# Recovering Missing Node Features with Local Structure-based Embeddings

###### Abstract

Node features bolster graph-based learning when exploited jointly with network
structure. However, a lack of nodal attributes is prevalent in graph data. We
present a framework to _recover completely missing node features for a set of
graphs_ , where we only know the signals of a subset of graphs. Our approach
incorporates prior information from both graph topology and existing nodal
values. We demonstrate an example implementation of our framework where we
assume that node features depend on local graph structure. Missing nodal
values are estimated by aggregating known features from the most similar
nodes. Similarity is measured through a node embedding space that preserves
local topological features, which we train using a Graph AutoEncoder. We
empirically show not only the accuracy of our feature estimation approach but
also its value for downstream graph classification. Our success embarks on and
implies the need to emphasize the relationship between node features and graph
structure in graph-based learning.

Index Terms—  Graph Signal Processing, Local Structure Embeddings, Missing
Feature Generation

Fig. 1: Embeddings for nodes in graphs from the AIDS molecule dataset. Each
point is a node embedding based on local structural characteristics, such as
degree. Nodes corresponding to graphs of different classes are shifted in the
embedding space, implying that local structure is correlated with molecule
class for the AIDS dataset.

##  1 Introduction

Fig. 2: Schematic of the proposed methodology. First, we compute a feature
matrix ${\mathbf{F}}^{(i)}$ for each graph ${\mathcal{G}}^{(i)}$ based on
structural characteristics. Then, we train a GAE on ${\mathbf{F}}^{(i)}$ to
produce node embeddings ${\mathbf{Z}}^{(i)}$ and corresponding graph
embeddings. The latter are used to measure similarity among graphs, and find
the most similar graphs to ${\mathcal{G}}^{(i)}$, collected in
${\mathcal{Q}}^{(i)}$. The node embeddings are used to find, among the nodes
of the graphs in ${\mathcal{Q}}^{(i)}$, those nodes which are the closest to
each node $n$ from ${\mathcal{G}}^{(i)}$, collected in
${\mathcal{N}}^{(i,j)}_{n}$. Finally, we estimate node features in
${\mathcal{G}}^{(i)}$ by averaging the features of the closest graphs and
their nodes, resulting in realistic yet accurate node feature estimates.

For practical applications in chemistry [1, 2], medicine [3], and many others
[4], data can be naturally represented as interconnected entities using
graphs. Supervised learning on graphs aims to predict characteristics using
both graph structure and, in some cases, node features, also known as graph
signals. These features can improve graph-based predictions when jointly used
with graphs, not only when the nodal values are semantically relevant but also
when the observations on nodes and the graph structure are dependent [5].

The relationship between node features and their underlying graph is well-
studied for node-level tasks. These tasks typically entail predicting nodal
characteristics for a single graph where a subset of features is known.
Classifying nodes in the semi-supervised learning setting requires the
influence of the underlying graph on nodal values to propagate known
information to unlabeled nodes [6]. Graph signal reconstruction is a common
task in graph signal processing in which partially observed features and a
known graph signal model are applied together for downstream tasks, such as
approximating hidden node values [7, 8, 9]. In these cases, a portion of the
features is known, even if a minority, and the graph topology informs how
existing values provide information about those hidden.

The task of recovering completely missing graph signals for a given graph is
far less explored. As nodal values are critical for predictions and necessary
for the implementation of graph neural networks (GNNs), we require a method to
accurately estimate unknown node features [5, 10]. Unlike node-level
predictions for which we have partial nodal observations, we cannot use the
underlying graph structure to propagate existing information and infer missing
values [6, 11]. In such cases, the graph may belong to a family of graphs
whose structural and nodal characteristics are related. Many graph-level tasks
consist of such data, such as predicting molecular structures and identifying
characteristics of social networks [2, 4]. Existing works often characterize
graph families by shared random graph models [12] or latent embedding spaces
[6]. However, these works enforce a global relationship between the graphs and
their signals, requiring knowledge of the entire graph.

Even under the assumption of a shared graph family, the relationship between
each of the graphs and its node features is largely decoupled for graph-level
learning tasks. For example, methods that interpolate between labeled graphs
for improving classification typically treat graphs and node features
separately [13, 14, 15]. Approaches that do aim to replace missing graph
signals typically rely on values that possess solely topological features with
no additional nodal information [5, 16], and many use unrelated values that do
not incorporate structure [17, 18]. We empirically show that such approaches
are suboptimal, even when we include structural information.

Given a set of graphs where only a subset has known node features, we present
a framework to recover completely missing node features. In particular, we
propose _node feature recovery for graph-level tasks incorporating both graph
topology and known feature values_. We demonstrate an implementation of this
approach assuming that node features depend on local graph structural
characteristics, differently from previous approaches that largely rely on
aggregating information from a node’s neighborhood, for example via low-pass
graph filters [19, 20]. Thus, we train a Graph AutoEncoder (GAE) to learn a
node embedding space that preserves the local structural characteristics for
each node, visualized in Fig. 1. In this setting, feature values are assumed
to be closer when node neighborhoods are similar, so nodes with known features
can effectively provide the most realistic feature estimates for those with
similar local topologies.

Our contributions are as follows.

  * (i)

We present an approach to learn completely missing node features whose values
are assumed to be dependent on graph structure, and we exhibit the approach in
practice through the setting where features depend on local structure.

  * (ii)

We demonstrate that for many graph classification benchmark datasets, local
node structure is indeed indicative of class.

  * (iii)

We empirically validate the ability of our method to not only accurately learn
missing node features using a set of graphs with known features, but we also
demonstrate the value of recovering accurate node features for downstream
tasks.

| MUTAG | AIDS | PROTEINS | ENZYMES | ogbg-molbbbp | ogbg-molbace
---|---|---|---|---|---|---
Zeros | $1.00\pm 0.00$ | $1.00\pm 0.00$ | $1.00\pm 0.00$ | $1.00\pm 0.00$ | $1.00\pm 0.00$ | $1.00\pm 0.00$
Ones | $2.450\pm 0.00$ | $6.083\pm 0.00$ | $1.414\pm 0.00$ | $1.414\pm 0.00$ | $0.806\pm 0.00$ | $0.803\pm 0.00$
Random | $1.510\pm 0.004$ | $3.550\pm 0.003$ | $0.965\pm 0.002$ | $0.961\pm 0.003$ | $0.896\pm 0.000$ | $0.895\pm 0.000$
Degree | $0.938\pm 0.001$ | $1.414\pm 0.006$ | $0.899\pm 0.003$ | $0.891\pm 0.001$ | $0.975\pm 0.000$ | $0.984\pm 0.000$
LSE-NG ($\bar{Q}=1$) | $0.631\pm 0.040$ | $0.790\pm 0.005$ | $0.739\pm 0.010$ | $0.715\pm 0.005$ | $0.288\pm 0.002$ | $0.264\pm 0.002$
LSE-NG ($\bar{Q}=3$) | $0.607\pm 0.012$ | $0.746\pm 0.005$ | $0.719\pm 0.006$ | $0.716\pm 0.007$ | $0.251\pm 0.003$ | $0.228\pm 0.001$
LSE-NN ($\bar{Q}=1$) | $\mathbf{0.119\pm 0.017}$ | $\mathbf{0.555\pm 0.005}$ | $0.708\pm 0.010$ | $\mathbf{0.654\pm 0.013}$ | $0.177\pm 0.002$ | $\mathbf{0.113\pm 0.001}$
LSE-NN ($\bar{Q}=3$) | $0.128\pm 0.024$ | $0.562\pm 0.005$ | $\mathbf{0.696\pm 0.007}$ | $0.664\pm 0.011$ | $\mathbf{0.175\pm 0.004}$ | $0.126\pm 0.002$
Table 1: Normalized feature generation error
$\|{\mathbf{X}}-\hat{{\mathbf{X}}}\|_{F}^{2}/\|{\mathbf{X}}\|_{F}^{2}$. For
LSE-NG and LSE-NN, we predict missing node features for each graph from the
$\bar{Q}$ nearest graphs in the dataset with respect to the graph embeddings
belonging to the same class. The top performing methods are bolded.

##  2 Background

In this section, we provide the necessary background on graph-based learning,
along with a review of existing approaches on node representation learning and
addressing missing node features. We start with some basic notation. A graph
${\mathcal{G}}=\\{{\mathcal{V}},{\mathcal{E}}\\}$ comprises a set of nodes
${\mathcal{V}}=\\{1,\ldots,N\\}$ and a set of edges
${\mathcal{E}}=\\{(n_{1},n_{2})|n_{1},n_{2}\in{\mathcal{V}}\\}$. Graphs can be
conveniently represented by the so-called adjacency matrix ${\mathbf{A}}$. For
edges in ${\mathcal{G}}$, $(n_{1},n_{2})\in{\mathcal{E}}$ iff
$A_{n_{1},n_{2}}=1$. In machine learning and signal processing setups, data is
often associated with each of the nodes. In particular, let
${\mathbf{X}}\in{\mathbb{R}}^{N\times F}$ be a data matrix, whose entry
$X_{n,f}$ represents the value of feature (signal) $f$ at node $n$. The $n$th
row of ${\mathbf{X}}$ is typically referred to as the data features associated
with node $n$ and the $f$th column of ${\mathbf{X}}$ as the $f$th graph
signal. On top of these _node features_ , one can also associate a number of
_topological features_ , such as centrality values or clustering coefficients,
with each of the nodes [21].

