ∂ x 5 u − u ∂ x u = 0 , 0 < x < 1 , t > 0 , u ( x , 0 ) = φ ( x ) , 0 < x < 1 , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x u ( 0 , t ) = h 4 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) t > 0 ,

admits a unique solution u ∈ Y T 1 s .

By Proposition 2.3,

sup 0 ≤ t ≤ T 1 ‖ u ‖ H s ( 0 , 1 ) ≤ β s ( ‖ ( φ , h → ) ‖ X T s ) ,

which implies that

‖ u ‖ C ( [ 0 , T 1 ] , H s ( 0 , 1 ) ) ≤ β s ( ‖ ( φ , h → ) ‖ X T s ) .

Taking ( u ( T 1 , x ) , h → ) as data, applying Proposition 3.1 to determine T 2 ∈ ( T 1 , T ] and the solution u ( t , x ) ∈ Y T 2 s . This procedure can be repeated until T n = T , where we arrive the global solution.□

4 Concluding comments

In 2019, we show in [11] a general local well-posedness of initial-boundary value problem of fifth-order KdV equation

(4.1) ∂ t u − ∂ x 5 u = u ∂ x u , 0 < x < 1 , t > 0 , u ( x , 0 ) = φ ( x ) , 0 < x < 1 , boundary value , t > 0 ,

with the following 16 possible admissible boundary values:

a u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 0 , t ) = h 3 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) ; b ∂ x u ( 0 , t ) = h 1 ( t ) ∂ x u ( 1 , t ) = h 2 ( t ) , ∂ x 2 u ( 1 , t ) = h 3 ( t ) , ∂ x 4 u ( 0 , t ) = h 4 ( t ) , ∂ x 4 u ( 1 , t ) = h 5 ( t ) ; c u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x 2 u ( 1 , t ) = h 3 ( t ) , ∂ x 3 u ( 0 , t ) = h 4 ( t ) , ∂ x 3 u ( 1 , t ) = h 5 ( t ) ; d ∂ x 2 u ( 1 , t ) = h 1 ( t ) , ∂ x 3 u ( 0 , t ) = h 2 ( t ) , ∂ x 3 u ( 1 , t ) = h 3 ( t ) , ∂ x 4 u ( 0 , t ) = h 4 ( t ) , ∂ x 4 u ( 1 , t ) = h 5 ( t ) ; e u ( 0 , t ) = h 1 ( t ) , ∂ x u ( 0 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x 2 u ( 1 , t ) = h 4 ( t ) , ∂ x 4 u ( 1 , t ) = h 5 ( t ) ; f u ( 1 , t ) = h 1 ( t ) , ∂ x u ( 0 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x 2 u ( 1 , t ) = h 4 ( t ) , ∂ x 4 u ( 0 , t ) = h 5 ( t ) ; g u ( 0 , t ) = h 1 ( t ) , ∂ x 2 u ( 1 , t ) = h 2 ( t ) , ∂ x 3 u ( 0 , t ) = h 3 ( t ) , ∂ x 3 u ( 1 , t ) = h 4 ( t ) , ∂ x 4 u ( 1 , t ) = h 5 ( t ) ; h u ( 1 , t ) = h 1 ( t ) , ∂ x 2 u ( 1 , t ) = h 2 ( t ) , ∂ x 3 u ( 0 , t ) = h 3 ( t ) , ∂ x 3 u ( 1 , t ) = h 4 ( t ) ∂ x 4 u ( 0 , t ) = h 5 ( t ) ; i u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 0 , t ) = h 3 ( t ) , ∂ x 2 u ( 1 , t ) = h 4 ( t ) , ∂ x 3 u ( 1 , t ) = h 5 ( t ) ; j u ( 0 , t ) = h 1 ( t ) , u ( 0 , t ) = h 2 ( t ) ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x 2 u ( 1 , t ) = h 4 ( t ) , ∂ x 3 u ( 0 , t ) = h 5 ( t ) ; k ∂ x u ( 0 , t ) = h 1 ( t ) , ∂ x 2 u ( 1 , t ) = h 2 ( t ) , ∂ x 3 u ( 1 , t ) = h 3 ( t ) , ∂ x 4 u ( 0 , t ) = h 4 ( t ) , ∂ x 4 u ( 1 , t ) = h 5 ( t ) ; l ∂ x u ( 1 , t ) = h 1 ( t ) , ∂ x 2 u ( 1 , t ) = h 2 ( t ) , ∂ x 3 u ( 0 , t ) = h 3 ( t ) , ∂ x 4 u ( 0 , t ) = h 4 ( t ) , ∂ x 4 u ( 1 , t ) = h 5 ( t ) ; m u ( 0 , t ) = h 1 ( t ) , ∂ x u ( 0 , t ) = h 2 ( t ) , ∂ x 2 u ( 1 , t ) = h 3 ( t ) , ∂ x 3 u ( 1 , t ) = h 4 ( t ) , ∂ x 4 u ( 1 , t ) = h 5 ( t ) ; n u ( 0 , t ) = h 1 ( t ) , ∂ x u ( 1 , t ) = h 2 ( t ) , ∂ x 2 u ( 1 , t ) = h 3 ( t ) , ∂ x 3 u ( 0 , t ) = h 4 ( t ) , ∂ x 4 u ( 1 , t ) = h 5 ( t ) ; o u ( 1 , t ) = h 1 ( t ) , ∂ x u ( 0 , t ) = h 2 ( t ) , ∂ x 2 u ( 1 , t ) = h 3 ( t ) , ∂ x 3 u ( 1 , t ) = h 4 ( t ) , ∂ x 4 u ( 0 , t ) = h 5 ( t ) ; p u ( 1 , t ) = h 1 ( t ) , ∂ x u ( 1 , t ) = h 2 ( t ) , ∂ x 2 u ( 1 , t ) = h 3 ( t ) , ∂ x 3 u ( 0 , t ) = h 4 ( t ) , ∂ x 4 u ( 0 , t ) = h 5 ( t ) .

