# HS-BAN: A Benchmark Dataset of Social Media Comments for Hate Speech
Detection in Bangla

Nauros Romim, Mosahed Ahmed
Dept. of Electrical & Electronic Engineering
Shahjalal University of Science and Technology
Sylhet-3114
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Md. Saiful Islam,  Arnab Sen Sharma
Dept. of Computer Science and Engineering
Shahjalal University of Science and Technology
Sylhet-3114
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Hriteshwar Talukder
Dept. of Electrical & Electronic Engineering
Shahjalal University of Science and Technology
Sylhet-3114
<EMAIL_ADDRESS>
Mohammad Ruhul Amin
Computer and Information Science
Fordham University
New York, USA
<EMAIL_ADDRESS>
###### Abstract

In this paper, we present HS-BAN, a binary class hate speech (HS) dataset in
Bangla language consisting of more than 50,000 labeled comments, including
40.17% hate and rest are non hate speech. While preparing the dataset a strict
and detailed annotation guideline was followed to reduce human annotation
bias. The HS dataset was also preprocessed linguistically to extract different
types of slang currently people write using symbols, acronyms, or alternative
spellings. These slang words were further categorized into traditional and
non-traditional slang lists and included in the results of this paper. We
explored traditional linguistic features and neural network-based methods to
develop a benchmark system for hate speech detection for the Bangla language.
Our experimental results show that existing word embedding models trained with
informal texts perform better than those trained with formal text. Our
benchmark shows that a Bi-LSTM model on top of the FastText informal word
embedding achieved 86.78% F1-score. We will make the dataset available for
public use.

##  1 Introduction

Hate speech (HS) has been rapidly spreading in recent years through various
social media sites. HS detection in the user’s comment section remains a
difficult challenge due to lack of formal language syntax, spelling mistakes,
and use of various slang and non-standard acronyms.[1]. Researchers have
started working in this very challenging domain, but much of their efforts
concentrated on English language[2]. There are almost 46 million
Facebook111https://www.statista.com/statistics/268136/top-15-countries-based-
on-number-of-facebook-users/ and 29 million
Youtube222https://www.statista.com/forecasts/1146236/youtube-users-in-
bangladesh Bangladeshi users but there has been a severe lack of large,
linguistically diverse Bangla HS datasets.

Existing datasets for Bangla HS have some significant drawbacks. First, as
Table A.1 of Appendix A shows, the datasets are not large enough. Most of them
contain less than 10k sentences. Secondly, the majority portions of those
datasets come from one or two domains, which makes them domain-dependent [3],
[4] and [5]. Thirdly, annotating HS is inherently a complex and challenging
task[1]. [6] showed how annotator’s bias on HS can influence the annotation
process and subsequently affect classification task. To combat this, [7]
created a stringent and detailed annotation guideline, providing specific
points on what constitutes HS and what not. Although [8] followed a guideline
to prepare a small Bangla HS dataset, it lacks inter-annotator scores, which
makes it difficult to judge annotation quality. To the best of our knowledge,
only [9] mentioned Cohen’s kappa score in their paper.

In this paper, we manifest HS-BAN, the largest binary class Bangla HS dataset
comprised of more than 50,000 comments crawled from Facebook and YouTube. We
present a few examples from the dataset in Table 1. To ensure cross-domain
generalization, we collected comments from seven categories as presented in
Section 2. A strict annotation guideline was followed to reduce human
annotation bias for labeling the dataset achieving an inter-annotator
agreement score of 0.658. In Section 3, we present the different natural
language processing approaches we used to extract linguistic features by
processing noisy social media text. Finally, in Section 4, we present the
benchmarking outcome for hate speech classification showing that a Bi-LSTM
model on top of the FastText informal word embedding achieved 86.78% F1-score.
The dataset and source code will be made publicly available to foster future
research.

##  2 Dataset

Table 1: Examples of the annotate HS-BAN dataset

### Data Collection:

We collected more than 100,000 comments from different YouTube channels and
Facebook pages in seven categories: sports, entertainment, crime, politics,
religion, celebrity, and miscellaneous. For each category except
miscellaneous, we compiled a list of controversial events that happened
recently (2017-2020) in Bangladesh that falls under those categories. Then
public comments were extracted on those issues by using an open-source tool
called Facepager333https://github.com/strohne/Facepager. Furthermore, for the
miscellaneous category, we searched for videos on YouTube related to Bangla
TikTok, roasting, or similar videos as the comment section in these videos
tend to be very toxic. We removed duplicate, highly similar comments based on
Jaccard Index > 0.8 to reduce repetitiveness and ensure a diverse vocabulary.

### Annotation:

We prepared a detailed annotation guideline based on the community guidelines
of Facebook and YouTube (Appendix B). We worked with 50 annotators who are
undergraduate students within the age range from 20 to 25 years. They know
about the sensitive nature of the task and they willingly volunteered in this
research. Three annotators tagged each comment, and the majority vote decided
the final decision. The inter-annotator agreement score [10] of 0.658
indicates that our dataset has moderate agreement among annotators.

### Dataset Statistics:

Our dataset has $50,314$ comments in total; among them, 20,209 of the dataset
comments are HS and 30,105 are Not-HS, which implies that our dataset is
moderately imbalanced. A more detailed statistics of our dataset is presented
in Table 2. We can see that each category is also moderately imbalanced,
especially the politics and celebrity categories, whereas crime has the
highest hate speech percentage. Observing the average word count (AWC) for
each category, we can find that, celebrity category has the highest AWC per
sentence with the most standard deviation while miscellaneous has the lowest.

Category | Total | HS% | AWC | Vocabulary
---|---|---|---|---
Sports | 5937 | 40.02 | $13.51\pm 18.27$ | 16299
Entertainment | 6843 | 41.31 | $12.30\pm 13.31$ | 16915
Crime | 4969 | 43.71 | $16.48\pm 21.53$ | 15272
Religion | 4978 | 38.31 | $13.25\pm 21.54$ | 13717
Politics | 2665 | 33.06 | $12.90\pm 17.93$ | 9059
Celebrity | 2394 | 30.49 | $22.96\pm 33.59$ | 14220
Miscellaneous | 22,528 | 41.35 | $11.35\pm 14.16$ | 33646
Total | 50314 | 40.17 | $13.07\pm 17.97$ | 72660

Table 2:  Category-wise Comment distribution

##  3 Experimental Setup

We removed all hashtags, numbers, and non-Bangla words. However, we kept emoji
and punctuation (e&p) in order to confirm [1]’s claim whether e&p can be good
indicators for HS. F1 score and Matthews correlation coefficient (MCC) were
used to evaluate all models. We split the dataset into train(80%) and
test(20%) datasets using stratified sampling so that each category in both
dataset contains an equal ratio of hate to not-hate comments. The training
dataset was further divided into train(80%) and development(20%) datasets for
feature selection and hyper-parameter tuning purposes.

### Machine Learning Models:

We trained linear Support Vector Machine (SVM)[11] with Term Frequency Inverse
Document Frequency (TF-IDF) weighted score as a feature. Regularizer C,
penalty, and loss were fine-tuned to find the best hyperparameter combination.
And for the deep learning model, we fine-tuned Convolutional Neural Networks
(CNN) as described in [12] to get the best F1 score in the development set. We
also implemented Bi-directional Long Short Term Memory (Bi-LSTM) as described
in [13].

### Word Embedding Models:

We experimented with BengFastText(BFT) [14] pretrained on 250 million Bangla
articles and FastText[15], a multilingual model pretrained on 157 languages.
Note that both are pretrained on formal texts; thus denoted as formal
embedding. Additionally, 1.47 million Bangla comments from Facebook and
YouTube on eight different categories: education, entertainment, health,
influencer, religion, politics, sports, technology were collected. Then four
different word-embedding models were trained using those datasets. Two
word2vec[16] models denoted as W2V(SG) (Word2Vec skip-gram) and W2V(CBOW)
(Word2Vec continuous bag of words) and two FastText embeddings denoted as
FT(SG) (FastText skip-gram) and FT(CBOW) (FastText continuous bag of words)
were trained using informal texts, hence denoted as informal embedding.

### Transformer Models:

We experimented with some transformer-based models: monolingual Bangla-
BERT[17], multilingual BERT-cased and uncased[18]. The hyperparameters were
set according to literature [9] except for mBERT-uncased. In this case, the
learning rate was changed from its original value of 5e-5 to 3e-5 as the
validation loss during the initial training phase was terrible.

##  4 Result and Analysis

### Baseline analysis:

We present our experimental findings in table 3. We experimented with
different combinations of char and word ngram, and only the most notable
combinations were presented. It becomes apparent that in our dataset, e&p do
not serve as valuable features. Word n-gram with e&p had the worst F1 score,
and for char n-gram (1,6) and (2,6), when e&p are present, the F1 score drops.
For this reason, in hyper-parameter tuning and neural network experiments, e&p
were removed in the prepossessing step. The experimental results show that
char n-gram as a feature performs better than word n-gram. Our dataset
contains many spelling mistakes, which means the same words can have
inconsistent and multiple spelling variations for which SVM peforms poorly
[2]. The performance drops with experiments combining char n-gram with word
n-gram further validated the fact. Overall, char ngram (1,6) is the best
performing feature with F1 score of 85.86.

### Formal vs Informal Embeddings:

We observed that, SVM with char (1-6) gram as feature outperformed neural
networks with FastText and BFT as embedding by a good margin. [19] found
similar observations for their informal sentiment analysis dataset. They
proposed that pre-trained models based on formal text do not perform well on
the informal dataset. Two FastText informal embedding FT(SG) and FT(CBOW)
outperformed SVM in three out of four instances. Only CNN+FT(CBOW) achieved a
lower F1 score. Most notably, they were trained on a small 1.47 million
informal sentences compared to embeddings such as BFT and FastText but still
managed to outperform the bigger embeddings by a large margin. From the
experiments, we conclude, word embedding trained from scratch on informal
corpus will perform better for the informal dataset collected from social
media.

FT(SG) and FT(CBOW) performed comparatively better than both W2V(SG) and
W2V(CBOW) as presented in Table 3. According to [20], FastText is generally a
better choice compared to Word2Vec for our dataset. We can also see that
W2V(SG) and FT(SG) outperformed W2V(CBOW) and FT(CBOW), respectively. From
this, we can summarize that word embedding created on the SG method tends to
perform better than the CBOW method for our dataset. Overall, combining Bi-
LSTM with FT(SG) or Bi-LSTM+FT(SG) achieved the best result with an F1 score
of 86.85%.

### Transformer vs Other:

All BERT variants performed significantly worse compared to other models.
Possible reasons could be that mBERT is under-tuned for Bangla informal text
as it was trained with Bangla Wikipedia for which we observe the lower
performance. The experimental success with informal word embedding over formal
embedding support the last assumption.

| Model | F1 | MCC
---|---|---|---
Baseline | Unigram (U) | 82.89 | 72.95
U+e&p | 82.38 | 72.17
Bigram (B) | 63.12 | 49.23
Trigram (T) | 24.85 | 23.60
U+B | 82.11 | 71.32
U+B+T | 81.16 | 70.38
Char 1-6 gram(C(1,6)) | 85.97 | 77.15
C(1,6)+e&p | 85.86 | 77
C(2,6) | 85.89 | 77.04
C(2,6)+e&p | 85.76 | 76.86
U+B+C(1,6) | 84.71 | 75.04
U+B+C(2,6) | 84.73 | 75.10
Formal embedding | CNN+BFT | 71.58 | 53.06
Bi-LSTM+BFT | 76.09 | 61.51
CNN+FastText | 84 | 72.89
Bi-LSTM+FastText | 83.89 | 73.43
Informal embedding | CNN+W2V(CBOW) | 80.03 | 68.42
Bi-LSTM+W2V(CBOW) | 80.48 | 69.46
CNN+W2V(SG) | 81.07 | 69.23
Bi-LSTM+W2V(SG) | 80.93 | 69.25
CNN+FT(CBOW) | 85.05 | 75.15
Bi-LSTM+FT(CBOW) | 86.73 | 77.76
CNN+FT(SG) | 86.58 | 77.33
Bi-LSTM+FT(SG) | 86.85 | 77.90
Transformer | mBERT-cased | 81.66 | 72.81
mBERT-uncased | 82.56 | 73.33
Bangla-BERT | 83.68 | 74.63
Table 3: Benchmarking the hate speech classification models.