Node representation learning. Learning node representations (embeddings) has
been a prevalent topic of research in the GSP literature almost since its
inception [22, 23]. Since the adjacency matrix ${\mathbf{A}}$ is an
alternative representation of ${\mathcal{G}}$, each node can be (perfectly)
represented in an $N$-dimensional vector space using the corresponding row of
${\mathbf{A}}$. As a result, node representation algorithms typically aim to
learn representations in a lower-dimensional space, that is, they aim at
learning a matrix ${\mathbf{Z}}\in{\mathbb{R}}^{N\times P}$ with $P<N$. The
ultimate goal when designing ${\mathbf{Z}}$ is to sufficiently characterize
nodal behavior in the context of the graph application at hand. Countless
approaches to learn nodal representations include algorithms from random walks
[24] to GNNs [6, 16].

Most of these approaches learn node embeddings based on node proximity: the
closer nodes $n$ and $n^{\prime}$ are in the graph, the more similar their
embeddings ${\mathbf{z}}_{n}$ and ${\mathbf{z}}_{n^{\prime}}$ are. Recent
works have begun to emphasize learning node embeddings based on the role of
each node in the graph, guided by its topological features [21, 5]. Under this
setting, we may transform structural similarities between nodes into geometric
relationships in the embedding space. Inspired by this concept, we draw on
such structure-based embeddings to identify which nodes are similar for
sharing feature values.

Missing node features. Previous works dealing with missing feature data
consider partially missing node features, where only some entries of the
feature matrix ${\mathbf{X}}$ are observed. In such formulations, the task,
known as feature imputation [25, 26, 27] or graph signal interpolation [7, 8,
9], is to learn the missing entries of ${\mathbf{X}}$. The full matrix
${\hat{\mathbf{X}}}$ (which contains now both the given and estimated values)
is then applied to a downstream task, usually for node-level tasks on a single
graph, such as node classification. Feature imputation has been approached by
graph spectral approaches [7], kernel approaches [9], propagating the known
features [25, 8] or using GNNs [26, 27]. Differently from these works, we aim
to solve a more difficult graph-level version of this problem, where for a
subset of (or all the) features, we do not have access to any of the nodes,
and we must infer the entire set of nodal values (either multiple columns of
${\mathbf{X}}$ or the full matrix itself) from data associated with other
graphs. As detailed in Sec. 3, this paper considers the case of having access
to a set of graphs, a subset of which have no observed node features (that is,
no access to any of the values of ${\mathbf{X}}$), which is common in social
networks, for example [10].

In the setting of completely missing node features, other works replace these
features with carefully crafted random matrices [17, 18], position-dependent
values [28, 29, 30], or structural properties, such as the degree [5, 16],
surprisingly showing that GNNs are still able to obtain great performance
without meaningful node features and only using the graph structure. However,
random and constant features are independent of class labels, and we
empirically demonstrate that using random or structural node features is
suboptimal. Moreover, our approach using a realistic estimate of node features
exhibits superior performance.

| MUTAG | AIDS | PROTEINS | ENZYMES | ogbg-molbbbp | ogbg-molbace
---|---|---|---|---|---|---
True Features |  83.25 $\pm$ 9.52 |  97.00 $\pm$ 1.49 |  72.74 $\pm$ 3.75 |  36.17 $\pm$ 4.97 |  85.53 $\pm$ 2.38 |  74.38 $\pm$ 3.49
Zeros |  78.75 $\pm$ 9.34 |  95.92 $\pm$ 1.87 |  70.09 $\pm$ 5.25 |  25.50 $\pm$ 7.15 |  77.15 $\pm$ 2.63 |  53.33 $\pm$ 5.25
Ones |  82.25 $\pm$ 9.55 |  94.45 $\pm$ 1.86 |  69.25 $\pm$ 5.26 |  24.08 $\pm$ 5.10 |  77.77 $\pm$ 3.26 |  53.55 $\pm$ 4.50
Random |  81.50 $\pm$ 8.23 |  94.30 $\pm$ 2.80 |  70.75 $\pm$ 4.66 |  22.92 $\pm$ 6.34 |  77.61 $\pm$ 3.46 |  53.73 $\pm$ 4.20
Degree |  81.00 $\pm$ 12.41 |  90.60 $\pm$ 4.88 |  69.65 $\pm$ 5.38 |  25.42 $\pm$ 6.32 |  79.71 $\pm$ 3.40 |  56.34 $\pm$ 4.77
Not using ${\mathcal{T}}_{\mathrm{miss}}$ | 84.75 $\pm$ 8.73 |  95.62 $\pm$ 1.56 |  71.28 $\pm$ 4.87 |  21.83 $\pm$ 6.75 |  83.27 $\pm$ 2.13 |  67.23 $\pm$ 5.27
LSE-NG ($\bar{Q}=1$) |  83.00 $\pm$ 7.48 |  95.75 $\pm$ 1.65 |  71.68 $\pm$ 3.76 |  24.08 $\pm$ 5.01 |  82.94 $\pm$ 1.90 |  69.02 $\pm$ 4.10
LSE-NG ($\bar{Q}=3$) |  81.50 $\pm$ 9.89 |  95.17 $\pm$ 1.70 | 72.79 $\pm$ 4.89 |  24.42 $\pm$ 5.30 |  83.72 $\pm$ 1.37 |  70.28 $\pm$ 2.94
LSE-NN ($\bar{Q}=1$) |  81.50 $\pm$ 9.63 | 96.10 $\pm$ 1.44 |  71.15 $\pm$ 4.57 |  23.83 $\pm$ 6.10 |  83.01 $\pm$ 1.54 | 71.63 $\pm$ 4.12
LSE-NN ($\bar{Q}=3$) |  78.25 $\pm$ 8.26 |  95.97 $\pm$ 1.63 |  70.53 $\pm$ 4.07 | 26.00 $\pm$ 6.13 | 84.08 $\pm$ 2.10 |  70.33 $\pm$ 3.57
Table 2: Accuracy in the test dataset obtained by the baselines and by the
approach presented in this work in the downstream task (graph classification).
The best performances (excluding those obtained using the true features) are
bolded.

##  3 Methodology

We introduce our proposed approach to estimate missing node features using
structural information and known node features, which is visualized in Fig. 2.
While the schematic in Fig. 2 illustrates the use of local structure for
sharing nodal values, other assumptions can easily be made to associate nodes
for feature learning.

Consider a graph dataset
${\mathcal{T}}=\\{({\mathcal{G}}^{(i)},{\mathbf{X}}^{(i)},y^{(i)})\\}_{i=1}^{T}$,
where for the $i$-th sample we have the graph ${\mathcal{G}}^{(i)}$ with
$N_{i}$ nodes, ${\mathbf{X}}^{(i)}\in\mathbb{R}^{N_{i}\times F}$ is a matrix
of node features of length $F$, and $y^{(i)}$ is the associated label. Let
${\mathcal{T}}_{\mathrm{miss}}\subset{\mathcal{T}}$ be a subset of
${\mathcal{T}}$ with missing features, where for every
$({\mathcal{G}},{\mathbf{X}},y)\in{\mathcal{T}}_{\mathrm{miss}}$, we only know
the duplex $({\mathcal{G}},y)$, and define
${\mathcal{T}}_{\mathrm{full}}={\mathcal{T}}\backslash{\mathcal{T}}_{\mathrm{miss}}$.
Our focus in this work is to recover the missing features ${\mathbf{X}}$ for
every $({\mathcal{G}},{\mathbf{X}},y)\in{\mathcal{T}}_{\mathrm{miss}}$,
resulting in a set $\hat{{\mathcal{T}}}_{\mathrm{miss}}$ consisting of
triplets $({\mathcal{G}},\hat{{\mathbf{X}}},y)$ with approximated features
$\hat{{\mathbf{X}}}$. Subsequently, we may use
$\hat{{\mathcal{T}}}={\mathcal{T}}_{\mathrm{full}}\cup\hat{{\mathcal{T}}}_{\mathrm{miss}}$
for downstream tasks such as graph classification.

Our approach consists of two steps: we first learn a node embedding space
preserving graph structural information through which we compute node
similarity, and then we predict the values of missing node features using
nearby node embeddings. While we select local structural characteristics as
the topological features of interest, note that our framework is amenable to
any choice of embedding space that allows us to relate nodes based on
similarity.

Node embedding space. We first obtain a latent space with which we compute
node similarity. We train a GAE to generate node embeddings characterized by
node roles, that is, their local structure [21]. More precisely, the GAE
consists of a graph convolutional network (GCN) as the encoder to learn from
structural characteristics, both global and local, and a multilayer perceptron
(MLP) as the decoder to invert the embedding process. For a graph
${\mathcal{G}}^{(i)}$, the GCN encoder takes as input ${\mathcal{G}}^{(i)}$
and a corresponding feature matrix
${\mathbf{F}}^{(i)}\in\mathbb{R}^{N_{i}\times F}$ containing structural
information from $F$ features. We apply the features in [21], including local
characteristics such as node degree and clustering coefficient, although any
features may be used to emphasize different structural behavior. The
parameters $\boldsymbol{\Theta}$ of the GAE $f_{\boldsymbol{\Theta}}$ are
trained to minimize the loss between the output of the GAE and the input
feature matrix

$\min_{\boldsymbol{\Theta}}\|{\mathbf{F}}^{(i)}-f_{\boldsymbol{\Theta}}({\mathbf{F}}^{(i)},{\mathcal{G}}^{(i)})\|_{F}^{2},$

where $f_{\boldsymbol{\Theta}}$ represents the GAE, whose output is computed
as
$f_{\boldsymbol{\Theta}}({\mathbf{F}},{\mathcal{G}}^{(i)})=\mathrm{MLP}_{\boldsymbol{\Theta}_{2}}(\mathrm{GNN}_{\boldsymbol{\Theta}_{1}}({\mathbf{F}},{\mathcal{G}}^{(i)}))$,
where
$\boldsymbol{\Theta}=\\{\boldsymbol{\Theta}_{1},\boldsymbol{\Theta}_{2}\\}$
and the GNN is defined via the following recursion [6]

${\mathbf{H}}^{(\ell)}=\sigma\left({\tilde{\mathbf{A}}}{\mathbf{H}}^{(\ell-1)}\boldsymbol{\Theta}^{(\ell)}\right),$
(1)

where ${\mathbf{H}}^{(\ell)}$ are the hidden features at layer $\ell$;
$\boldsymbol{\Theta}_{1}=\\{\boldsymbol{\Theta}^{(\ell)}\\}_{\ell=1}^{L}$ are
the learnable parameters of the $L$ layers; and $\sigma$ is a pointwise non-
linearity. We let ${\mathbf{A}}$ denote the adjacency matrix of the input
graph ${\mathcal{G}}^{(i)}$, and we define
$\hat{{\mathbf{A}}}={\mathbf{A}}+{\mathbf{I}}$,
$\hat{{\mathbf{D}}}=\mathrm{diag}(\hat{{\mathbf{A}}}{\bf 1})$, and
${\tilde{\mathbf{A}}}={\hat{\mathbf{D}}}^{-1/2}{\hat{\mathbf{A}}}{\hat{\mathbf{D}}}^{-1/2}$.
Note that the matrix multiplication in (1) can be understood as a low-pass
graph filtering [19], where the nodes average their own value with the values
of their neighbors. As a result, the embeddings based on (1) will promote
similar representations for nodes whose local structural features are similar.
A visualization of the resultant node embeddings from graphs in the molecular
classification dataset AIDS [31] is shown in Fig. 1.