However, not all of the mentioned cases are suitable for the method proposed in this article. What we are interested in is which other cases can employ the method presented in this article to extend their local solutions into global solutions?

The boundary conditions (a), (c), (i), and (j) share a common characteristic: when the boundary conditions are homogeneous, the corresponding solutions exhibit global L 2 -boundedness. In fact,

d d t ∫ 0 1 u 2 d x = 2 ∫ 0 1 u ( ∂ x 5 u + u ∂ x u ) d x = 2 u ∂ x 4 u ∣ 0 1 − 2 ∂ x u ∂ x 3 u ∣ 0 1 + ( ∂ x 2 u ) 2 ∣ 0 1 + 2 3 u 3 0 1 ≤ 0 ,

when boundary value takes (a) or (c) or (i) or (j). Thus, we have

(4.2) ‖ u ( x , t ) ‖ L 2 ( 0 , 1 ) = ∫ 0 1 u 2 d x ≤ ∫ 0 1 φ 2 d x = ‖ φ ( x ) ‖ L 2 ( 0 , 1 ) .

It is not difficult to see that the method presented in this article is also applicable to (a), (c), (i), and (j). Indeed, making minor adjustments to the proofs presented in this article is all that is needed to achieve the goal.

Finally, are the remaining cases globally well-posed? How can this be proven? The answer lies in adding nonlinear feedback. We will discuss this issue elsewhere.

Acknowledgement

The first author is sponsored by Qing-Lan project. The authors are deeply grateful for the suggestions made by the anonymous reviewer.

  1. Conflict of interest: The authors state no conflict of interest.

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Received: 2023-02-03
Revised: 2023-09-04
Accepted: 2023-11-14
Published Online: 2023-12-16

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2023-0158
Keywords for this article
fifth-order KdV equation; initial-boundary value problem; a priori estimate; global well-posedness
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Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
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  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
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  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
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  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
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  60. On semigroups of transformations that preserve a double direction equivalence
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  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
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