Category | Total | HS% | F1 avg | F1 improved
---|---|---|---|---
Sports | 4748 | 40.01 | 84.99 | 87.70
Entertainment | 5468 | 41.32 | 87.09 | 88.51
Crime | 3973 | 43.71 | 92.27 | 92.77
Religion | 3978 | 38.31 | 85.92 | 88.67
Politics | 2137 | 33.04 | 80.69 | 86.39
Celebrity | 1909 | 30.490 | 78.84 | 83.02
Miscellaneous | 18,021 | 41.35 | 85.99 | 88.00

Table 4: Category wise result comparison

### Category-wise Result:

From Table 4 we can see that for each category, the F1 scores of 5 top-
performing models: SVM, Bi-LSTM+FT(SG), CNN+FT(SG), Bi-LSTM+FT(CBOW), and
CNN+FT(CBOW) were averaged and presented in the F1 avg column. Politics and
Celebrity categories achieved the lowest average F1 scores. Two reasons might
contribute to this: both categories have the lowest training data and the
lowest HS%. To check this hypothesis, we dropped some NHS in each category so
that each of them has an equal number of HS and NHS. Then, the average F1
score for all five models was calculated and shown in the F1 improved column.
Note that F1 scores for Politics and Celebrity improved the most. On the other
hand, Crime has the best HS% and achieved the most F1 avg score. And so, it
also improved the least (only 0.5) after balancing.

### Influence of slang words:

The dataset vocabulary was extracted, and slang words were annotated with the
help of a linguistic expert. These slang words were divided into two types:
traditional slang (TS) and non-traditional slang (NTS). NTS is not usually
used as slang words but can be interpreted as such depending on the context.
Figure C.1 in Appendix C shows some examples of both types of slang words.
Then we looked at how models predicted comments that contain at least one such
slang word. For simplicity, we ignored comments that contained both TS and NTS
words. Our initial hypothesis was that models would face more trouble
detecting sentences NTS as these words are often used as HS as well as NHS.
Table 5 shows this result. TS acc shows how models predicted comments
containing TS and NTS acc is for NTS. We can see that models performed better
when comments contained TS words rather than NTS. NTS words are often vague in
meaning, and so they pose special difficulty for the models.

Model | TS acc | NTS acc
---|---|---
SVM | 84.27 | 79.53
Bi-LSTM+FT(SG) | 84.87 | 79.40
Bi-LSTM+FT(CBOW) | 85.08 | 77.97
CNN+FT(SG) | 84.95 | 77.59
CNN+FT(CBOW) | 83.39 | 74.87
Table 5: Influence of traditional and non traditional slang words

##  5 Conclusion

In this paper, we present HS-BN, the largest Bangla HS dataset collected from
social media comments. We followed various schemes to ensure linguistic
diversity and reduce repetitiveness. We found that emoji and punctuation do
not affect HS detection. We also found that word embedding trained on informal
text outperforms available embedding trained on formal text. We also observed
that traditional slangs make the detection of hate speech much easier than
that of comments with non-traditional slangs. In the future, we plan to
research on the interpretability and generalizability of hate speech detection
model.

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## Appendix

##  A: Bangla HS dataset comparision

| Dataset size | No of HS |  Annotation guideline |  Agreement score
---|---|---|---|---
(Chakraborty and Seddiqui, 2019) [5] | 5,644 | 2500 | No | No
(Emon et al., 2019) [4] | 4,700 | 3137 | No | No
(Awal et al., 2018) [21] | 2665 | 1214 | No | No
(Ishmam and Sharmin, 2019) [3] | 5,126 | 3178 | No | No
(Banik, 2019) [22] | 10,219 | 4255 | No | No
(Karim et al., 2020a) [9] | 6115 | 6115 | No | Yes
(Romim et al., 2021) [8] | 30,000 | 10,000 | Yes | No
Our | 50,314 | 20,209 | Yes | Yes
Table A.1: Overview of binary hate speech dataset of Bangla social media

##  B: Annotation Guidelines

We chose three criteria for a comment to be labeled as HS. These are:

  * •

Deliberate attack

  * •

Directed towards a specific group of people

  * •

Motivated by an aspect of groups identity

But bellow we also present a more nuanced guideline for annotation, based on
the community standard of Facebook and YouTube.

Criteria for HS

Criteria for not HS

##  C: Traditional and non traditional slang words

(a) Traditional Slang WordCloud

(b) Non Traditional Slang WordCloud

Figure C.1: Comparison of traditional and non-traditional slang words.


justified

# Dynamical fluctuations in a minimal model of higher-order networks

Leonardo Di Gaetano  Department of Network and Data Science, Central European
University, 1100 Vienna, Austria  Giorgio Carugno  Department of Mathematics,
King’s College London, Strand, London WC2R 2LS, UK  Federico Battiston
<EMAIL_ADDRESS>Department of Network and Data Science, Central European
University, 1100 Vienna, Austria  Francesco Coghi<EMAIL_ADDRESS>Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes
Alfvéns väg 12, SE-106 91 Stockholm, Sweden

###### Abstract

Although higher-order interactions are known to affect the typical state of
dynamical processes giving rise to new collective behavior, how they drive the
emergence of rare events and fluctuations is still an open problem. We
investigate how fluctuations of a dynamical quantity of a random walk
exploring a higher-order network arise over time. By focusing on a minimal
model, we show that higher-order interactions always hamper the appearance of
rare events, although the same structure facilitates visits of certain nodes,
an event considered atypical on the corresponding system with only pairwise
interactions. If the structure of interactions is not fixed but is optimally
selected to favour a particular fluctuation, a phase transition emerges where
a random walk is typically both homogeneously spread over the network and
localised on a portion of it. Our study lays the groundwork for a wider and
general theory of fluctuations and rare events in higher-order networks.

The appearance of fluctuations in dynamical processes is pivotal in
determining the future evolution of many real-world systems [1]. The emergence
of rare events may be bolstered or hindered by the hosting complex
environment, which often can be conveniently modeled as a complex network [2,
3, 4]. Large fluctuations in complex networks have been studied across a
variety of processes, including percolation [5, 6, 7, 8], spreading [9, 10],
and transport [11, 12, 13, 14]. A stream of research has focused on random
walks as a versatile model of diffusion in discrete spaces [15, 16, 17, 18,
19] and on their rare event properties [20, 21, 22]. Large deviation theory
has revealed that low-degree nodes are more susceptible than hubs to the
appearance of atypical loads, possibly leading to dynamical phase transitions
[23, 24, 25, 26].

Despite their success, graphs can only provide a constrained description of
real-world systems, as links are inherently limited to model pairwise
interactions only [27, 28]. Yet, from social [29, 30, 31, 32] to biological
[33, 34, 35, 36] networks, in a wide variety of real-word systems interactions
may occur among three or more units at a time. In recent years, hypergraphs
[37] have emerged as a versatile tool to model systems with such higher-order
interactions. Interestingly, taking into account higher-order interactions has
shown to lead to new collective phenomena in a variety of dynamical processes
[38], including diffusion [39, 40], contagion [41, 42, 43], synchronization
[44, 45, 46, 47, 48] and evolutionary games [49, 50, 51]. While such studies
have focused on characterising dynamical behavior at the typical state,
understanding fluctuations and rare events statistics driven by the presence
of higher-order interactions is to this day still an open problem.

To this end, in this work we propose a study of fluctuations and rare events
on higher-order networks using large-deviation theory tools. We focus on
random walks on higher-order networks and on a particular time-additive
observable that monitors the time the random walker spends in a certain region
(core) of the hypergraph. Our study reveals how fluctuations arise in time for
a random walk on a fixed hypergraph structure (quenched case), and which
higher-order structure is optimal to achieve them (annealed case). We show
that in the quenched case fluctuations of the core occupation time are always
hampered by the higher-order structure, while the same higher-order
interactions are best to visit nodes that would not be typically visited on
the corresponding system with only pairwise interactions. In the annealed
case, when the structure of interactions is not a-priori fixed, a phase
transition arises between a typically homogeneously-spread random walk and a
random walk more localised on subportions of the hypergraph.

## Model

We introduce a hypergraph $G=(V,E)$, where $V$ represents the set of nodes,
and $E=\\{E_{1},E_{2},\dots,E_{M}\\}$ the set of hyperedges, i.e., $E_{l}$ is
an unordered collection of nodes belonging to the same hyperedge $l$. We
consider in particular an illustrative structure consisting of a core node,
labelled $0$, connected with peripheral nodes through a varying number of
higher-order connections, labelled by $i\in\\{1,\dots,N-1\\}$. As shown in
Fig. 1, the graph is composed by $|V|=N$ nodes, a fully connected pairwise
structure, i.e. ${N\choose 2}$ binary edges $E_{i(2N-i-1)/2+j}=\\{i,j\\}$ for
$(i,j)\in[0,N-1]^{2}$ and $i<j$, and a binomially-distributed random number
$H$ of parameter $p\in[0,1]$ of three-body interactions
$E_{N(N-1)/2+i}=\\{0,i,j\\}$ where $i$ is an odd node and $j-i=1$, i.e., all
triangular interactions are centered in $0$. Intuitively, the higher the
number of higher-order interactions the better connected the core node is with
the periphery of the hypergraph. For simplicity, in the following we constrain
the higher-order structure so that each peripheral node can participate in at
most one three-body interaction. As we will show, for this symmetric model,
non-pairwise interactions affect the statistics of the core occupation time
only through their total number $\eta$. In particular, the probability of
drawing a hypergraph with a number of three-body interactions $H=\eta$ is
given by

$\mathbb{P}(\eta)\coloneqq\mathbb{P}(H=\eta)={N_{\triangle}\choose\eta}p^{\eta}(1-p)^{N_{\triangle}-\eta}\quad.$
(1)

This minimal model is no special with regards to our findings below—it only
simplifies calculations. We have checked this numerically with hypergraphs
that allow for overlapping three-body interactions and since results do not
qualitatively change we have decided to not include them. In summary, for the
model we consider here, $G$ comes as an instance of an ensemble of hypergraphs
whose higher-order structure is fully described by two parameters only,
namely, $N$ and $p$.

Figure 1: Sketch of a realisation of the hypergraph $G$. Dashed-lines
representing binary interactions define the fully-connected binary backbone of
the hypergraph. In pink, two higher-order interactions connect the core node
$0$ with the peripheral nodes $1$, $2$, $3$, and $4$. The stochastic dynamics
is represented by arrows departing from certain nodes and pointing onto others
(different thicknesses refer to different jump probabilities and depend on
whether nodes share only pairwise or also higher-order interactions).