Given the GAE, we obtain node embeddings for graph ${\mathcal{G}}^{(i)}$ as
${\mathbf{Z}}^{(i)}=\mathrm{GNN}_{\boldsymbol{\Theta}_{1}}({\mathbf{F}}^{(i)},{\mathcal{G}}^{(i)})\in{\mathbb{R}}^{N_{i}\times
P}$, where $P$ is the dimension of the embedding space. We further define
graph embeddings ${\mathbf{z}}^{(i)}\in{\mathbb{R}}^{P}$ by computing the
average across the node dimension, that is,
${\mathbf{z}}^{(i)}=\frac{1}{N_{i}}\sum_{k=1}^{N_{i}}{\mathbf{z}}^{(i)}_{k}\in{\mathbb{R}}^{P}$
where ${\mathbf{z}}^{(i)}_{k}\in{\mathbb{R}}^{P}$ is $k$th row of
${\mathbf{Z}}^{(i)}$.

With the node and graph embeddings for every graph in ${\mathcal{T}}$, we
associate ${\mathcal{G}}^{(i)}\in{\mathcal{T}}_{\mathrm{miss}}$ with nearby
graphs and nodes with respect to the embedding space. Let
${\mathcal{Q}}^{(i)}\subset{\mathcal{T}}_{\mathrm{full}}$ be the set of the
$\bar{Q}$ nearest graphs to ${\mathcal{G}}^{(i)}$ with the same label, that
is, for every ${\mathcal{G}}^{(j)}\in{\mathcal{Q}}^{(i)}$, we have that
$y^{(i)}=y^{(j)}$. More specifically, all the graphs in ${\mathcal{Q}}^{(i)}$
are closer to ${\mathcal{G}}^{(i)}$ than those with the same label but not in
${\mathcal{Q}}^{(i)}$ using as distance the error between their embeddings
$\|{\mathbf{z}}^{(i)}-{\mathbf{z}}^{(j)}\|_{2}$. For nearby nodes, we
similarly let ${\mathcal{N}}^{(i,j)}_{n}$ be the set containing the $\bar{N}$
nodes of graph ${\mathcal{G}}^{(j)}\in{\mathcal{Q}}^{(i)}$ closest to node $n$
of graph ${\mathcal{G}}^{(i)}\in{\mathcal{T}}_{\mathrm{miss}}$. That is, all
of the $\bar{N}$ nodes from graph ${\mathcal{G}}^{(j)}$ in
${\mathcal{N}}^{(i,j)}_{n}$ are closer (in terms of the same distance
previously defined) to node $n$ in graph ${\mathcal{G}}^{(i)}$ than those not
in ${\mathcal{N}}^{(i,j)}_{n}$.

Predicting missing graph signals. Once we are able to compute node similarity,
we predict the missing node features using the nearest graphs and nodes. We
propose to generate the features ${\hat{\mathbf{X}}}^{(i)}$ of
${\mathcal{G}}^{(i)}\in{\mathcal{T}}_{\mathrm{miss}}$ as the average of the
features in the closest nodes and graphs to ${\mathcal{G}}^{(i)}$ in
${\mathcal{T}}_{\mathrm{full}}$ with respect to their embeddings. Let
${\mathbf{C}}^{(i,j)}\in{\mathbb{R}}^{N_{i}\times N_{j}}$ be the
transformation matrix mapping the features from
${\mathcal{G}}^{(j)}\in{\mathcal{Q}}^{(i)}$ to ${\mathcal{G}}^{(i)}$, where
the entry at the $n$th row and $\ell$th column is

$C^{(i,j)}_{n,\ell}=\left\\{\begin{array}[]{lc}\frac{1}{|{\mathcal{N}}_{n}^{(i,j)}|}&n\in\\{1,...,N_{i}\\},\ell\in{\mathcal{N}}_{n}^{(i,j)}\\\
0&\mathrm{otherwise}\end{array}\right.$ (2)

For each node $n$ in ${\mathcal{G}}^{(i)}$, the product
${\mathbf{C}}^{(i,j)}{\mathbf{X}}^{(j)}$ computes the average of the features
from the nodes in ${\mathcal{N}}_{n}^{(i,j)}$, that is, the closest nodes to
$n$ from ${\mathcal{G}}^{(j)}$. Given the set of closest graphs
${\mathcal{Q}}^{(i)}$ to ${\mathcal{G}}^{(i)}$, we compute our node feature
estimates for ${\mathcal{G}}^{(i)}$ as

${\hat{\mathbf{X}}}^{(i)}=\frac{1}{\bar{Q}}\sum_{j\in{\mathcal{Q}}^{(i)}}{\mathbf{C}}^{(i,j)}{\mathbf{X}}^{(j)}.$
(3)

These features complete the triplet
$({\mathcal{G}}^{(i)},\hat{{\mathbf{X}}}^{(i)},y^{(i)})$ for every estimate in
$\hat{{\mathcal{T}}}_{\mathrm{miss}}$, allowing us to use these graphs for
downstream tasks such as graph classification.

##  4 Results

We showcase the capabilities of our proposed approach for missing feature
generation. We demonstrate our method in comparison with several baselines in
numerical experiments, both for node feature learning and downstream graph
classification.

Datasets. We use six real-world data benchmarks: MUTAG, AIDS, PROTEINS and
ENZYMES from the TUDataset collection [31], and ogbg-molbace and ogbg-molbppp
from the OGBG collection [32]. Not only are these standard benchmark datasets
for classification tasks, they also provide node features, which allows us to
test the hypothesis that node features are relevant for the classification
task as well as evaluate our proposed approach. The datasets contain either
graphs representing molecules (MUTAG, AIDS, ogbg-molbace and ogbg-molbppp),
where nodes represent atoms and edges represent chemical bonds, or proteins
(PROTEINS) and enzymes (ENZYMES), where nodes represent structural elements
and edges encode node proximity.

Experimental setup. We split the dataset with 10% of the data for validation;
10% for testing; 30% for training, or ${\mathcal{T}}_{\mathrm{full}}$; and 50%
as missing, that is, the set of graphs with missing node features
${\mathcal{T}}_{\mathrm{miss}}$. We conduct 15 random realizations of these
splits, and the results presented in Tables 1 and 2 list the mean and standard
deviation of the metric of interest (to be defined next) across every
realization.

We demonstrate the efficacy of our method on both feature generation and
downstream graph classification. Our approach that uses the local structure-
based node embeddings of nearest neighbors, denoted “LSE-NN”, is compared to
several baselines. Alternatives to “LSE-NN” include classical approaches:
“Degree” denoting node degree, “Ones” for features of all ones, “Zeros” for
features of all zeros, “Random” with features sampled uniformly at random on
$[0,1]$.

Our method “LSE-NN” exploits not only similar graphs for estimating missing
node features but also nodes with similar local structures. We highlight the
benefits of such an approach by also comparing it to a modification denoted
“LSE-NG”. For this variant, we obtain the nearest graphs ${\mathcal{Q}}^{(i)}$
for ${\mathcal{G}}^{(i)}$ as for “LSE-NN”, but we then assign feature values
to each node $k$ in ${\mathcal{G}}^{(i)}$ by taking node features uniformly at
random from the nodes of graphs ${\mathcal{G}}^{(j)}\in{\mathcal{Q}}^{(i)}$.
This is equivalent to replacing ${\mathbf{C}}^{(i,j)}$ in (3) with a random
permutation matrix ${\mathbf{P}}^{(i,j)}\in\\{0,1\\}^{N_{i}\times N_{j}}$.
Thus, “LSE-NG” predicts node features using similar graphs but does not align
nodes by local structure.

Feature generation performance. We first compare the ability of each method to
recover the original node features. The results presented in Table 1 show the
node feature estimation error as
$\|\hat{{\mathbf{X}}}-{\mathbf{X}}\|_{F}^{2}/\|{\mathbf{X}}\|_{F}^{2}$, where
$\hat{{\mathbf{X}}}$ denotes the estimated features and ${\mathbf{X}}$ the
true ones. We see that our architecture consistently beats the alternatives,
achieving a lower error in every dataset considered, in some cases by a large
margin. This shows that the true node features, which are assumed to be the
optimal features for downstream tasks, are best recovered by our architecture,
without the need for partial observations on the graphs with missing features.

Graph classification performance. We also assess the utility of the estimated
features for graph classification. The results are shown in Table 2, where we
present label prediction accuracy using a GNN model trained with the estimated
features. We choose the Graph Isomorphism Network (GIN) [16] as our GNN
architecture, a standard model for graph classification. For this task, we
also add an additional baseline “Not using ${\mathcal{T}}_{\mathrm{miss}}$”,
where we train the GIN only on the subset of graphs
${\mathcal{T}}_{\mathrm{full}}$ with known node features, ignoring the graphs
in ${\mathcal{T}}_{\mathrm{miss}}$ with missing features. In all cases but the
MUTAG dataset, the best performance is achieved using the learned GAE
embeddings, either from randomly copying nodes from the nearest graphs “LSE-
NG”, which enjoys the best performance on PROTEINS dataset, or by using the
nearest nodes “LSE-NN”, which obtains superior performance for all other
datasets. Thus, not only do we infer missing node features accurately, but the
estimates are sufficiently realistic to bolster classification performance
when we do not observe the node features of many graphs.

##  5 Conclusion

In this work, we proposed a framework to recover completely missing node
features for a set of graphs. We implemented this framework for estimating
features that are characterized primarily by local graph structure. To this
end, we presented a node embedding space using only local topological
features. The embedding space provided a node similarity metric with which we
estimated missing node features using similar nodes from nearby graphs. Our
estimates aid graph classification when features are missing, emphasizing the
need for accurate nodal characteristics. In the future, we will generalize to
applications such as graph data augmentation, where we can generate synthetic
graphs with realistic node features. Our work connecting node features and
graph structure can bolster the success of graph-based learning by exploiting
not only structural information but also values explicitly embedded therein.

## References

  * [1] J. Gilmer, S. S. Schoenholz, P. F. Riley, O. Vinyals, and G. E. Dahl, “Neural message passing for quantum chemistry,” in Intl. Conf. on Machine Learning (ICML), vol. 70, pp. 1263–1272, PMLR, 2017.
  * [2] D. K. Duvenaud, D. Maclaurin, J. Iparraguirre, R. Bombarell, T. Hirzel, A. Aspuru-Guzik, and R. P. Adams, “Convolutional networks on graphs for learning molecular fingerprints,” in Advances in Neural Info. Process. Syst., vol. 28, 2015.
  * [3] S. I. Ktena, S. Parisot, E. Ferrante, M. Rajchl, M. Lee, B. Glocker, and D. Rueckert, “Distance metric learning using graph convolutional networks: Application to functional brain networks,” in Medical Intl. Joint Conf. on Artif. Intell. and Computer Assisted Intervention (MICCAI), vol. 10433, pp. 469–477, 2017.
  * [4] J. Kim and M. Hastak, “Social network analysis: Characteristics of online social networks after a disaster,” Intl. J. of Info. Management, vol. 38, no. 1, pp. 86–96, 2018.
  * [5] H. Cui, Z. Lu, P. Li, and C. Yang, “On positional and structural node features for graph neural networks on non-attributed graphs,” in ACM Inter. Conf. on Info. & Knowledge Man., CIKM, p. 3898–3902, 2022.
  * [6] T. N. Kipf and M. Welling, “Semi-supervised classification with graph convolutional networks,” in Intl. Conf. on Learning Representations (ICLR), 2017.
  * [7] S. Chen, R. Varma, A. Sandryhaila, and J. Kovačević, “Discrete signal processing on graphs: Sampling theory,” IEEE Trans. Signal Process., vol. 63, no. 24, pp. 6510–6523, 2015.
  * [8] D. Ramirez, A. G. Marques, and S. Segarra, “Graph-signal reconstruction and blind deconvolution for structured inputs,” Signal Process., vol. 188, p. 108180, 2021.
  * [9] D. Romero, M. Ma, and G. B. Giannakis, “Kernel-based reconstruction of graph signals,” IEEE Trans. Signal Process., vol. 65, pp. 764–778, 2016.
  * [10] C. Cai and Y. Wang, “A simple yet effective baseline for non-attributed graph classification,” arXiv preprint arXiv:1811.03508, 2018.
  * [11] X. Chen, S. Chen, J. Yao, H. Zheng, Y. Zhang, and I. W. Tsang, “Learning on attribute-missing graphs,” IEEE Trans. Pattern Analysis and Machine Intell., vol. 44, no. 2, pp. 740–757, 2022.
  * [12] M. Navarro and S. Segarra, “Joint network topology inference via a shared graphon model,” IEEE Trans. Signal Process., vol. 70, pp. 5549–5563, 2022\.
  * [13] X. Han, Z. Jiang, N. Liu, and X. Hu, “G-Mixup: Graph data augmentation for graph classification,” in Intl. Conf. on Machine Learning (ICML), vol. 162, pp. 8230–8248, PMLR, 2022.
  * [14] X. Ma, X. Chu, Y. Wang, Y. Lin, J. Zhao, L. Ma, and W. Zhu, “Graph interpolation via fast Fused-Gromovization,” arXiv preprint arXiv:2306.15963, 2023.
  * [15] M. Navarro and S. Segarra, “GraphMAD: Graph mixup for data augmentation using data-driven convex clustering,” in IEEE Intl. Conf. Acoust., Speech and Signal Process. (ICASSP), pp. 1–5, 2023.
  * [16] K. Xu, W. Hu, J. Leskovec, and S. Jegelka, “How powerful are graph neural networks?,” in Intl. Conf. on Learning Representations (ICLR), 2019.
  * [17] R. Abboud, İ. İ. Ceylan, M. Grohe, and T. Lukasiewicz, “The surprising power of graph neural networks with random node initialization,” in Intl. Joint Conf. on Artif. Intell., 2021.
  * [18] R. Sato, M. Yamada, and H. Kashima, “Random features strengthen graph neural networks,” in SIAM Intl. Conf. on Data Mining (SDM), 2021.
  * [19] E. Isufi, F. Gama, D. I. Shuman, and S. Segarra, “Graph filters for signal processing and machine learning on graphs,” arXiv preprint arXiv:2211.08854, 2022.
  * [20] H. Liu, A. Scaglione, and H.-T. Wai, “Blind graph matching using graph signals,” arXiv preprint arXiv:2306.15747, 2023.
  * [21] X. Guo, W. Zhang, W. Wang, Y. Yu, Y. Wang, and P. Jiao, “Role-oriented graph auto-encoder guided by structural information,” in Database Syst. for Advanced Applications, pp. 466–481, 2020.
  * [22] H. Cai, V. W. Zheng, and K. C.-C. Chang, “A comprehensive survey of graph embedding: Problems, techniques, and applications,” IEEE Trans. Knowledge and Data Engineering, vol. 30, pp. 1616–1637, 2017.
  * [23] P. Goyal and E. Ferrara, “Graph embedding techniques, applications, and performance: A survey,” Knowledge-Based Systems, vol. 151, pp. 78–94, 2018\.
  * [24] B. Perozzi, R. Al-Rfou, and S. Skiena, “DeepWalk: Online learning of social representations,” in Intl. Conf. on Knowledge Discovery and Data Mining (SIGKDD), pp. 701–710, ACM, 2014.
  * [25] E. Rossi, H. Kenlay, M. I. Gorinova, B. Chamberlain, X. Dong, and M. M. Bronstein, “On the unreasonable effectiveness of feature propagation in learning on graphs with missing node features,” arXiv preprint arXiv:2111.12128, 2021.
  * [26] J. You, X. Ma, D. Ding, M. Kochenderfer, and J. Leskovec, “Handling missing data with graph representation learning,” Advances in Neural Info. Process. Syst., 2020.
  * [27] I. Spinelli, S. Scardapane, and A. Uncini, “Missing data imputation with adversarially-trained graph convolutional networks,” Neural Netw., vol. 129, pp. 249–260, 2020.
  * [28] J. You, R. Ying, and J. Leskovec, “Position-aware graph neural networks,” in Intl. Conf. on Machine Learning (ICML), vol. 97, pp. 7134–7143, PMLR, 2019\.
  * [29] P. Li, Y. Wang, H. Wang, and J. Leskovec, “Distance encoding: Design provably more powerful neural networks for graph representation learning,” in Advances in Neural Info. Process. Syst., vol. 33, pp. 4465–4478, 2020.
  * [30] J. You, J. M. Gomes-Selman, R. Ying, and J. Leskovec, “Identity-aware graph neural networks,” AAAI Conf. on Artif. Intell., vol. 35, no. 12, pp. 10737–10745, 2021.
  * [31] C. Morris, N. M. Kriege, F. Bause, K. Kersting, P. Mutzel, and M. Neumann, “TUDataset: A collection of benchmark datasets for learning with graphs,” in Intl. Conf. on Machine Learning (ICML), 2020.
  * [32] W. Hu, M. Fey, M. Zitnik, Y. Dong, H. Ren, B. Liu, M. Catasta, and J. Leskovec, “Open Graph Benchmark: Datasets for Machine Learning on Graphs,” arXiv preprint arXiv:2005.00687, vol. abs/2005.00687, 2020.


# On the stability of the Kawahara equation with a distributed infinite memory

Roberto de A. Capistrano–Filho  Departamento de Matemática, Universidade
Federal de Pernambuco (UFPE), 50740-545, Recife-PE, Brazil.
<EMAIL_ADDRESS>,  Boumediène Chentouf  Kuwait University,
Faculty of Science, Department of Mathematics, Safat 13060, Kuwait
<EMAIL_ADDRESS>and  Isadora Maria de Jesus  Instituto de
Matemática, Universidade Federal de Alagoas (UFAL), Maceió-AL and Departamento
de Matemática, Universidade Federal de Pernambuco (UFPE), 50740-545, Recife-
PE, Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This article will deal with the stabilization problem for the higher-order
dispersive system, commonly called the Kawahara equation. To do so, we
introduce a damping mechanism via a distributed memory term in the equation to
prove that the solutions of the Kawahara equation are exponentially stable,
provided that specific assumptions on the memory kernel are fulfilled. This is
possible thanks to the energy method that permits to provide a decay estimate
of the system energy.