We consider on $G$ a discrete-time random walk $X=\\{X_{l}\\}_{l=1}^{n}$,
where $X_{l}$ is the node where the random walk sits at time $l$ [40]. The
random walk is characterized by an unbiased dynamics encoded in the transition
matrix $\Pi=\left\\{\pi_{ij}\right\\}$ whose entries are

$\pi_{ij}=\frac{k_{ij}^{H}}{\sum_{l=1}^{N}k_{il}^{H}}\ ,$ (2)

where $k_{ij}^{H}$ represents the hyperdegree, i.e., the number of nodes,
excluding $i$, that are present in the hyperedges that are common to $i$ and
$j$. As the random walk explores the graph, it collects information in the
form of the time-additive observable

$T_{n}=\frac{1}{n}\sum_{l=1}^{n}\delta_{X_{\ell},0}\ ,$ (3)

which measures the fraction of time the random walk has spent on the core node
$0$ up to time $n$. Typically, for a very long time, for a number $H=\eta$ of
higher-order interactions the fraction of time the walker spends in $0$ can be
calculated [40] and reads

$T_{\eta,\text{typ}}=\frac{4\eta+N-1}{8\eta+(N-1)^{2}}\ .$ (4)

The higher the number of triangular interactions, the better connected is the
core with the periphery of the graph, and the longer the time the random walk
will spend on $0$.

We consider dynamical fluctuations in two different physical scenarios. First,
we study the mean behavior of rare events of $T_{n}$ over the ensemble of
hypergraphs (quenched case). Then, at the expense of an entropic cost
associated with the logarithm of $P(\eta)$ in (1), we let the random walk
choose the optimal hypergraph that generates a particular atypical fluctuation
of $T_{n}$ (annealed case).

## Quenched fluctuations

Here, we study fluctuations of $T_{n}$, which play a crucial role in the
finite-time evolution of real-world systems, using large deviation theory [52,
53, 54]. This is possible because the leading scaling behavior of the
probability distribution
$\mathbb{P}_{\eta,n}(t)\coloneqq\mathbb{P}_{\eta,n}(T_{n}=t)$ is exponential
in time, i.e.,

$\mathbb{P}_{\eta,n}(t)=e^{-nI_{\eta}(t)+o(n)}\ ,$ (5)

where $I_{\eta}(t)$ is the non-negative large-deviation rate function and
$o(n)$ denotes sub-linear corrections in $n$. Further notice the subscript
$\eta$ which refers to the fact that, at this stage, we are studying
fluctuations of $T_{n}$ on a fixed graph with $\eta$ higher-order
interactions. To understand rare events, we need to calculate $I_{\eta}$ in
(5). However, evaluating $I_{\eta}$ directly is often non-trivial, hence we
resort to a change of ensemble in order to get more meaningful information on
fluctuations. To this end, we introduce the Scaled Cumulant Generating
Function (SCGF)

$\Psi_{\eta}(s)=\lim_{n\rightarrow\infty}\ln
G_{\eta,n}(s)=\lim_{n\rightarrow\infty}\ln\mathbb{E}\left[e^{nsT_{n}}\right]\
,$ (6)

which characterizes the leading exponential behavior of the moment generating
function $G_{\eta,n}(s)$ associated with $T_{n}$. For finite and connected
hypergraphs, $\Psi_{\eta}(s)$ is analytic, and one can calculate $I_{\eta}(t)$
with a procedure described in the Gartner–Ellis theorem [55, 52, 53, 54] that
makes use of the Legendre–Fenchel transform

$I_{\eta}(t)=\sup_{s\in\mathbb{R}}\left(st-\Psi_{\eta}(s)\right)\ ,$ (7)

which links the Laplace parameter $s$ with a fluctuation $T_{n}=t$ via

$t=\Psi_{\eta}^{\prime}(s)\ .$ (8)

Because the random walk $X$ is ergodic, the SCGF can be obtained spectrally as

$\Psi_{\eta}(s)=\ln\zeta_{s}\ ,$ (9)

where $\zeta_{s}$, computed numerically, is the dominant eigenvalue of the so-
called tilted matrix

$\Pi_{s}=\left\\{(\pi_{s})_{ij}\right\\}=\left\\{\pi_{ij}e^{s\delta_{0,j}}\right\\}\
.$ (10)

To account for average properties of the ensemble of hypergraphs considered,
one can take a quenched average over the disorder—here characterized by the
number $\eta$ of higher-order interactions—of the function $\Psi_{\eta}$.
Recalling that $H$ is a binomially distributed random variable with parameter
$p$ and that the maximum number of higher-order interactions is
$N_{\triangle}=\text{ceil}\left[(N-2)/2\right]$, the quenched average can
explicitly be written as

$\Psi_{\text{q}}(s)=\sum_{\eta=0}^{N_{\triangle}}\mathbb{P}(\eta)\Psi_{\eta}(s)\
,$ (11)

where q stands for quenched. 111Remarkably, the quenched average (11) takes
such a simplified form because for a fixed number $\eta$ of higher-order
interactions we have only one possible transition matrix. However, we note
that more complicated models might lead to different disorder configurations
and therefore different transition matrices. In the latter case, to
disentangle disorder and dynamics one would need to carefully study
combinatorially how many different configurations arise by fixing $\eta$.
Given $\Psi_{\text{q}}(s)$ in (11), the quenched rate function
$I_{\text{q}}(t)$ can be obtained via a Legendre–Fenchel transform of
$\Psi_{q}$ (rather than $\Psi_{\eta}$) in (7).

To understand the role of higher-order interactions, we first look at whether
fluctuations, of the same magnitude but for different values of $p$, are more
or less likely to appear on higher-order networks generated with different
values of $p$. To understand this, we re-scale $t$ in $I_{\text{q}}(t)$ with
the typical fraction of time spent in $0$ by the random walk at a fixed
parameter $p$, namely $T_{\text{typ}}$, obtained by averaging (4) over
$\mathbb{P}(\eta)$. In Fig. 2(a) we plot the rate functions
$I_{q}(\tilde{t}=t/T_{\text{typ}})$ for different values of $p$ and compare
them with the rate function for a graph with no higher-order interactions
($p=0$ case). Because of the re-scaling, all rate functions are $0$ at the
typical value $\tilde{t}=1$. The likelihood is encoded in the shape of the
rate function branches, the higher (lower) the branch the exponentially-less
(more) likely is a fluctuation $\tilde{t}\neq 1$ to appear. We note that for
$p>0$, higher-order interactions always hinder fluctuations away from the
typical value, making it harder for the random walk to visit a core-localized
or periphery-delocalized phase in the fluctuations. This is because with
increasing $p$ the average number of higher-order interactions pointing to
node $0$ grows generating a ‘containment’ effect on the dynamics. As a direct
consequence escaping from node $0$ is made harder and dynamical fluctuations
are (exponentially) suppressed.

In Fig. 2(b), we also look at how, for a fluctuation $t$ of $T_{n}$ for the
random walk on the graph with only pairwise interactions, the large deviation
rate function $I_{q}(t)$ changes with increasing $p$. On the one hand, what is
typical for the case $p=0$ becomes more and more atypical with increasing $p$.
On the other hand, rare values of $T_{n}$ greater than the typical one for the
case $p=0$, can become typical just by increasing the number of higher-order
interactions. On the opposite, rare values of $T_{n}$ smaller than the typical
one, become only more atypical by introducing higher-order interactions. In
other words, by introducing higher-order interactions on a fully-pairwise
network we make it easier for the random walk to spend more (less) time on the
core (peripheral) node(s).

Figure 2: (a) Rate functions $I_{q}(\tilde{t})$ for different probabilities
$p$ of having higher-order interactions in the hypergraph. The higher the $p$,
the narrower the rate functions for $|\tilde{t}|>1$. (b) Heatmap representing
how the rate function $I_{q}(t)$ behaves as a function of $t$ and $p$ (for
visualisation purposes we plot $\sqrt{I_{q}}$). The light-blue line represents
the typical value $T_{\text{typ}}$ which linearly increases with $p$. Plots
obtained for a hypergraph with $N=1000$ nodes.

## Annealed fluctuations

We have hitherto considered the higher-order interaction structure of the
hypergraph and the dynamics as two distinct objects and have studied how the
dynamics itself is influenced by a fixed underlying structure. However, in a
variety of real-world systems, structure and dynamics are characterised by a
tight interplay [57]. In such a scenario, fluctuations arising for a dynamical
observable, such as $T_{n}$, will be sustained by an optimal, i.e., the most
likely one in probability terms, underlying structure. Motivated by this, in
the following we make use of our model to investigate how and which higher-
order interactions can be considered optimal for realising a fluctuation of
our dynamical observable.

In practice, we calculate annealed averages of the moment generating function
$G_{\eta,n}$ over the disorder as follows:

$G_{n}(s)=\sum_{\eta=0}^{N_{\triangle}}\mathbb{P}(\eta)G_{\eta,n}(s)\ ,$ (12)

where we remind the reader that fixing $s$ corresponds to fixing $t$ (on
average) according to (8). By considering both the long-time and large-graph
(and hence large $N_{\triangle}$) asymptotics, and introducing the re-scaled
parameter $h=\eta/N_{\triangle}\in[0,1]$, we approximate $G_{n}(s)$ in (12)
with its saddle-point solution for $h$ as

$G_{n}(s)\approx
e^{n\left(\ell^{-1}\log\mathbb{P}(h^{*})+\Psi_{\eta^{*}}(s)\right)}\ ,$ (13)

where we call $\ell=n/N_{\triangle}$ annealing parameter and $h^{*}$ is the
saddle-point solution, i.e. the most probable fraction of higher-order
interactions appearing that generates the fluctuation
$\Psi_{\eta^{*}}^{\prime}(s)=t$. We only consider the leading exponential
behaviour in $n$ of (13), i.e.,

$\hat{\Psi}_{\ell}(s)\coloneqq\ell^{-1}\log\mathbb{P}(h^{*})+\Psi_{\eta^{*}}(s)\
,$ (14)

whose Legendre–Fenchel transform $\hat{I}_{\ell}(t)$ is obtained replacing
$\Psi_{\eta}$ with $\hat{\Psi}_{\ell}$ in (7). The annealing parameter $\ell$
plays a key role. On the one hand, in the limit $\ell\rightarrow 0$ the
hypergraph size blows up and, because the disorder is self-averaging, all
probability concentrates around the typical number of higher-order
interactions, hence recovering the quenched average (11) for a fixed $p$.
[Notice however that it is only in the limit that the quenched average is
recovered because the disorder is self-averaging. Indeed, for small but finite
$\ell$ spurious contributions of $O(\ell^{-1})$ appear at exponent in (13) and
shift $\hat{I}_{\ell}$ upwards.] On the other hand, for
$\ell\rightarrow\infty$ the hypergraph size, although large, does not blow up
and therefore disorder and dynamics ‘interact’ at the saddle-point solution of
(13) selecting the most likely structure that realises the fluctuation
associated to $s$. We refer to
$\Psi_{a}(s)\coloneqq\hat{\Psi}_{\ell\rightarrow\infty}(s)$ as the annealed
SCGF and to $I_{a}(t)$ as the annealed rate function which characterises the
leading exponential behaviour of the probability distribution of $T_{n}$ in
such a scenario. Notice that all intermediate values of $\ell$ do not
necessarily define a good SCGF.222More explicitly, a large deviation principle
for $T_{n}$, similar in form to (5), does not hold for all values of $\ell$.
In particular, for $\ell$ finite and small, one has $N>n$ and therefore the
ergodicity assumption necessary to derive $\Psi_{\eta^{*}}(s)$ falls. It is
only for $n>N$ and $n$ large that we get a good approximation—exact in the
limit $n\rightarrow\infty$—for the distribution of $T_{n}$.