###### Key words and phrases:

Kawahara equation, Infinite memory, Well-posedness, Stability, Energy method

###### 2020 Mathematics Subject Classification:

Primary: 35Q53, 93D20, 93D15 Secondary: 34K25

*Corresponding author.

##  1\. Introduction

###  1.1. Model under consideration and objective

The fifth order nonlinear dispersive equation

(1.1) $\pm
2\partial_{t}u+3u\partial_{x}u-\nu\partial^{3}_{x}u+\frac{1}{45}\partial^{5}_{x}u=0,$

models numerous physical phenomena. Considering suitable assumptions on the
amplitude, wavelength, wave steepness, and so on, the properties of the
asymptotic models for water waves have been extensively studied in the last
years, through (1.1), to understand the full water wave system. For a rigorous
justification of various asymptotic surface and internal waves models, we
suggest the reader consult the following references [1, 4, 24].

On the other hand, we can formulate the waves as a free boundary problem of
the incompressible, irrotational Euler equation in an appropriate non-
dimensional form with at least two parameters $\delta:=\frac{h}{\lambda}$ and
$\varepsilon:=\frac{a}{h}$, non-dimensional, where the water depth, the
wavelength and the amplitude of the free surface are parameterized as
$h,\lambda$ and $a$, respectively. In turn, if we introduce another non-
dimensional parameter $\mu$, so-called the Bond number, which measures the
importance of gravitational forces compared to surface tension forces, then
the physical condition $\delta\ll 1$ characterizes the waves, which are called
long waves or shallow water waves. On the other hand, there are several long-
wave approximations depending on the relations between $\varepsilon$ and
$\delta$. For instance, if we consider $\varepsilon=\delta^{4}\ll 1$ and
$\mu=\frac{1}{3}+\nu\varepsilon^{\frac{1}{2}},$ and in connection with the
critical Bond number $\mu=\frac{1}{3}$, we have the so-called Kawahara
equation, represented by (1.1), and derived by Hasimoto and Kawahara in [20,
23].

The main concern of this article is to deal with the well-posedness and
stability of an initial-boundary-value problem related to (1.1). Specifically,
we are concerned with a fifth-order dispersive partial differential equation
with a memory term

(1.2)
$\begin{cases}\partial_{t}u(x,t)+\partial_{x}^{3}u(x,t)-a_{0}\partial_{x}^{5}u(x,t)+u(x,t)\partial_{x}u(x,t)+a_{1}\partial_{x}u(x,t)\\\
\quad\quad+(-1)^{k}\displaystyle\int_{0}^{\infty}f(s)\partial_{x}^{2k}u(x,t-s)ds=0,&\
(x,t)\in I\times(0,\infty),\\\
u(0,t)=u(L,t)=\partial_{x}u(0,t)=\partial_{x}u(L,t)=\partial_{x}^{2}u(L,t)=0,&t>0,\\\
u(x,-t)=u_{0}(x,t),&x\in I,\ t{\geq}0.\end{cases}$

Here $u$ represents the amplitude of the dispersive wave, $k\in\\{0,1,2\\}$,
$L>0$, $I=(0,L)$, while $a_{1}\in\mathbb{R}$ and $a_{0}>0$ are physical
parameter of the dispersive equation. Moreover, $u_{0}$ is the initial
condition and $f$ is the memory kernel satisfying
$f:\mathbb{R}_{+}:=[0,\infty)\to\mathbb{R}$ so as there exists a positive
constant $c_{0}$ such that:

(1.3) $f\in C^{2}(\mathbb{R}_{+}),\quad f^{\prime}<0,\quad 0\leq
f^{\prime\prime}\leq-c_{0}f^{\prime},\quad
f(0)>0\quad\hbox{and}\quad\lim_{s\to\infty}f(s)=0.$

Thereafter, the energy associated with the system (1.2) is

(1.4)
$E(t)=\dfrac{1}{2}\left(\|u(t)\|^{2}+\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}(\cdot,s)\|^{2}ds\right),\
t\in\mathbb{R}_{+}.$

Observe that $E^{\prime}<0$ and hence the energy of our system is decreasing
(see Lemma 2.1). This means that the localized damping mechanism and the
memory term constitute a damping mechanism and consequently one has to study
the decay of the solutions of (1.2). Notwithstanding, it has been noticed that
the stability property of solutions of numerous physical systems may be lost
when a memory effect occurs [26]. Thus, our concern is to provide an answer to
the following questions:

Does the energy $E(t)$ decay to 0 as $t$ is sufficiently large? If so, can we
provide a decay rate?

###  1.2. Historical background

Let us first present a review of the main results available in the literature
for the analysis of the Kawahara equation in a bounded interval. A pioneer
work is due to Silva and Vasconcellos [29, 30], where the authors studied the
stabilization of global solutions of the linear Kawahara equation in a bounded
interval under the effect of a localized damping mechanism. The second
endeavor, in this line, was completed by Capistrano-Filho et al. [3], where
the generalized Kawahara equation in a bounded domain $Q_{T}=(0,T)\times(0,L)$
considered the following system

(1.5)
$\left\\{\begin{array}[]{llr}\partial_{t}u+\partial_{x}u+\partial^{3}_{x}u-\partial^{5}_{x}u+u^{p}\partial_{x}u+a(x)u=0,&\mbox{in}\
Q_{T},\\\
u(t,0)=u(t,L)=\partial_{x}u(t,0)=\partial_{x}u(t,L)=\partial^{2}_{x}u(t,L)=0,&\mbox{on}\
[0,T],\\\ u(0,x)=u_{0}(x),&\mbox{in}\ [0,L],\end{array}\right.$

with $p\in[1,4)$ and $a(x)$ is a nonnegative function and positive only on an
open subset of $(0,L)$. It is proven that the solutions of the above system
decay exponentially.

The internal controllability problem has been tackled by Chen [9] for the
Kawahara equation with homogeneous boundary conditions. Using Carleman
estimates associated with the linear operator of the Kawahara equation with an
internal observation, a null controllable result was shown when the internal
control is effective in a subdomain $\omega\subset(0,L)$. In [6], considering
the system (1.5) with an internal control $f(t,x)$ and homogeneous boundary
conditions, the equation is exactly shown to be controllable in
$L^{2}$-weighted Sobolev spaces and, additionally, controllable by regions in
$L^{2}$-Sobolev space.

Recently, a new tool for the control properties of the Kawahara operator was
proposed. In [8], the authors showed a new type of controllability for the
Kawahara equation, what they called overdetermination control problem. A
boundary control was designed so that the solution to the problem under
consideration satisfies an integral condition.

The last works on the stabilization of the Kawahara equation with a localized
time-delayed interior control. In [7, 10], under suitable assumptions on the
time delay coefficients, the authors were able to prove that solutions of the
Kawahara system are exponentially stable. The results were obtained using the
Lyapunov approach and a compactness-uniqueness argument. More recently, the
authors in [5] gave an analysis to better understand the stabilization issue
for the Kawahara equation. Indeed, it is shown that the Kawahara equation
under the action of a time-delayed boundary control system remains
exponentially stable under a condition on the length of the spatial domain.
Such a desirable property is proved using two different approaches. It is also
worth mentioning that the stability of the solutions to the Kawahara equation
has been extensively studied in the context of periodic or non-periodic
bounded domain [17, 18, 19, 22] and also in the case when the spacial variable
lies in $(-\infty,\infty)$ or $[0,\infty)$ [12, 13, 14, 16, 21, 25].

We end the literature review by mentioning that the occurrence of a memory
phenomenon in the Kawahara problem (1.2) could be explained in practice by the
fact that numerous compressible and incompressible fluids are intrinsically
viscoelastic and therefore the influence of the past values of the amplitude
of the dispersive wave of the fluid is unavoidable [2, 11, 15, 27].

Regarding the main contribution of this paper, we can claim that we can go one
step further in the study of the stabilization problem for the fifth-order
Korteweg-de-Vries type system. Compared to the recent works [3, 5, 7, 10],
where damping mechanisms and delay controls are used, this paper closes the
gap since it is the first work to treat exponential stability using only
infinite memory. It is also noteworthy that the current work shows that a
memory term plays a role of a damping control in the sense that it leads to
the stability of the system without any additional damping such as $a(x)u$
used in [3, 10, 31] to get the stability property of the system. Finally, note
that our results remain valid if $a_{1}=0$ and hence the drift term
$\partial_{x}u(x,t)$ can be omitted.

###  1.3. Notations and main result

Throughout this article, $C$ denotes a constant that can be different from one
step to another in the demonstrations presented here. Let us use $\langle\ ,\
\rangle$ and $\|\cdot\|$ to denote the standard real inner product in
$L^{2}(I)$ and its corresponding norm given by

$\langle u,v\rangle=\int_{0}^{L}u(x)v(x)dx\quad\mbox{\ and \
}\quad\|u\|=\left(\int_{0}^{L}|u(x)|^{2}dx\right)^{\frac{1}{2}}.$

As in [15], we use the history approach by introducing the following variable
$\eta^{t}$ and its initial data $\eta^{0}$ defined by

(1.6) $\eta^{t}(x,s)=\int_{t-s}^{t}u(x,\tau)d\tau\mbox{\ and \
}\eta^{0}(x,s)=\int_{0}^{s}u_{0}(x,\tau)d\tau,\ x\in I,s,t\in\mathbb{R}_{+}.$

Direct and formal computations show that the functional $\eta^{t}$ satisfies

(1.7)
$\begin{cases}\partial_{t}\eta^{t}(x,s)+\partial_{s}\eta^{t}(x,s)=u(x,t),&x\in
I,\ s,t\in\mathbb{R}_{+},\\\
\eta^{t}(0,s)=\eta^{t}(L,s)=0,&s,t\in\mathbb{R}_{+},\\\ \eta^{t}(x,0)=0,&x\in
I,\ t\in\mathbb{R}_{+}.\\\ \end{cases}$

In order to express the memory term in (1.2) in terms of $\eta^{t},$ pick
$g:=-f^{\prime}.$ Thus, according to (1.3), we get

(1.8) $g\in C^{1}(\mathbb{R}_{+}),\ g>0,\ 0\leq-g^{\prime}\leq c_{0}g,\
g_{0}=\int_{0}^{\infty}g(s)ds=f(0)>0\mbox{ and
}\lim_{s\rightarrow\infty}g(s)=0.$

On the other hand, integrating by parts with respect to $s$ and using that
$\eta^{t}(x,0)=0$ and the limit (1.3), we have that

(1.9)
${\displaystyle\int_{0}^{\infty}g(s)\partial_{x}^{2k}\eta^{t}(x,s)ds=\int_{0}^{\infty}f(s)\partial_{x}^{2k}u(x,t-s)ds.}$

Note that, with (1.9) in hands, we rewrite (1.2) as follows

(1.10)
${\displaystyle\partial_{t}u+\partial_{x}^{3}u-a_{0}\partial_{x}^{5}u+u\partial_{x}u+a_{1}\partial_{x}u+a(x)u+(-1)^{k}\int_{0}^{\infty}g(s)\partial_{x}^{2k}\eta^{t}(x,s)ds=0.}$

Thereafter, we introduce a variable $U$ and its initial data $U_{0}$ defined
by

(1.11) $U=(u,\eta^{t})^{T}\quad\mbox{ and }\quad
U_{0}(x,s)=(u_{0}(x),\eta^{0}(x,s))^{T}$

with

(1.12) $u\in L^{2}(I)\mbox{\ \ and \ \ }\eta^{t}\in
L_{g}:=\left\\{v:\mathbb{R}_{+}\longrightarrow
H_{k};\int_{0}^{\infty}g(s)\|\partial_{x}^{k}v(s)\|^{2}ds<+\infty\right\\},$

where the space $H_{k}$ is defined as

$H_{k}=\begin{cases}L^{2}(I),\;\;\mbox{ if }k=0,\\\ H_{0}^{1}(I),\;\;\mbox{ if
}k=1,\\\ H_{0}^{2}(I),\;\;\mbox{ if }k=2.\end{cases}$

Furthermore, we will consider in the set $L_{g}$, defined above, the inner
product and norm are given by

(1.13) $\langle
v,w\rangle_{L_{g}}=\int_{0}^{\infty}g(s)\langle\partial_{x}^{k}v(s),\partial_{x}^{k}w(s)\rangle
ds\mbox{ \ \ and\ \
}\|v(s)\|_{L_{g}}=\left(\int_{0}^{\infty}g(s)\|\partial_{x}^{k}v(s)\|^{2}ds\right)^{\frac{1}{2}},$

respectively and we define the energy space as $\mathcal{H}=L^{2}(I)\times
L_{g},$ which will be equipped with the following inner product and its
corresponding norm

$\langle(v_{1},v_{2}),(w_{1},w_{2})\rangle_{\mathcal{H}}=\langle
v_{1},w_{1}\rangle+\langle v_{2},w_{2}\rangle_{L_{g}}\mbox{ \ \ and\ \
}\|(v(s),w(s))\|_{\mathcal{H}}=\left(\|v(s)\|^{2}+\|w(s)\|_{L_{g}}^{2}\right)^{\frac{1}{2}}.$

Additionally, to get our stability results we assume the following additional
hypothesis to $g$. There exists a function
$\xi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that

(1.14) $\xi\in C^{1}(\mathbb{R}^{+}),\ \xi^{\prime}\leq 0,\
\int_{0}^{\infty}\xi(s)ds=\infty\mbox{\ and \ }g^{\prime}\leq-\xi g.$

In the sequel, $M_{P}$ is the smallest positive constant satisfying the
Poincaré’s Inequality

$\|v\|^{2}\leq M_{P}\|\partial_{x}v\|^{2},$

for all $v\in H_{0}^{1}(I)$. Furthermore, let us denote by $M_{S}$ the
positive constant of the Sobolev embedding $H^{1}(I)\hookrightarrow
L^{\infty}(I)$

$\|v\|_{L^{\infty}(I)}^{2}\leq M_{S}\|v\|_{H^{1}(I)}^{2},\quad v\in H^{1}(I).$

With this in hand, we will announce the main result of this article,
precisely, the stability result for the solutions of (1.2). For that, let us
reformulate our problem (1.2) and (1.8) as an abstract initial value problem,
namely,

(1.15) $\begin{cases}\partial_{t}U(t)=\mathcal{A}U,\\\ U(0)=U_{0}.\end{cases}$

The main result of the article can be read as follows.

######  Theorem 1.1.

Assume that $a_{0}>0$. Also, suppose that (1.3) and (1.14) are verified. If
$U_{0}\in\mathcal{H}$ satisfies

(1.16)
$a_{1}M_{P}^{2}+\dfrac{2}{3}M_{P}(M_{P}+1)\sqrt{L}M_{S}\|U_{0}\|<5a_{0},$

then there exist positive constants $c$ and $\tilde{c}$ such that the solution
$U$ of (1.15) satisfies the following stability estimates

  1. $(i)$

If $\xi$ is a constant function, we have

(1.17) $E(t)\leq\tilde{c}e^{-ct},\ t\in\mathbb{R}_{+}.$

  2. $(ii)$

If $\xi$ is not a constant function, yields that

(1.18)
$E(t)\leq\tilde{c}e^{-c\int_{0}^{t}\xi(\tau)d\tau}\left(1+\int_{0}^{t}e^{c\int_{0}^{\sigma}\xi(\tau)d\tau}\xi(\sigma)\int_{\sigma}^{\infty}g(s)h(\sigma,s)dsd\sigma\right),t\in\mathbb{R}_{+},$

where

$h(t,s)=t^{2}+t+\left\|\int^{t-s}_{0}\partial_{x}^{k}u_{0}(\cdot,\tau)d\tau\right\|,$

for $0\leq t\leq s$.

######  Remark 1.2.

Functions $g$ satisfying (1.8) and (1.14) are very wide and contain, for
example, the ones which converge to zero exponentially like

$g(s)=d_{1}e^{-q_{1}s},$

where $\xi(s)=q_{1}=\xi_{0}$ with $d_{1}>0$ and $q_{1}>0$, or polynomial like

$g(s)=d_{1}(1+s)^{-q_{1}},$

where $\xi(s)=\dfrac{q_{1}}{s+1}$, $\xi_{0}=q_{1}$ with $d_{1}>0$ and
$q_{1}>1$, or between them like the following one

$g(s)=d_{1}e^{-q_{1}(s+1)^{p_{1}}},$

where $\xi(s)=q_{1}p_{1}(s+1)^{p_{1}-1}$, $\xi_{0}=q_{1}p_{1},$ with
$d_{1}>0$, $q_{1}>0$ and $p_{1}\in(0,1)$.

Our work is outlined as follows: Section 2 is devoted to presenting
preliminary results which are essential for the rest of the article. In
Section 3 we prove the well-posedness of the damping-memory problem (1.2).
After that, the main result of the article, namely, Theorem 1.1 is shown in
Section 4.

##  2\. Preliminaries

As mentioned before, (1.2) and (1.8) can be seen as the following Cauchy
problem:

(2.1) $\begin{cases}\partial_{t}U(t)=\mathcal{A}U\\\ U(0)=U_{0},\end{cases}$

with the operator $\mathcal{A}$ given by

$\mathcal{A}(U)=\left(\begin{array}[]{c}-\partial_{x}^{3}u+a_{0}\partial_{x}^{5}u-u\partial_{x}u-a_{1}\partial_{x}u-(-1)^{k}\displaystyle\int_{0}^{\infty}g(s)\partial_{x}^{2k}\eta^{t}(\cdot,s)ds\\\
u-\partial_{s}\eta^{t}\end{array}\right).$

Here, let us consider the domain of $\mathcal{A}$ as follows

$D(\mathcal{A})=\\{U\in\mathcal{H};\mathcal{A}(U)\in\mathcal{H},u\in
H^{2}_{0}(I),\partial_{x}^{2}u(L)=0,\eta^{t}(x,0)=0\\}.$

Additionally, for $T>0,$ we introduce the space

$\mathcal{B}=C([0,T];L^{2}(I))\cap L^{2}(0,T;H^{2}(I))$

whose considered norm is

$\|\cdot\|_{\mathcal{B}}=\|\cdot\|_{C([0,T];L^{2}(I))}+\|\cdot\|_{L^{2}(0,T;H^{2}(I))}.$

The next lemma gives us a formal calculation of the derivative of
$E(t)=\dfrac{1}{2}\|U(t)\|^{2}_{\mathcal{H}}$, defined by (1.4), which will be
important in the work (the computations will be rigorously justified later).