In Fig. 3(a) we plot $\hat{I}_{\ell}$ for several values of $\ell$. As
expected, for small $\ell$ we retrieve the quenched rate function $I_{q}$ (for
the same parameter $p=0.5$ used here) which is realised by the typical number
of higher-order interactions
$\eta^{*}=h^{*}N_{\triangle}\sim\text{ceil}\left[N_{\triangle}/2\right]$
throughout all fluctuations (see Fig. 3(b)). As we increase $\ell$, the
function $\hat{I}_{\ell}$ tends to flatten, and in the limit
$\ell\rightarrow\infty$ the annealed rate function $I_{a}$ develops a plateau
of zeros.333That an annealed rate function is a lower bound of a quenched one
is known in the mathematics literature [63, 64, 65, 66]. Intuitively, this is
consequence of picking an optimal structure to generate fluctuations in the
dynamics rather than having it fixed as in the quenched case. We characterise
this behaviour as a phase transition, which can be well understood by looking
at the optimal values $\eta^{*}$ that are selected in the limit
$\ell\rightarrow\infty$. Indeed, as one slightly moves from the typical
scenario $s=0$ and looks into fluctuations for either $s<0$ or $s>0$ the
structure optimally realising such fluctuations abruptly changes from,
respectively, a graph with no higher-order interactions to a structure that
maximizes their number. This comes as a consequence of our previous
observations: considering the case with no higher-order interactions as the
reference one, fluctuations on the right of the typical value, i.e., $s>0$,
are boosted by having more triangles on the graph, also hampering fluctuations
on the left, i.e., $s<0$ (see Fig. 2(b)). Despite the non-singleton set of
zeros of $I_{a}$, we remark that not all values of $t$ can be considered
typical: only those created by the most likely disorder configurations—sitting
at the extreme values of the plateau—will be.

Figure 3: (a) Quenched rate function $I_{q}$ (marked $Q$ in the legend),
$\hat{I}_{\ell}$ functions for different $\ell$, and annealed rate function
$I_{a}$ (marked as $A$ in the legend). Monte-Carlo simulations for the three
cases with no (left-most), maximum (right-most), and half-maximum (center)
number of higher-order interactions are plotted as coloured dots. (b)The
optimal value $\eta^{*}$ for the number of higher-order interactions plotted
as a function of $s$ (the fluctuation parameter). Results obtained for a
hypergraph with $N=1000$ nodes and $p=0.5$.

To validate our argument, always in Fig. 3(a) we plot Monte-Carlo (MC)
simulations as coloured dots for three different scenarios of random walks
exploring a graph with no (left-most black), max (right-most black), and half-
max (orange) number of higher-order interactions. By nature, MC simulations
for finite time $n$ underestimates the true probability (the dots do not touch
the abscissa) and can only show small fluctuations from the typical value.
However, simulations seem to fully agree with our description. Orange dots
well describe the shape of the quenched rate function $I_{q}$ close to its
minimum. Bearing in mind that $I_{a}$ is only the convex hull of the real
annealed rate function, black dots describe as well the two different
extreme—although always typical—behaviour appearing as a consequence of
annealing the dynamics through the disorder of higher-order interactions. The
structural phase transition signals the appearance of multiple typical states
between two extreme values for the random walk exploring our hypergraph $G$,
namely (i) a regime where the random walk is well spread over the whole graph,
and (ii) a regime where the random walk spends three times as much time on the
core node because of many higher-order interactions driving it there.

## Conclusion

In summary, we have presented an original study of dynamical fluctuations for
a stochastic process exploring a network with higher-order interactions. For
simplicity we have considered a model of a fully-connected pairwise graph with
additional random higher-order interactions connecting a core with peripheral
nodes. By using spectral methods of large deviation theory we have derived the
leading exponential scaling of fluctuations for a dynamical observable, here
considered to be the mean fraction of time the random walk spends on the core.
We have first focused on the (quenched) case of a fixed topology, finding that
a network with higher-order interactions always inhibit random walk
fluctuations. However, this same network facilitates visits of the core, an
event considered atypical on the fully-pairwise network. We have then
considered the (annealed) case where the random walk dynamics select the
optimal structure that realises a particular fluctuation. Our analysis reveals
the emergence of a phase transition characterised by two typical regimes: a
random walk visiting all nodes homogeneously, and another walk spending a much
bigger portion of time on the core node of the graph.

An interesting open problem is to characterise the appearance of fluctuations
for co-evolving systems, a setting where both the random walk and topology
evolve on a similar time scale [60], to better understand which physical
processes lead to the selection of optimal higher-order structures. Similarly,
additional insights might originate from considering more complex higher-order
structures, such as hypergraphs displaying a broad distribution of higher-
order interactions [61] or community structure [62]. Finally, an interesting
question is whether considering not only complex higher-order structures, but
also how complex pairwise connectivity patterns enhances or hinders dynamical
fluctuations, and—in the case of the former—which exact organization maximizes
the appearance of rare events. Higher-order interactions are known to give
rise to new collective behavior, affecting the typical state of several
processes, including spreading [41] and synchronization [47]. Hence, beyond
diffusive dynamics, our study lays the groundwork for a wider and general
theory of rare events in higher-order networks, a necessary step towards
understanding the finite-time evolution of real-world systems.

The authors are grateful to Pierpaolo Vivo and Hugo Touchette for valuable
feedback during the writing stage of the manuscript.

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# Non-uniform continuity of the Fokas-Olver-Rosenau-Qiao equation in Besov
spaces

Xing<EMAIL_ADDRESS>
College of Information and Management Science, Henan Agricultural University,
Zhengzhou, Henan, 450002, China

Abstract: In this paper, we prove that the solution map of Fokas-Olver-
Rosenau-Qiao equation (FORQ) is not uniformly continuous on the initial data
in Besov spaces. Our result extends the previous non-uniform continuity in
Sobolev spaces (Nonlinear Anal., 2014) [11] to Besov spaces and is consistent
with the present work (J. Math. Fluid Mech., 2020) [17] on Novikov equation up
to some coefficients when dropping the extra term $(\partial_{x}u)^{3}$ in
FORQ.

Keywords: Fokas-Olver-Rosenau-Qiao equation, Non-uniform continuous
dependence, Besov spaces

MSC (2010): 35B30; 35G25; 35Q53

##  1 Introduction

In this paper, we are concerned with the following Fokas-Olver-Rosenau-Qiao
equation (FORQ)

$\displaystyle\left\\{\begin{array}[]{ll}u_{t}-u_{xxt}+3u^{2}u_{x}-u_{x}^{3}-4uu_{x}u_{xx}+2u_{x}u_{xx}^{2}-u^{2}u_{xxx}+u_{x}^{2}u_{xxx}=0,~{}~{}t>0,~{}x\in\mathbb{R},\\\
u(0,x)=u_{0},~{}~{}x\in\mathbb{R}.\end{array}\right.$ (1.3)

Eq. (1.3) written in a slightly different form was first derived by Fokas [3]
as an integrable generalisation of the modified KdV equation. Soon after,
Fuchssteiner [5] and Olver-Rosenau [19] independently obtained similar
versions of this equation by performing a simple explicit algorithm based on
the bi-Hamiltonian representation of the classically integrable system.
Several years later, the concise form written above was recovered by Qiao [20]
from the two-dimensional Euler equations by using an approximation procedure.

The entire integrable hierarchy related to the FORQ equation was proposed by
Qiao [21]. It also has bi-Hamiltonian structure, which was first derived in
[19] and then in [20], admits Lax pair [20] and peakon travelling wave
solutions that are orbitally stable [6, 22, 18]. For more discussion about Lax
integrability and peakon solutions of FORQ we refer to [2], where this
equation is also referred as the modified Camassa-Holm equation.

The local well-posedness and ill-posedness for the Cauchy problem of the FORQ
equation (1.3) in Sobolev spaces and Besov spaces were studied in the series
of papers [11, 12, 13, 4]. It was showed by Himonas-Mantzavinos [11] that the
FORQ is well-posed in Sobolev space $H^{s}$ with $s>\frac{5}{2}$ in the sense
of Hadamard. Fu et al.[4] established the local well-posedness in Besov space
$B_{p,r}^{s}$ with $s>\max\\{2+\frac{1}{p},\frac{5}{2}\\}$, $1\leq
p,r\leq\infty$. After the non-uniform dependence for some dispersive equations
was studied by Kenig et al. [14], the issue of non-uniform continuity of
solutions on initial data has attracted much more attention, such as on
classical Camassa-Holm equation [7, 8, 9, 16] and on famous Novikov equation
[10, 17]. It was further proved in [11] that the dependence on initial data is
sharp, i.e. the data-to-solution map is continuous but not uniformly
continuous.

For studying the non-uniform continuity of the FORQ equation, it is more
convenient to express (1.3) in the following equivalent nonlocal form

$\begin{cases}u_{t}+u^{2}\partial_{x}u=\frac{1}{3}(\partial_{x}u)^{3}-\frac{1}{3}(1-\partial_{x}^{2})^{-1}[(\partial_{x}u)^{3}]-\partial_{x}(1-\partial_{x}^{2})^{-1}[\frac{2}{3}u^{3}+u(\partial_{x}u)^{2}],\\\
u(0,x)=u_{0},\;\qquad t>0,~{}x\in\mathbb{R}.\end{cases}$ (1.4)

When removing $(\partial_{x}u)^{3}$ from (1.4), (1.4) becomes the following
Novikov equation up to some coefficients

$\begin{cases}u_{t}+u^{2}\partial_{x}u=-\frac{1}{3}(1-\partial_{x}^{2})^{-1}[(\partial_{x}u)^{3}]-\partial_{x}(1-\partial_{x}^{2})^{-1}[\frac{2}{3}u^{3}+u(\partial_{x}u)^{2}],\\\
u(0,x)=u_{0},\;\qquad t>0,~{}x\in\mathbb{R}.\end{cases}$ (1.5)

Recently, Li, Yu and Zhu [17] have proved that the solution map of Novikov
equation is not uniformly continuous dependence on the initial data in the
Besov spaces $B_{p,r}^{s}(\mathbb{R})$,
$s>\max\\{1+\frac{1}{p},\frac{3}{2}\\}$, $1\leq p,r\leq\infty$. It is noticed
that well-posedness for Novikov equation holds for
$s>\max\\{1+\frac{1}{p},\frac{3}{2}\\}$ while well-posedness for FORQ holds
for $s>\max\\{2+\frac{1}{p},\frac{5}{2}\\}$. This difference between the well-
posedness index of these equations may be explained by the presence of the
extra term $(\partial_{x}u)^{3}$ in FORQ, which is not quasi-linear and is
absent from Novikov equation [11].

Up to the present, there is no result for the non-uniform continuous
dependence of FORQ in Besov space and it seems more difficult due to the
presence of the extra term $(\partial_{x}u)^{3}$, which elevates two
regularities, compared with Novikov equation and the method developed for the
Novikov equation in [17] will make it more complex. In this paper, we will
follow a different route to bypass this problem. Firstly, consider a new
system satisfied by $(1-\partial_{x})u:\triangleq v$. Secondly, for any
bounded set $v_{0}$ in working space, the corresponding solution
$\mathbf{S}_{t}(v_{0})$ can be approximated by a function of first degree of
time $t$ with convective term and nonlocal term being coefficients. With
suitable choice of initial data, the difference between two solutions will
produce a term from convective term which will not be small for small time,
and thus we obtain the non-uniform continuous dependence of FORQ. These will
be described in more detail later.

Now, we state our main result.

###### Theorem 1.1

Let $s>\max\\{2+\frac{1}{p},\frac{5}{2}\\}$, $1\leq p,r\leq\infty$. The
solution map $u_{0}\rightarrow\mathbf{S}_{t}(u_{0})$ of the initial value
problem (1.4) is not uniformly continuous from any bounded subset of
$B_{p,r}^{s}(\mathbb{R})$ into $\mathcal{C}([0,T];B_{p,r}^{s}(\mathbb{R}))$.
More precisely, there exist two sequences $u^{1,n}(0,x)$ and $u^{2,n}(0,x)$
such that

$\displaystyle\|u^{1,n}(0,x),u^{2,n}(0,x)\|_{B_{p,r}^{s}}\lesssim
1,\qquad\qquad\qquad\lim_{n\rightarrow\infty}\|u^{1,n}(0,x)-u^{2,n}(0,x)\|_{B_{p,r}^{s}}=0,$

but

$\displaystyle\liminf_{n\rightarrow\infty}\|\mathbf{S}_{t}(u^{1,n}(0,x))-\mathbf{S}_{t}(u^{2,n}(0,x))\|_{B_{p,r}^{s}}\gtrsim
t,\qquad t\in[0,T_{0}],$

with small positive time $T_{0}$ for $T_{0}\leq T$.