######  Lemma 2.1.

Let us consider $\ I=(0,L)$ and $a_{0}>0.$ Assume that (1.3) hold, then the
derivative in time of the energy functional $E$ satisfies

(2.2)
$E^{\prime}(t)=-\dfrac{1}{2}a_{0}\left[(\partial^{2}_{x}u)(0)\right]^{2}+\dfrac{1}{2}\int_{0}^{\infty}g^{\prime}(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds.$

###### Proof.

Observe that it follows from (1.3) that

(2.3)
$E^{\prime}(t)=\left\langle\partial_{t}u,u\right\rangle+\frac{1}{2}\partial_{t}\left(\int_{0}^{\infty}g(s)\left\|\partial_{x}^{k}\eta^{t}(\cdot,s)\right\|^{2}ds\right).$

We will analyze each part of the $E^{\prime}(t)$ separately. First, note that
by multiplying (1.10) by $u$, integrating by parts in and using the boundary
condition of (1.2), we have

(2.4)
$\int_{0}^{L}u\partial_{t}udx=-a_{0}\frac{1}{2}\left(\partial_{x}^{2}u(0)\right)^{2}-(-1)^{k}\int_{0}^{L}u\int_{0}^{\infty}g(s)\partial_{x}^{2k}\eta^{t}(x,s)dsdx.$

Now, multiplying (1.7) by $(-1)^{k}\partial_{x}^{2k}g(s)\eta^{t}$ and again,
integrating by parts in $I\times\mathbb{R}_{+}$, we get

(2.5)
$\begin{split}\int_{0}^{L}\int_{0}^{\infty}(-1)^{k}g(s)\partial_{x}^{2k}\eta^{t}\partial_{t}\eta^{t}dsdx&+\int_{0}^{L}\int_{0}^{\infty}(-1)^{k}g(s)\partial_{x}^{2k}\eta^{t}\partial_{s}\eta^{t}dsdx\\\
&=\int_{0}^{L}\int_{0}^{\infty}(-1)^{k}ug(s)\partial_{x}^{2k}\eta^{t}dsdx\end{split}$

thanks to the boundary conditions of (1.7). When $k=0$, (2.2) holds directly
from (2.3), (2.4) and (2.5). For the case $k=1$ or $k=2$, note that once again
integrating by parts $k-$times with respect to the variable $x$, in the two
left-hand parcels of (2.5) and, only one time with respect to the variable $s$
in the second left-hand parcel of (2.5), we find that

(2.6)
$\begin{split}\frac{1}{2}\partial_{t}\left(\int_{0}^{\infty}g(s)\left\|\partial_{x}^{k}\eta^{t}\right\|^{2}ds\right)=\frac{1}{2}\int_{0}^{\infty}g^{\prime}(s)\left\|\partial_{x}^{k}\eta^{t}\right\|^{2}ds+(-1)^{k}\int_{0}^{L}u\int_{0}^{\infty}g(s)\partial_{x}^{2k}\eta^{t}dsdx,\end{split}$

since we have that $\eta^{t}(x,0)=0$ and that the limit (1.8) holds. Hence, in
this case, to get (2.2) just add (2.4) and (2.6). ∎

######  Remark 2.2.

Let us give some comments.

  * (i)

Since $a_{0}>0$ and due to the assumptions on $g^{\prime}$ (see (1.8)), it
follows from (2.2) that $E^{\prime}(t)\leq 0$. Hence the memory is acting as a
mechanism of damping feedback.

  * (ii)

Note that the integral term of (2.2) is well-defined. In fact, observe that
since $0\leq-g^{\prime}\leq c_{0}g$, we have

(2.7)
$\left|\int_{0}^{\infty}g^{\prime}(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds\right|=-\int_{0}^{\infty}g^{\prime}(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds\leq
c_{0}\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds=c_{0}\|\partial_{x}^{k}\eta^{t}\|_{L_{g}}^{2}<\infty,$

for any $\eta^{t}\in L_{g}$, showing our claim.

##  3\. Well-posedness of the memory problem

In this section, we will study the well-posedness of the system (1.2).
Precisely, we will initially study the well-posedness of the linearized system
associated with (1.2), and then, we will show that the system with source term
is well-posed and, finally, we prove that the original nonlinear system (1.2)
is well-posed.

###  3.1. Well-posedness: The linearized problem

In this subsection, we give the details about the well-posedness of the
linearized system associated with (1.2), namely

(3.1)
$\begin{cases}\partial_{t}u+\partial_{x}^{3}u-a_{0}\partial_{x}^{5}u+a_{1}\partial_{x}u+(-1)^{k}\displaystyle\int_{0}^{\infty}g(s)\partial_{x}^{2k}\eta^{t}(x,s)ds=0,&\
(x,t)\in I\times(0,\infty),\\\
\partial_{t}\eta^{t}(x,s)+\partial_{s}\eta^{t}(x,s)-u(x,t)=0,&x\in I,\
s,t\in\mathbb{R}_{+},\\\ \eta^{t}(0,s)=\eta^{t}(L,s)=\eta^{t}(x,0)=0,&x\in I,\
s,t\in\mathbb{R}_{+},\\\
u(0,t)=u(L,t)=\partial_{x}u(0,t)=\partial_{x}u(L,t)=\partial_{x}^{2}u(L,t)=0,&t>0,\\\
u(x,0)=u_{0}(x),&x\in I,\end{cases}$

with some initial data $(u_{0},\eta^{0}).$ Note that the system (3.1) can be
written in an abstract form in $\mathcal{H}$ as follows

(3.2) $\begin{cases}\partial_{t}\Phi(t)=A\Phi(t),\ t>0\\\
\Phi(0)=\Phi_{0},\end{cases}$

with $\Phi=(u,\eta^{t}),\ \Phi_{0}=(u_{0},\eta^{0})$ and $A$ is a linear
operator giving by

(3.3)
$A(\Phi)=\left(\begin{array}[]{c}-\partial_{x}^{3}u+a_{0}\partial_{x}^{5}u-a_{1}\partial_{x}u-(-1)^{k}\displaystyle\int_{0}^{\infty}g(s)\partial_{x}^{2k}\eta^{t}(x,s)ds\\\
u-\partial_{s}\eta^{t}\end{array}\right)$

with domain

(3.4) $D(A)=\\{\Phi\in\mathcal{H};A(\Phi)\in\mathcal{H},u\in
H^{2}_{0}(I),\partial_{x}^{2}u(L)=0,\eta^{t}(x,0)=0\\}.$

In turn, recall that, in this section, the generic positive constant $C$ is
independent of the initial data $\Phi_{0}$ but may depend on $T,g_{0}$ and the
system’s parameters $a_{i}(i=0,1)$. The following result ensures the well-
posedness of the linearized system.

######  Theorem 3.1.

Let us consider $\ I=(0,L),\ a_{1}\in\mathbb{R}$ and $\ a_{0}>0.$ Suppose that
(1.3) is verified, then the following assertions can be verified:

  * (i)

The linear operator $A$ defined by (3.3) generates a $C_{0}$-semigroup of
contractions $S(t)$. Moreover, given an initial data $\Phi_{0}\in D(A)$, the
problem (3.2) admits a unique classical solution

(3.5) $\Phi\in C(\mathbb{R}_{+};D(A))\cup C^{1}(\mathbb{R}_{+},\mathcal{H}).$

In turn, if $\Phi_{0}\in\mathcal{H},$ then (3.2) have a unique mild solution

(3.6) $\Phi\in C(\mathbb{R}_{+};\mathcal{H}).$

  * $(ii)$

For any $\Phi_{0}\in\mathcal{H}$ and $T>0,$ the following estimates holds

(3.7) $\|u\|_{L^{2}(0,T;H^{2}(0,L))}^{2}\leq
C\|(u_{0},\eta^{0})\|_{\mathcal{H}}^{2},$

for some positive constant $C$. Additionally, the mapping

(3.8)
$\Delta:\Phi_{0}=(u_{0},\eta^{0})^{T}\in\mathcal{H}\rightarrow\Phi(\cdot):=S(\cdot)\Phi_{0}\in\mathcal{B}\times
C([0,T];L_{g})$

is continuous.

###### Proof.

In order to show (i), consider $\Phi=(u,\eta^{t})\in D(A).$ Thanks to (2.2)
and (3.2), we find

(3.9) $\begin{split}\langle
A(\Phi),\Phi\rangle_{\mathcal{H}}=\langle\partial_{t}\Phi,\Phi\rangle_{\mathcal{H}}=\left(\frac{1}{2}\|\Phi\|^{2}_{\mathcal{H}}\right)^{\prime}=E^{\prime}(t)<0.\end{split}$

Thus, $A$ is dissipative thanks to Remark 2.2. On the other hand, we can check
that the adjoint operator of $A$ is defined by

(3.10)
$A^{*}\Psi=\left(\begin{array}[]{c}\partial_{x}^{3}v-a_{0}\partial_{x}^{5}v+a_{1}\partial_{x}v+(-1)^{k}\displaystyle\int_{0}^{\infty}g(s)\partial_{x}^{2k}\zeta^{t}(x,s)ds\\\
-v+\dfrac{g^{\prime}(s)}{g(s)}\zeta^{t}+\partial_{s}\zeta^{t}\end{array}\right)$

with domain

(3.11) $D(A^{*})=\\{\Psi\in\mathcal{H};A^{*}(\Psi)\in\mathcal{H},v\in
H^{2}_{0}(I),\partial_{x}^{2}v(0)=0,\zeta^{t}(x,0)=0\\}.$

The same line of thought may be applied to obtain

(3.12) $\langle
A^{*}(\Psi),\Psi\rangle_{\mathcal{H}}=-a_{0}\dfrac{(\partial_{x}^{2}v(L))^{2}}{2}+\dfrac{1}{2}\int_{0}^{\infty}g^{\prime}(s)\|\partial_{x}^{k}\zeta^{t}\|^{2}ds\leq
0,\forall\Psi\in D(A^{*})$

and hence $A^{*}$ is also dissipative. Now, since $A$ is densely defined and
closed, the assertion in (i) is a direct consequence of the semigroup theory
of linear operator, for details see [28].