###### Remark 1.1

Since $B_{2,2}^{s}=H^{s}$ for any $s\in\mathbb{R}$, our result extends the
previous non-uniform continuity in Sobolev spaces [11] to Besov spaces.

###### Remark 1.2

When dropping the extra term $(\partial_{x}u)^{3}$ in FORQ, we can get the
same result on Novikov equation up to some coefficients, which is consistent
with the present work [17] on Novikov equation. The method we use in proving
Theorem 1.1 is different from [17] and is more general.

Notations: Given a Banach space $X$, we denote the norm of a function on $X$
by $\|\|_{X}$, and

$\displaystyle\|\cdot\|_{L_{T}^{\infty}(X)}=\sup_{0\leq t\leq
T}\|\cdot\|_{X}.$

The symbol $A\lesssim B$ means that there is a uniform positive constant $C$
independent of $A$ and $B$ such that $A\leq CB$.

##  2 Littlewood-Paley analysis

In this section, we will review the definition of Littlewood-Paley
decomposition and nonhomogeneous Besov space, and then list some useful
properties. For more details, the readers can refer to [1].

There exists a couple of smooth functions $(\chi,\varphi)$ valued in $[0,1]$,
such that $\chi$ is supported in the ball
$\mathcal{B}\triangleq\\{\xi\in\mathbb{R}^{d}:|\xi|\leq\frac{4}{3}\\}$,
$\varphi$ is supported in the ring
$\mathcal{C}\triangleq\\{\xi\in\mathbb{R}^{d}:\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\\}$
and $\varphi\equiv 1$ for $\frac{4}{3}\leq|\xi|\leq\frac{3}{2}$. Moreover,

$\forall\,\,\xi\in\mathbb{R}^{d},\,\,\chi(\xi)+{\sum\limits_{j\geq
0}\varphi(2^{-j}\xi)}=1,$
$\forall\,\,\xi\in\mathbb{R}^{d}\setminus\\{0\\},\,\,{\sum\limits_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)}=1,$
$|j-j^{\prime}|\geq 2\Rightarrow\textrm{Supp}\,\
\varphi(2^{-j}\cdot)\cap\textrm{Supp}\,\,\varphi(2^{-j^{\prime}}\cdot)=\emptyset,$
$j\geq
1\Rightarrow\textrm{Supp}\,\,\chi(\cdot)\cap\textrm{Supp}\,\,\varphi(2^{-j}\cdot)=\emptyset.$

Then, we can define the nonhomogeneous dyadic blocks $\Delta_{j}$ and
nonhomogeneous low frequency cut-off operator $S_{j}$ as follows:

$\Delta_{j}{u}=0,\,\,if\,\,j\leq-2,\quad\Delta_{-1}{u}=\chi(D)u=\mathcal{F}^{-1}(\chi\mathcal{F}u),$
$\Delta_{j}{u}=\varphi(2^{-j}D)u=\mathcal{F}^{-1}(\varphi(2^{-j}\cdot)\mathcal{F}u),\,\,if\,\,j\geq
0,$ $S_{j}{u}={\sum\limits_{j^{\prime}=-\infty}^{j-1}}\Delta_{j^{\prime}}{u}.$

######  Definition 2.1 ([1])

Let $s\in\mathbb{R}$ and $1\leq p,r\leq\infty$. The nonhomogeneous Besov space
$B^{s}_{p,r}(\mathbb{R}^{d})$ consists of all tempered distribution $u$ such
that

$\displaystyle||u||_{B^{s}_{p,r}(\mathbb{R}^{d})}\triangleq\Big{|}\Big{|}(2^{js}||\Delta_{j}{u}||_{L^{p}(\mathbb{R}^{d})})_{j\in\mathbb{Z}}\Big{|}\Big{|}_{\ell^{r}(\mathbb{Z})}<\infty.$

In the following, we list some basic lemmas and properties about Besov space
which will be frequently used in proving our main result.

###### Lemma 2.1

([1]) (1) Algebraic properties: $\forall s>0,$ $B_{p,r}^{s}(\mathbb{R}^{d})$
$\cap$ $L^{\infty}(\mathbb{R}^{d})$ is a Banach algebra.
$B_{p,r}^{s}(\mathbb{R}^{d})$ is a Banach algebra $\Leftrightarrow
B_{p,r}^{s}(\mathbb{R}^{d})\hookrightarrow
L^{\infty}(\mathbb{R}^{d})\Leftrightarrow s>\frac{d}{p}$ or $s=\frac{d}{p},$
$r=1$.
(2) For any $s>0$ and $1\leq p,r\leq\infty$, there exists a positive constant
$C=C(d,s,p,r)$ such that

$\|uv\|_{B^{s}_{p,r}(\mathbb{R}^{d})}\leq
C\Big{(}\|u\|_{L^{\infty}(\mathbb{R}^{d})}\|v\|_{B^{s}_{p,r}(\mathbb{R}^{d})}+\|v\|_{L^{\infty}(\mathbb{R}^{d})}\|u\|_{B^{s}_{p,r}(\mathbb{R}^{d})}\Big{)}.$

(3) Let $m\in\mathbb{R}$ and $f$ be an $S^{m}-$ multiplier (i.e.,
$f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is smooth and satisfies that
$\forall\alpha\in\mathbb{N}^{d}$, there exists a constant
$\mathcal{C}_{\alpha}$ such that
$|\partial^{\alpha}f(\xi)|\leq\mathcal{C}_{\alpha}(1+|\xi|)^{m-|\alpha|}$ for
all $\xi\in\mathbb{R}^{d}$). Then the operator $f(D)$ is continuous from
$B_{p,r}^{s}(\mathbb{R}^{d})$ to $B_{p,r}^{s-m}(\mathbb{R}^{d})$.
(4) For any $s\in\mathbb{R}$, $(1-\partial_{x})^{-1}$ is an isomorphic mapping
from $B_{p,r}^{s-1}(\mathbb{R}^{d})$ into $B_{p,r}^{s}(\mathbb{R}^{d})$.

###### Lemma 2.2

([1, 15]) Let $1\leq p,r\leq\infty$ and
$\sigma>-\min\\{\frac{1}{p},1-\frac{1}{p}\\}.$ There exists a constant
$C=C(p,r,\sigma)$ such that for any smooth solution to the following linear
transport equation:

$\quad\partial_{t}f+v\partial_{x}f=g,\quad\;f|_{t=0}=f_{0}.$

We have

$\displaystyle\sup_{s\in[0,t]}\|f(s)\|_{B^{\sigma}_{p,r}(\mathbb{R})}\leq
Ce^{CV_{p}(v,t)}\Big{(}\|f_{0}\|_{B^{\sigma}_{p,r}(\mathbb{R})}+\int^{t}_{0}\|g(\tau)\|_{B^{\sigma}_{p,r}(\mathbb{R})}\mathrm{d}\tau\Big{)},$
(2.6)

with

$\displaystyle V_{p}(v,t)=\begin{cases}\int_{0}^{t}\|\nabla
v(s)\|_{B^{\frac{1}{p}}_{p,\infty}(\mathbb{R})\cap
L^{\infty}(\mathbb{R})}\mathrm{d}s,&\ \ \mathrm{if}\;\sigma<1+\frac{1}{p},\\\
\int_{0}^{t}\|\nabla v(s)\|_{B^{\sigma}_{p,r}(\mathbb{R})}\mathrm{d}s,&\ \
\mathrm{if}\;\sigma=1+\frac{1}{p}\mbox{ and }r>1,\\\ \int_{0}^{t}\|\nabla
v(s)\|_{B^{\sigma-1}_{p,r}(\mathbb{R})}\mathrm{d}s,&\ \
\mathrm{if}\;\sigma>1+\frac{1}{p}\ \mathrm{or}\ \\{\sigma=1+\frac{1}{p}\mbox{
and }r=1\\}.\end{cases}$

##  3 Reformulation of the System

Due to the presence of the extra term $(\partial_{x}u)^{3}$ in FORQ, it seems
difficult to deal with Eq. (1.4) directly. Therefore, we shall first
differentiate FORQ with respect to $x$ and then simplify the resulting
expression, we obtain

$\displaystyle\partial_{t}(\partial_{x}u)$ $\displaystyle=$
$\displaystyle(\partial_{x}u)^{2}\partial_{x}^{2}u-2u(\partial_{x}u)^{2}-u^{2}\partial_{x}^{2}u+[\frac{2}{3}u^{3}+u(\partial_{x}u)^{2}]$
(3.1)
$\displaystyle-(1-\partial_{x}^{2})^{-1}\partial_{x}[\frac{1}{3}(\partial_{x}u)^{3}]-(1-\partial_{x}^{2})^{-1}[\frac{2}{3}u^{3}+u(\partial_{x}u)^{2}].$

Let $v=(1-\partial_{x})u$, we have from (1.4) and (3.1) that

$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}v=(v^{2}-2uv)\partial_{x}v-\frac{1}{3}u^{3}-\frac{1}{3}v^{3}-\Phi_{1}(v)-\Phi_{2}(v),\\\
u=(1-\partial_{x})^{-1}v,\\\ v(0,x)=(1-\partial_{x})u_{0}(x)\triangleq
v_{0},\end{array}\right.$ (3.5)

where the nonlocal terms $\Phi_{1}(v),\Phi_{2}(v)$ are defined by

$\displaystyle\Phi_{1}(v)=(1-\partial_{x}^{2})^{-1}[\frac{8}{3}u^{3}-\frac{1}{3}v^{3}-3u^{2}v],\;\;\;\Phi_{2}(v)=\partial_{x}(1-\partial_{x}^{2})^{2}[\frac{1}{3}v^{3}-u^{2}v].$

Since $(1-\partial_{x})^{-1}$ is an isomorphic mapping from
$B_{p,r}^{s-1}(\mathbb{R})$ into $B_{p,r}^{s}(\mathbb{R})$, the non-uniform
continuous dependence of $u$ in $B_{p,r}^{s}$ then can be transformed into
that of $v$ in $B_{p,r}^{s-1}.$

##  4 Non-uniform continuous dependence

In this section, we will give the proof of Theorem 1.1. However, as explained
above, we will directly consider Eq. (3.5) satisfied by $v$.

Firstly, we establish the estimates of the difference between the solution
$\mathbf{S}_{t}(v_{0})$ and initial data $v_{0}$ in different Besov norms.
That is

###### Proposition 4.1

Assume that $||v_{0}||_{B^{s-1}_{p,r}}\lesssim 1$. Under the assumptions of
Theorem 1.1, we have

$\displaystyle||\mathbf{S}_{t}(v_{0})-v_{0}||_{B^{s-2}_{p,r}}\lesssim
t||v_{0}||^{2}_{B^{s-2}_{p,r}}||v_{0}||_{B^{s-1}_{p,r}},$
$\displaystyle||\mathbf{S}_{t}(v_{0})-v_{0}||_{B^{s-1}_{p,r}}\lesssim
t\big{(}||v_{0}||^{3}_{B^{s-1}_{p,r}}+||v_{0}||^{2}_{B^{s-2}_{p,r}}||v_{0}||_{B^{s}_{p,r}}\big{)},$
$\displaystyle||\mathbf{S}_{t}(v_{0})-v_{0}||_{B^{s}_{p,r}}\lesssim
t\big{(}||v_{0}||^{2}_{B^{s-1}_{p,r}}||v_{0}||_{B^{s}_{p,r}}+||v_{0}||^{2}_{B^{s-2}_{p,r}}||v_{0}||_{B^{s+1}_{p,r}}\big{)}.$

Proof For simplicity, denote $v(t)=\mathbf{S}_{t}(v_{0})$. Firstly, according
to the local well-posedness result [4, 11], there exists a positive time
$T=T(||u_{0}||_{B^{s}_{p,r}},s,p,r)$ such that the solution $u(t)$ belongs to
$\mathcal{C}([0,T];B_{p,r}^{s})$. Moreover, by Lemmas 2.1-2.2, for all
$t\in[0,T]$ and $\gamma\geq s-2$, there holds

$\displaystyle||u(t)||_{B^{\gamma}_{p,r}}\leq
C||u_{0}||_{B^{\gamma}_{p,r}}\;\;\text{or}\;\;||v(t)||_{B^{\gamma-1}_{p,r}}\leq
C||v_{0}||_{B^{\gamma-1}_{p,r}}.$ (4.1)

Now we shall estimate the different Besov norms of the term $v(t)-v_{0}$,
which can be bounded by $t$ multiplying the corresponding Besov norms of
initial data $v_{0}$.

It follows by differential mean value theorem and the Minkowski inequality
that

$\displaystyle||v(t)-v_{0}||_{B^{s-1}_{p,r}}$
$\displaystyle\lesssim\int^{t}_{0}||\partial_{\tau}v||_{B^{s-1}_{p,r}}\mathrm{d}\tau$
$\displaystyle\lesssim\int^{t}_{0}||(v^{2}-2uv)\partial_{x}v||_{B^{s-1}_{p,r}}\mathrm{d}\tau+\int^{t}_{0}||\frac{1}{3}v^{3}||_{B^{s-1}_{p,r}}\mathrm{d}\tau$
$\displaystyle\quad+\int^{t}_{0}||\frac{1}{3}u^{3}||_{B^{s-1}_{p,r}}\mathrm{d}\tau+\int^{t}_{0}||\Phi_{1}(v)||_{B^{s-1}_{p,r}}\mathrm{d}\tau+\int^{t}_{0}||\Phi_{2}(v)||_{B^{s-1}_{p,r}}\mathrm{d}\tau.$

Here, we shall only have to estimate
$||(v^{2}-2uv)\partial_{x}v||_{B^{s-1}_{p,r}}$ and
$||\frac{1}{3}v^{3}||_{B^{s-1}_{p,r}}$, since the other terms can be processed
in a similar more relaxed way and have the same bound as
$||v^{3}||_{B^{s-1}_{p,r}}$.

Using the fact that $B_{p,r}^{s-2}$ is an Banach algebra with
$s-2>\max\\{\frac{1}{p},\frac{1}{2}\\}$, together with the product estimates
(2) in Lemma 2.1, one has

$\displaystyle||v^{3}||_{B^{s-1}_{p,r}}$ $\displaystyle\lesssim$
$\displaystyle||v||^{3}_{B^{s-1}_{p,r}},$
$\displaystyle||(v^{2}-2uv)\partial_{x}v||_{B^{s-1}_{p,r}}$
$\displaystyle\lesssim$
$\displaystyle||(v^{2}-2uv)||_{L^{\infty}}||\partial_{x}v||_{B^{s-1}_{p,r}}+||(v^{2}-2uv)||_{B^{s-1}_{p,r}}||\partial_{x}v||_{L^{\infty}}$
$\displaystyle\lesssim$
$\displaystyle||v||^{2}_{B^{s-2}_{p,r}}||v||_{B^{s}_{p,r}}+||v||^{3}_{B^{s-1}_{p,r}},$

where we have used the relation $u=(1-\partial_{x})^{-1}v$, and
$(1-\partial_{x})^{-1}$ is a $S^{-1}-$ multiplier, which is continuous from
$B_{p,r}^{s^{\prime}-1}(\mathbb{R})$ to $B_{p,r}^{s^{\prime}}(\mathbb{R})$,
thus $||u||_{B^{s^{\prime}}_{p,r}}\lesssim||v||_{B^{s^{\prime}-1}_{p,r}}$ for
any $s^{\prime}\in\mathbb{R}$.

Therefore, in view of (4.1), for $t\in[0,T]$, we have

$\displaystyle||v(t)-v_{0}||_{B^{s-1}_{p,r}}$ $\displaystyle\lesssim
t||v||^{2}_{L_{t}^{\infty}(B^{s-2}_{p,r})}||v||_{L_{t}^{\infty}(B^{s}_{p,r})}+||v||^{3}_{L_{t}^{\infty}(B^{s-1}_{p,r})}$
$\displaystyle\lesssim
t(||v_{0}||^{2}_{B^{s-2}_{p,r}}||v_{0}||_{B^{s}_{p,r}}+||v_{0}||^{3}_{B^{s-1}_{p,r}}).$

Following the same procedure of estimate as above, we have

$\displaystyle||v(t)-v_{0}||_{B^{s-2}_{p,r}}$ $\displaystyle\lesssim$
$\displaystyle\int^{t}_{0}||\partial_{\tau}v||_{B^{s-2}_{p,r}}\mathrm{d}\tau$
$\displaystyle\lesssim$
$\displaystyle\int^{t}_{0}||(v^{2}-2uv)\partial_{x}v||_{B^{s-2}_{p,r}}\mathrm{d}\tau+\int^{t}_{0}||\frac{1}{3}v^{3}||_{B^{s-2}_{p,r}}\mathrm{d}\tau$
$\displaystyle+\int^{t}_{0}||\frac{1}{3}u^{3}||_{B^{s-2}_{p,r}}\mathrm{d}\tau+\int^{t}_{0}||\Phi_{1}(v)||_{B^{s-2}_{p,r}}\mathrm{d}\tau+\int^{t}_{0}||\Phi_{2}(v)||_{B^{s-2}_{p,r}}\mathrm{d}\tau$
$\displaystyle\lesssim$ $\displaystyle
t||v||^{2}_{L_{t}^{\infty}(B^{s-2}_{p,r})}||v||_{L_{t}^{\infty}(B^{s-1}_{p,r})}$
$\displaystyle\lesssim$ $\displaystyle
t||v_{0}||^{2}_{B^{s-2}_{p,r}}||v_{0}||_{B^{s-1}_{p,r}},$

and

$\displaystyle||v^{3}||_{B^{s}_{p,r}}$ $\displaystyle\lesssim$
$\displaystyle||v^{2}||_{L^{\infty}}||v||_{B^{s}_{p,r}}+||v^{2}||_{B^{s}_{p,r}}||v||_{L^{\infty}}$
$\displaystyle\lesssim$
$\displaystyle||v||^{2}_{B^{s-1}_{p,r}}||v||_{B^{s}_{p,r}},$
$\displaystyle||(v^{2}-2uv)\partial_{x}v||_{B^{s}_{p,r}}$
$\displaystyle\lesssim$
$\displaystyle||(v^{2}-2uv)||_{L^{\infty}}||\partial_{x}v||_{B^{s}_{p,r}}+||(v^{2}-2uv)||_{B^{s}_{p,r}}||\partial_{x}v||_{L^{\infty}}$
$\displaystyle\lesssim$
$\displaystyle||v||^{2}_{B^{s-2}_{p,r}}||v||_{B^{s+1}_{p,r}}+||v||^{2}_{B^{s-1}_{p,r}}||v||_{B^{s}_{p,r}},$

hence,

$\displaystyle||v(t)-v_{0}||_{B^{s}_{p,r}}$ $\displaystyle\lesssim$
$\displaystyle\int^{t}_{0}||\partial_{\tau}v||_{B^{s}_{p,r}}\mathrm{d}\tau$
$\displaystyle\lesssim$
$\displaystyle\int^{t}_{0}||(v^{2}-2uv)\partial_{x}v||_{B^{s}_{p,r}}\mathrm{d}\tau+\int^{t}_{0}||\frac{1}{3}v^{3}||_{B^{s}_{p,r}}\mathrm{d}\tau$
$\displaystyle+\int^{t}_{0}||\frac{1}{3}u^{3}||_{B^{s}_{p,r}}\mathrm{d}\tau+\int^{t}_{0}||\Phi_{1}(v)||_{B^{s}_{p,r}}\mathrm{d}\tau+\int^{t}_{0}||\Phi_{2}(v)||_{B^{s}_{p,r}}\mathrm{d}\tau$
$\displaystyle\lesssim$ $\displaystyle
t(||v||^{2}_{L^{\infty}_{t}(B^{s-2}_{p,r})}||v||_{L^{\infty}_{t}(B^{s+1}_{p,r})}+||v||^{2}_{L^{\infty}_{t}(B^{s-1}_{p,r})}||v||_{L^{\infty}_{t}(B^{s}_{p,r})})$
$\displaystyle\lesssim$ $\displaystyle
t(||v_{0}||^{2}_{B^{s-2}_{p,r}}||v_{0}||_{B^{s+1}_{p,r}}+||v_{0}||^{2}_{B^{s-1}_{p,r}}||v_{0}||_{B^{s}_{p,r}}),$

Thus, we finish the proof of Proposition 4.1.

With the different Besov norms estimates of $v-v_{0}$ at hand, we have the
following core estimates, which implies that for any bounded initial data
$v_{0}$ in $B^{s-1}_{p,r}$, the corresponding solution $S_{t}(v_{0})$ can be
approximated by
$v_{0}+t(v_{0}^{2}-2u_{0}v_{0})\partial_{x}v_{0}+\frac{1}{3}tv_{0}^{3}+t[\frac{1}{3}u_{0}^{3}+\Phi_{1}(v_{0})+\Phi_{2}(v_{0})]$
near $t=0$.

###### Proposition 4.2

Assume that $||v_{0}||_{B^{s-1}_{p,r}}\lesssim 1$. Then under the assumptions
of Theorem 1.1, there holds

$\displaystyle||\mathbf{S}_{t}(v_{0})-v_{0}-t\mathbf{v}_{0}||_{B^{s-1}_{p,r}}\lesssim
t^{2}\big{(}||v_{0}||^{3}_{B^{s-1}_{p,r}}+||v_{0}||^{2}_{B^{s-2}_{p,r}}||v_{0}||_{B^{s}_{p,r}}+||v_{0}||^{4}_{B^{s-2}_{p,r}}||v_{0}||_{B^{s+1}_{p,r}}\big{)},$

where
$\mathbf{v}_{0}=(v_{0}^{2}-2u_{0}v_{0})\partial_{x}v_{0}+\frac{1}{3}v_{0}^{3}+\frac{1}{3}u_{0}^{3}+\Phi_{1}(v_{0})+\Phi_{2}(v_{0}).$

Proof Using differential mean value theorem and the Minkowski inequality, we
first arrive at

$\displaystyle||v(t)-v_{0}-t\mathbf{v}_{0}||_{B^{s-1}_{p,r}}$
$\displaystyle\lesssim$
$\displaystyle\int^{t}_{0}||\partial_{\tau}v-\mathbf{v}_{0}||_{B^{s-1}_{p,r}}\mathrm{d}\tau$
(4.2) $\displaystyle\lesssim$
$\displaystyle\int^{t}_{0}||v^{2}\partial_{x}v-v_{0}^{2}\partial_{x}v_{0}||_{B^{s-1}_{p,r}}\mathrm{d}\tau+\int^{t}_{0}||2uv\partial_{x}v-2u_{0}v_{0}\partial_{x}v_{0}||_{B^{s-1}_{p,r}}\mathrm{d}\tau$
$\displaystyle+\int^{t}_{0}||\frac{1}{3}v^{3}-\frac{1}{3}v_{0}^{3}||_{B^{s-1}_{p,r}}\mathrm{d}\tau+\int^{t}_{0}||\frac{1}{3}u^{3}-\frac{1}{3}u_{0}^{3}||_{B^{s-1}_{p,r}}\mathrm{d}\tau$
$\displaystyle+\int^{t}_{0}||\Phi_{1}(v)-\Phi_{1}(v_{0})||_{B^{s-1}_{p,r}}\mathrm{d}\tau+\int^{t}_{0}||\Phi_{2}(v)-\Phi_{2}(v_{0})||_{B^{s-1}_{p,r}}\mathrm{d}\tau.$

Using (3) in Lemma 2.1, it is sufficient to estimate
$||v^{2}\partial_{x}v-v_{0}^{2}\partial_{x}v_{0}||_{B^{s-1}_{p,r}}$,
$||2uv\partial_{x}v-2u_{0}v_{0}\partial_{x}v_{0}||_{B^{s-1}_{p,r}}$ and
$||v^{3}-v_{0}^{3}||_{B^{s-1}_{p,r}}$, since the other terms can be processed
in a similar more relaxed way and have the same bound as
$||v^{3}-v_{0}^{3}||_{B^{s-1}_{p,r}}$.

It should be noticed that according to (4.1),
$||v||_{B^{s-1}_{p,r}}\lesssim||v_{0}||_{B^{s-1}_{p,r}}\lesssim 1$, which will
be frequently used later.

Due to the fact that $B_{p,r}^{s-2}$ is an Banach algebra with
$s-2>\max\\{\frac{1}{p},\frac{1}{2}\\}$, combining with the product estimates
(2) in Lemma 2.1, we get

$\displaystyle||v^{2}\partial_{x}v-v_{0}^{2}\partial_{x}v_{0}||_{B^{s-1}_{p,r}}$
$\displaystyle=$
$\displaystyle||(v^{2}-v_{0}^{2})\partial_{x}v+v_{0}^{2}(\partial_{x}v-\partial_{x}v_{0})||_{B^{s-1}_{p,r}}$
$\displaystyle\lesssim$
$\displaystyle||(v^{2}-v_{0}^{2})\partial_{x}v||_{B^{s-1}_{p,r}}+||v_{0}^{2}(\partial_{x}v-\partial_{x}v_{0})||_{B^{s-1}_{p,r}}$
$\displaystyle\lesssim$
$\displaystyle||v^{2}-v_{0}^{2}||_{L^{\infty}}||\partial_{x}v||_{B^{s-1}_{p,r}}+||v^{2}-v_{0}^{2}||_{B^{s-1}_{p,r}}||\partial_{x}v||_{L^{\infty}}$
$\displaystyle+||v_{0}^{2}||_{L^{\infty}}||\partial_{x}v-\partial_{x}v_{0}||_{B^{s-1}_{p,r}}+||v_{0}^{2}||_{B^{s-1}_{p,r}}||\partial_{x}v-\partial_{x}v_{0}||_{L^{\infty}}$
$\displaystyle\lesssim$
$\displaystyle||v-v_{0}||_{B^{s-2}_{p,r}}||v||_{B^{s}_{p,r}}+||v-v_{0}||_{B^{s-1}_{p,r}}+||v-v_{0}||_{B^{s}_{p,r}}||v_{0}||^{2}_{B^{s-2}_{p,r}},$
$\displaystyle||2uv\partial_{x}v-2u_{0}v_{0}\partial_{x}v_{0}||_{B^{s-1}_{p,r}}$
$\displaystyle=$
$\displaystyle||2(u-u_{0})v\partial_{x}v+2u_{0}(v\partial_{x}v-v_{0}\partial_{x}v_{0})||_{B^{s-1}_{p,r}}$
$\displaystyle\lesssim$
$\displaystyle||u-u_{0}||_{B^{s-1}_{p,r}}||v\partial_{x}v||_{B^{s-1}_{p,r}}+||u_{0}||_{B^{s-1}_{p,r}}||v\partial_{x}v-v_{0}\partial_{x}v_{0}||_{B^{s-1}_{p,r}}$
$\displaystyle\lesssim$
$\displaystyle||v-v_{0}||_{B^{s-2}_{p,r}}||v||_{B^{s}_{p,r}}+||v-v_{0}||_{B^{s-1}_{p,r}}+||v-v_{0}||_{B^{s}_{p,r}}||v_{0}||^{2}_{B^{s-2}_{p,r}},$

and

$\displaystyle\|v^{3}-v_{0}^{3}\|_{B^{s-1}_{p,r}}=\|(v-v_{0})(v^{2}+vv_{0}+v_{0}^{2})\|_{B^{s-1}_{p,r}}\lesssim||v-v_{0}||_{B^{s-1}_{p,r}}.$

Taking the above estimates into (4.2), which together with Proposition 4.1
yield

$\displaystyle||v(t)-v_{0}-t\mathbf{v}_{0}||_{B^{s-1}_{p,r}}$
$\displaystyle\lesssim$
$\displaystyle\int_{0}^{t}(||v-v_{0}||_{B^{s-1}_{p,r}}+||v-v_{0}||_{B^{s-2}_{p,r}}||v||_{B^{s}_{p,r}}d\tau+||v-v_{0}||_{B^{s}_{p,r}}||v_{0}||^{2}_{B^{s-2}_{p,r}})d\tau$
$\displaystyle\lesssim$ $\displaystyle
t^{2}\big{(}||v_{0}||^{3}_{B^{s-1}_{p,r}}+||v_{0}||^{2}_{B^{s-2}_{p,r}}||v_{0}||_{B^{s}_{p,r}}+||v_{0}||^{4}_{B^{s-2}_{p,r}}||v_{0}||_{B^{s+1}_{p,r}}\big{)}.$

Thus, we complete the proof of Proposition 4.2. $\Box$

Now, we move on the proof of Theorem 1.1.

Proof of Theorem 1.1 Let $\hat{\phi}\in\mathcal{C}^{\infty}_{0}(\mathbb{R})$
be an even, real-valued and non-negative funtion on $\mathbb{R}$ and satisfy

$\displaystyle\hat{\phi}(x)=$ $\displaystyle 1,$ if $|x|\leq\frac{1}{4}$,
$\displaystyle\hat{\phi}(x)=$ $\displaystyle 0,$ if $|x|\geq\frac{1}{2}$.

Define the high frequency function $f_{n}$ and the low frequency function
$g_{n}$ by

$f_{n}=2^{-ns}\phi(x)\sin\Big{(}\frac{17}{12}2^{n}x\Big{)},\qquad
g_{n}=2^{-\frac{n}{2}}\phi(x),\quad n\gg 1.$

It has been showed in [16] that $\|f_{n}\|_{B_{p,r}^{\sigma}}\lesssim
2^{n(\sigma-s)}$.

Let

$v^{1,n}(0,x)=(1-\partial_{x})(f_{n}+g_{n}),\;v^{2,n}(0,x)=(1-\partial_{x})f_{n}.$

Consider Eq. (3.5) with initial data $v^{1,n}(0,x)$ and $v^{2,n}(0,x)$,
respectlively. Obviously, we have

$\displaystyle||v^{1,n}(0,x)-v^{2,n}(0,x)||_{B^{s-1}_{p,r}}=||(1-\partial_{x})g_{n}||_{B^{s-1}_{p,r}}\leq
C2^{-\frac{n}{2}},$

which means that

$\displaystyle\lim_{n\to\infty}||v^{1,n}(0,x)-v^{2,n}(0,x)||_{B^{s-1}_{p,r}}=0.$

It is easy to show that

$\displaystyle||v^{1,n}(0,x)||_{B^{s-2}_{p,r}}$ $\displaystyle\lesssim$
$\displaystyle||f_{n}+g_{n}||_{B^{s-1}_{p,r}}\lesssim||f_{n}+g_{n}||_{B^{s-\frac{1}{2}}_{p,r}}\lesssim
2^{-\frac{n}{2}},$ $\displaystyle||v^{1,n}(0,x)||_{B^{s+\sigma}_{p,r}}$
$\displaystyle\lesssim$
$\displaystyle||f_{n}+g_{n}||_{B^{s+\sigma+1}_{p,r}}\lesssim
2^{n(\sigma+1)}\qquad\mathrm{for}\qquad\sigma\geq-\frac{3}{2},$
$\displaystyle||v^{2,n}(0,x)||_{B^{s+t}_{p,r}}$ $\displaystyle\lesssim$
$\displaystyle||f_{n}||_{B^{s+t+1}_{p,r}}\lesssim
2^{n(t+1)}\qquad\qquad\;\;\mathrm{for}\qquad t\in\mathbb{R},$

which imply

$\displaystyle\big{(}||v^{1,n}(0,x)||^{3}_{B^{s-1}_{p,r}}+||v^{1,n}(0,x)||^{2}_{B^{s-2}_{p,r}}||v^{1,n}(0,x)||_{B^{s}_{p,r}}+||v^{1,n}(0,x)||^{4}_{B^{s-2}_{p,r}}||v^{1,n}(0,x)||_{B^{s+1}_{p,r}}\big{)}\lesssim
1,$
$\displaystyle\big{(}||v^{2,n}(0,x)||^{3}_{B^{s-1}_{p,r}}+||v^{2,n}(0,x)||^{2}_{B^{s-2}_{p,r}}||v^{2,n}(0,x)||_{B^{s}_{p,r}}+||v^{2,n}(0,x)||^{4}_{B^{s-2}_{p,r}}||v^{2,n}(0,x)||_{B^{s+1}_{p,r}}\big{)}\lesssim
1.$

Furthermore, since $v^{1,n}(0,x)$ and $v^{2,n}(0,x)$ are both bounded in
$B_{p,r}^{s-1}$, according to Proposition 4.2, we deduce that

$\displaystyle||\mathbf{S}_{t}(v^{1,n}(0,x))-\mathbf{S}_{t}(v^{2,n}(0,x))||_{B^{s-1}_{p,r}}$
(4.3) $\displaystyle\geq$ $\displaystyle
t\big{|}\big{|}([v^{1,n}(0,x)]^{2}-2u^{1,n}(0,x)v^{1,n}(0,x))\partial_{x}v^{1,n}(0,x)$
$\displaystyle-([v^{2,n}(0,x)]^{2}-2u^{2,n}(0,x)v^{2,n}(0,x))\partial_{x}v^{2,n}(0,x)$
$\displaystyle+\frac{1}{3}([v^{1,n}(0,x)]^{3}-[v^{2,n}(0,x)]^{3})+[\frac{1}{3}[u^{1,n}(0,x)]^{3}+\Phi_{1}(v^{1,n}(0,x))+\Phi_{2}(v^{1,n}(0,x))]$
$\displaystyle-[\frac{1}{3}[u^{2,n}(0,x)]^{3}+\Phi_{1}(v^{2,n}(0,x))+\Phi_{2}(v^{2,n}(0,x))]\big{|}\big{|}_{B^{s-1}_{p,r}}-\big{|}\big{|}(1-\partial_{x})g_{n}\big{|}\big{|}_{B_{p,r}^{s-1}}-Ct^{2}$
$\displaystyle\geq$
$\displaystyle~{}t\big{|}\big{|}([v^{1,n}(0,x)]^{2}-2u^{1,n}(0,x)v^{1,n}(0,x))\partial_{x}v^{1,n}(0,x)$
$\displaystyle-([v^{2,n}(0,x)]^{2}-2u^{2,n}(0,x)v^{2,n}(0,x))\partial_{x}v^{2,n}(0,x)\big{|}\big{|}_{B^{s-1}_{p,r}}-C2^{-\frac{n}{2}}-Ct^{2}.$

For the sake of simplicity and convenience, in the following we denote

$v^{i,n}(0,x)\triangleq v_{i},\;\;u^{i,n}(0,x)\triangleq u_{i},\;\;i=1,2.$

The coefficient of the first order term of $t$ in the last inequality in (4.3)
is simplified as

$\displaystyle(v_{1}^{2}-2u_{1}v_{1})\partial_{x}v_{1}-(v_{2}^{2}-2u_{2}v_{2})\partial_{x}v_{2}$
$\displaystyle=$
$\displaystyle(v_{1}^{2}\partial_{x}v_{1}-v_{2}^{2}\partial_{x}v_{2})-2(u_{1}v_{1}\partial_{x}v_{1}-v_{2}\partial_{x}v_{2})$
$\displaystyle=$ $\displaystyle I_{1}-I_{2}.$

Bring in the concrete form of $v_{1}$ and $v_{2}$ where necessary, we have

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle(v_{1}^{2}-v_{2}^{2})\partial_{x}v_{1}+v_{2}^{2}(\partial_{x}v_{1}-\partial_{x}v_{2})$
$\displaystyle=$
$\displaystyle(v_{1}^{2}-v_{2}^{2})\partial_{x}(1-\partial_{x})(f_{n}+g_{n})+v_{2}^{2}(\partial_{x}v_{1}-\partial_{x}v_{2})$
$\displaystyle=$
$\displaystyle(v_{1}^{2}-v_{2}^{2})\partial_{x}(f_{n}+g_{n})-(v_{1}^{2}-v_{2}^{2})\partial_{x}^{2}g_{n}-(v_{1}^{2}-v_{2}^{2})\partial_{x}^{2}f_{n}+v_{2}^{2}\partial_{x}(v_{1}-v_{2}),$
$\displaystyle I_{2}$ $\displaystyle=$ $\displaystyle
2u_{1}(v_{1}-v_{2})\partial_{x}v_{1}+2v_{2}(u_{1}\partial_{x}v_{1}-u_{2}\partial_{x}v_{2})$
$\displaystyle=$ $\displaystyle
2u_{1}(v_{1}-v_{2})\partial_{x}(1-\partial_{x})(f_{n}+g_{n})+2v_{2}u_{1}\partial_{x}(v_{1}-v_{2})+2v_{2}(u_{1}-u_{2})\partial_{x}v_{2}$
$\displaystyle=$ $\displaystyle
2u_{1}(v_{1}-v_{2})\partial_{x}(f_{n}+g_{n})-2u_{1}(v_{1}-v_{2})\partial_{x}^{2}g_{n}-2u_{1}(v_{1}-v_{2})\partial_{x}^{2}f_{n}$
$\displaystyle+2v_{2}u_{1}\partial_{x}(v_{1}-v_{2})+2v_{2}(u_{1}-u_{2})\partial_{x}v_{2}.$

Using Lemma 2.1, after simple calculation, we obtain

$\displaystyle\|(v_{1}^{2}-v_{2}^{2})\partial_{x}(f_{n}+g_{n})\|_{B_{p,r}^{s-1}}$
$\displaystyle\lesssim$
$\displaystyle\|v_{1}^{2}-v_{2}^{2}\|_{L^{\infty}}\|f_{n}+g_{n}\|_{B_{p,r}^{s}}+\|v_{1}^{2}-v_{2}^{2}\|_{B_{p,r}^{s-1}}\|\partial_{x}(f_{n}+g_{n})\|_{L^{\infty}}$
$\displaystyle\lesssim$ $\displaystyle 2^{-\frac{n}{2}},$
$\displaystyle\|(v_{1}^{2}-v_{2}^{2})\partial_{x}^{2}g_{n}\|_{B_{p,r}^{s-1}}$
$\displaystyle\lesssim$
$\displaystyle\|v_{1}^{2}-v_{2}^{2}\|_{B_{p,r}^{s-1}}\|\partial_{x}^{2}g_{n}\|_{B_{p,r}^{s-1}}\lesssim
2^{-\frac{n}{2}},$
$\displaystyle\|v_{2}^{2}\partial_{x}(v_{1}-v_{2})\|_{B_{p,r}^{s-1}}$
$\displaystyle\lesssim$
$\displaystyle\|v_{2}^{2}\|_{B_{p,r}^{s-1}}\|\partial_{x}(v_{1}-v_{2})\|_{B_{p,r}^{s-1}}\lesssim
2^{-\frac{n}{2}},$
$\displaystyle\|2u_{1}(v_{1}-v_{2})\partial_{x}(f_{n}+g_{n})\|_{B_{p,r}^{s-1}}$
$\displaystyle\lesssim$
$\displaystyle\|u_{1}\|_{B_{p,r}^{s-1}}\|(v_{1}-v_{2})\|_{B_{p,r}^{s-1}}\|\partial_{x}(f_{n}+g_{n})\|_{B_{p,r}^{s-1}}\lesssim
2^{-\frac{n}{2}},$
$\displaystyle\|2u_{1}(v_{1}-v_{2})\partial_{x}^{2}g_{n}\|_{B_{p,r}^{s-1}}$
$\displaystyle\lesssim$
$\displaystyle\|u_{1}\|_{B_{p,r}^{s-1}}\|(v_{1}-v_{2})\|_{B_{p,r}^{s-1}}\|\partial_{x}^{2}g_{n}\|_{B_{p,r}^{s-1}}\lesssim
2^{-\frac{n}{2}},$
$\displaystyle\|2v_{2}u_{1}\partial_{x}(v_{1}-v_{2})\|_{B_{p,r}^{s-1}}$
$\displaystyle\lesssim$
$\displaystyle\|v_{2}\|_{B_{p,r}^{s-1}}\|u_{1}\|_{B_{p,r}^{s-1}}\|\partial_{x}(v_{1}-v_{2})\|_{B_{p,r}^{s-1}}\lesssim
2^{-\frac{n}{2}},$
$\displaystyle\|2v_{2}(u_{1}-u_{2})\partial_{x}v_{2}\|_{B_{p,r}^{s-1}}$
$\displaystyle\lesssim$
$\displaystyle\|v_{2}\|_{B_{p,r}^{s-1}}\|u_{1}-u_{2}\|_{B_{p,r}^{s-1}}\|\partial_{x}v_{2}\|_{B_{p,r}^{s-1}}\lesssim
2^{-\frac{n}{2}}.$

While

$\displaystyle[2u_{1}(v_{1}-v_{2})-(v_{1}^{2}-v_{2}^{2})]\partial_{x}^{2}f_{n}$
$\displaystyle=$
$\displaystyle(1-\partial_{x})g_{n}[g_{n}+\partial_{x}(2f_{n}+g_{n})]\partial_{x}^{2}f_{n}$
$\displaystyle=$
$\displaystyle(1-\partial_{x})g_{n}(1+\partial_{x})g_{n}\partial_{x}^{2}f_{n}+2(1-\partial_{x})g_{n}\partial_{x}f_{n}\partial_{x}^{2}f_{n},$

using product law (2) in Lemma 2.1, we have that

$\displaystyle\|(1-\partial_{x})g_{n}\partial_{x}f_{n}\partial_{x}^{2}f_{n}\|_{B_{p,r}^{s-1}}$
$\displaystyle\lesssim$
$\displaystyle\|(1-\partial_{x})g_{n}\partial_{x}f_{n}\|_{B_{p,r}^{s-1}}\|\partial_{x}^{2}f_{n}\|_{L^{\infty}}+\|(1-\partial_{x})g_{n}\partial_{x}f_{n}\|_{L^{\infty}}\|\partial_{x}^{2}f_{n}\|_{B_{p,r}^{s-1}}$
$\displaystyle\lesssim$ $\displaystyle 2^{-\frac{n}{2}}\cdot 2^{0}\cdot
2^{n(2-s)}+2^{-\frac{n}{2}}\cdot 2^{n(1-s)}\cdot 2^{n}\lesssim
2^{-n(s-\frac{3}{2})}.$

Taking the above estimates into (4.3), we find that

$\displaystyle||\mathbf{S}_{t}(v^{1,n}(0,x))-\mathbf{S}_{t}(v^{2,n}(0,x))||_{B^{s-1}_{p,r}}\geq
t\big{|}\big{|}(1-\partial_{x})g_{n}(1+\partial_{x})g_{n}\partial_{x}^{2}f_{n}\big{|}\big{|}_{B^{s-1}_{p,r}}-C2^{-\frac{n}{2}}-Ct^{2}.$
(4.4)

For the term
$(1-\partial_{x})g_{n}(1+\partial_{x})g_{n}\partial_{x}^{2}f_{n}$, it can be
verified that
$\Delta_{j}\big{(}(1-\partial_{x})g_{n}(1+\partial_{x})g_{n}\partial_{x}^{2}f_{n}\big{)}=0,$
$j\neq n$ and
$\Delta_{n}\big{(}(1-\partial_{x})g_{n}(1+\partial_{x})g_{n}\partial_{x}^{2}f_{n}\big{)}=(1-\partial_{x})g_{n}(1+\partial_{x})g_{n}\partial_{x}^{2}f_{n}$
for $n\geq 5.$ Direct calculation shows that

$\displaystyle\big{|}\big{|}(1-\partial_{x})g_{n}(1+\partial_{x})g_{n}\partial_{x}^{2}f_{n}\big{|}\big{|}_{B^{s-1}_{p,r}}=2^{n(s-1)}\big{|}\big{|}(1-\partial_{x})g_{n}(1+\partial_{x})g_{n}\partial_{x}^{2}f_{n}\|_{L^{p}}$
$\displaystyle=$
$\displaystyle\|2^{-2n}(1-\partial_{x})\phi(1+\partial_{x})\phi\partial_{x}^{2}\phi\sin(\frac{17}{12}2^{n}x)+\frac{17}{12}2^{-n}(1-\partial_{x})\phi(1+\partial_{x})\phi\phi\cos(\frac{17}{12}2^{n}x)$
$\displaystyle+\frac{17}{12}2^{-n}(1-\partial_{x})\phi(1+\partial_{x})\phi\partial_{x}\phi\cos(\frac{17}{12}2^{n}x)+\frac{17}{12}(1-\partial_{x})\phi(1+\partial_{x})\phi\partial_{x}\phi\sin(\frac{17}{12}2^{n}x)\big{|}\big{|}_{L^{p}}$
$\displaystyle\geq$
$\displaystyle\frac{17}{12}\big{|}\big{|}\psi(x)\sin(\frac{17}{12}2^{n}x)\big{|}\big{|}_{L^{p}}-2^{-n}\rightarrow\frac{17}{12}\big{(}\frac{\int_{0}^{2\pi}|\sin
x|^{p}dx}{2\pi}\big{)}^{\frac{1}{p}}\big{|}\big{|}\psi(x)\big{|}\big{|}_{L^{p}},$

by the Riemann Theorem, where
$(1-\partial_{x})\phi(1+\partial_{x})\phi\partial_{x}\phi\triangleq\psi(x),$
which together with (4.4) yield

$\displaystyle\liminf_{n\rightarrow\infty}||\mathbf{S}_{t}(v^{1,n}(0,x))-\mathbf{S}_{t}(v^{2,n}(0,x))||_{B^{s-1}_{p,r}}\gtrsim
t\quad\text{for}\ t\ \text{small enough}.$

That is to say, the solution map $v_{0}\rightarrow S_{t}(v_{0})$ of the
initial value problem (3.5) depends not uniformly continuous on initial data
in $B_{p,r}^{s-1}$.

Since $u=(1-\partial_{x})^{-1}v$ and $(1-\partial_{x})^{-1}$ is an isomorphic
mapping from $B_{p,r}^{s-1}(\mathbb{R})$ into $B_{p,r}^{s}(\mathbb{R})$, hence
the non-uniform continuous dependence of $v$ in $B_{p,r}^{s-1}$ is consistent
with that of $u$ in $B_{p,r}^{s}.$

Thus, this completes the proof of Theorem 1.1.

## Acknowledgments

The author is very grateful to Dr. Jinlu Li for some useful suggestions. This
work is partially supported by the National Natural Science Foundation of
China (Grant No.11801090).

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