From now on we will show (ii). Let $\Phi_{0}=(u_{0},\eta^{0})\in\mathcal{H}$.
As we know that $S(t)$ is a $C_{0}$\- semigroup of contractions we have

(3.13)
$\|S(t)\Phi_{0}\|_{\mathcal{H}}=\|(u,\eta^{t})\|_{\mathcal{H}}\leq\|\Phi_{0}\|_{\mathcal{H}}=\|(u_{0},\eta^{0})\|_{\mathcal{H}},\quad\forall
t\in[0,T].$

Next, consider the function $p(x,t)$, to be chosen later, and consider a
regular solution $\Phi=(u,\eta^{t})$ of (3.2) with initial data $\Phi_{0}\in
D(A).$ In this case, $\Phi$ has the regularity (3.5). Then, multiplying the
equation (3.1) by $2xu$, integrating by parts over $[0,T]\times I$ and using
the boundary condition we have

(3.14)
$\begin{split}&4\int_{0}^{T}\|\partial_{x}u\|^{2}dt+5a_{0}\int_{0}^{T}\|\partial_{x}^{2}u\|^{2}dt\\\
&=\int_{0}^{L}xu_{0}^{2}dx-\int_{0}^{L}xu^{2}(x,T)dx-(-1)^{k}\int_{0}^{T}\int_{0}^{L}2xu\int_{0}^{\infty}g(s)\partial_{x}^{2k}\eta^{t}(x,s)dsdxdt+a_{1}\int_{0}^{T}\|u\|^{2}dt\\\
&\leq
L\|u_{0}\|^{2}+a_{1}\int_{0}^{T}\|u\|^{2}dt-(-1)^{k}\int_{0}^{T}\int_{0}^{L}2xu\int_{0}^{\infty}g(s)\partial_{x}^{2k}\eta^{t}(x,s)dsdxdt.\end{split}$

Let us treat the case $k=0$ and $k\in\\{1,2\\}$ separately.

Case 1: $k=0$.

First, note that

(3.15)
$\begin{split}-\int_{0}^{T}\int_{0}^{L}\int_{0}^{\infty}2g(s)xu\eta^{t}(x,s)dsdxdt\leq&L^{2}\int_{0}^{T}\int_{0}^{\infty}g(s)\|u\|^{2}dsdt+\int_{0}^{T}\|\eta^{t}(\cdot,s)\|^{2}_{L_{g}}dt\\\
=&L^{2}g_{0}\int_{0}^{T}\|u\|^{2}dt+\int_{0}^{T}\|\eta^{t}(\cdot,s)\|^{2}_{L_{g}}dt.\end{split}$

Thus, amalgamating (3.14) and (3.15), we deduce that

(3.16)
$\int_{0}^{T}\left(\|u\|^{2}+\|\partial_{x}u\|^{2}+\|\partial_{x}^{2}u\|^{2}\right)dt\leq
C\|(u_{0},\eta^{0})\|^{2}_{\mathcal{H}},$

where $C=C(T,L,a_{0},a_{1},g_{0})>0$, showing (3.7) for $\Phi\in D(A)$.
Finally, the result for $\Phi\in\mathcal{H}$ follows by a density argument.
This, together with (3.13) implies the continuity of $\Delta$.

Case 2: $k=1$.

Now, considering $k=1$, integrating the last term of (3.14) by parts and using
Hölder’s inequality and Young’s inequality we get

(3.17) $\begin{split}(4-\epsilon
g_{0}L^{2})\int_{0}^{T}\|\partial_{x}u\|^{2}dt+5a_{0}\int_{0}^{T}\|\partial_{x}^{2}u\|^{2}dt+2\int_{0}^{T}\int_{0}^{L}a(x)xu^{2}dxdt\\\
\leq
L\|u_{0}\|^{2}+(a_{1}+g_{0})\int_{0}^{T}\|u\|^{2}dt+\left(1+\dfrac{1}{\epsilon}\right)\int_{0}^{T}\|\eta^{t}\|_{L_{g}}^{2}dt.\end{split}$

Taking $\epsilon=\dfrac{3}{g_{0}L^{2}}>0$, we have

(3.18)
$\int_{0}^{T}\left(\|u\|^{2}+\|\partial_{x}u\|^{2}+\|\partial_{x}^{2}u\|^{2}\right)dt\leq
C\|(u_{0},\eta^{0})\|^{2}_{\mathcal{H}},$

where $C=C(T,L,a_{0},a_{1},g_{0})>0$ showing (3.7) for $\Phi\in D(A)$. By
density argument, (3.7) holds true for $\Phi\in\mathcal{H}$ and also the
continuity of the mapping $\Delta$ is verified.

Case 3: $k=2$.

One has merely to argue as in the previous case. The only difference is that
we need to handle the term involving $\|\partial_{x}^{2}u\|^{2}$ in addition
to $\|\partial_{x}u\|^{2}$. ∎

###  3.2. Well-posedness: The equation with source term

The goal of this part is to deal with the well-posedness of the system (3.1)
with a source term $\varphi(x,t)$

(3.19)
$\begin{cases}\partial_{t}u+\partial_{x}^{3}u-a_{0}\partial_{x}^{5}u+a_{1}\partial_{x}u+(-1)^{k}\int_{0}^{\infty}g(s)\partial_{x}^{2k}\eta^{t}(x,s)ds=\varphi(x,t),&\
(x,t)\in I\times(0,\infty),\\\
\partial_{t}\eta^{t}(x,s)+\partial_{s}\eta^{t}(x,s)-u(x,t)=0,&\ x\in I,\
s,t\in\mathbb{R}_{+},\\\ \eta^{t}(0,s)=\eta^{t}(L,s)=\eta^{t}(x,0)=0,&x\in I,\
s,t\in\mathbb{R}_{+},\\\
u(0,t)=u(L,t)=\partial_{x}u(0,t)=\partial_{x}u(L,t)=\partial_{x}^{2}u(L,t)=0,&t>0,\\\
u(x,0)=u_{0}(x),&x\in I.\end{cases}$

We have the following result:

######  Theorem 3.2.

Let us consider $T>0$ and $a_{0}>0$. If (1.3) is verified, then we have:

  1. $(i)$

If $\Phi_{0}=(u_{0},\eta^{0})^{T}\in\mathcal{H}$ and $\varphi\in
L^{1}(0,T;L^{2}(I)),$ then there exists a unique mild solution
$\Phi=(u,\eta^{t})^{T}$ of (3.19) such that $\Phi\in\mathcal{B}\times
C([0,T];L_{g}),$

(3.20) $\|(u,\eta^{t})\|_{C([0,T];\mathcal{H})}^{2}\leq
C_{0}\left(\|(u_{0},\eta^{0})\|_{\mathcal{H}}^{2}+\|\varphi\|_{L^{1}(0,T;L^{2}(I))}^{2}\right)$

and

(3.21) $\|u\|_{\mathcal{B}}^{2}\leq
C_{1}\left(\|(u_{0},\eta^{0})\|_{\mathcal{H}}^{2}+\|\varphi\|_{L^{1}(0,T;L^{2}(I))}^{2}\right),$

for some positive constants $C_{0},\ C_{1}$ independent of $\Phi_{0}$ and
$\varphi.$

  2. $(ii)$

Given $u\in L^{2}(0,T;H^{2}(I)),$ we have $u\partial_{x}u\in
L^{1}(0,T;L^{2}(I))$ and the map

(3.22) $\Theta:u\in L^{2}(0,T;H^{2}(I))\rightarrow u\partial_{x}u\in
L^{1}(0,T;L^{2}(I))$

is continuous.

###### Proof.

(i) Since $A$ generates a $C_{0}$-semigroup of contractions $S(t)$ and
$\varphi\in L^{1}(0,T;L^{2}(I))$ and in order to ensure the validity of the
computations, we shall work with a regular solution $\Phi$ of (3.19) stemmed
from an initial data $\Phi_{0}=(u_{0},\eta^{0})^{T}\in D(A).$ It is well-known
from the semigroups theory [28] that the solution of (3.19) satisfies

(3.23) $\|(u,\eta^{t})\|_{\mathcal{H}}\leq
C\left(\|(u_{0},\eta^{0})\|_{\mathcal{H}}+\int_{0}^{t}\|\varphi\|dt\right)\leq
C\left(\|(u_{0},\eta^{0})\|_{\mathcal{H}}+\|\varphi\|_{L^{1}(0,T;L^{2}(I))}\right),$

and consequently (3.20) holds. We also have, thanks to (3.23), that

(3.24) $\|u\|_{C([0,T];L^{2}(I))}\leq
C\left(\|(u_{0},\eta^{0})\|_{\mathcal{H}}+\|\varphi\|_{L^{1}(0,T;L^{2}(I))}\right).$

Therefore, to obtain the $H^{2}$-norm of the solution, that is, (3.21), we use
an analogous process as in the proof of (3.7), and hence we will omit it. On
the other hand, a density argument allows us to extend the results to any
initial condition $\Phi_{0}\in\mathcal{H}$.

(ii) First, consider $y,z\in L^{2}(0,T;H^{2}(I))$. We have

(3.25) $\begin{split}\|y\partial_{x}y\|_{L^{1}(0,T;L^{2}(I))}\leq
K\int_{0}^{T}\|y\|_{H^{2}(I)}\|\partial_{x}y\|dt\leq
K\int_{0}^{T}\|y\|_{H^{2}(I)}^{2}dt=K\|y\|_{L^{2}(0,T;H^{2}(I))}^{2},\end{split}$

where $K$ is the positive constant of the Sobolev embedding
$H^{2}\hookrightarrow L^{\infty}(I)$. Thus,

$y\partial_{x}y\in L^{1}(0,T;L^{2}(I)),$

for each $y\in L^{2}(0,T;H^{2}(I)).$

In turn, using triangle inequality together with Cauchy–Schwarz inequality, we
get

(3.26)
$\begin{split}\|\Theta(y)-\Theta(z)\|_{L^{1}(0,T;L^{2}(I))}\leq&K\int_{0}^{T}\|y-z\|_{H^{2}(I)}\|y\|_{H^{2}(I)}dt+K\int_{0}^{T}\|z\|_{H^{2}(I)}\|y-z\|_{H^{2}(I)}dt\\\
\leq&K\|y-z\|_{L^{2}(0,T;H^{2}(I))}\|y\|_{L^{2}(0,T;H^{2}(I))}\\\
&+K\|z\|_{L^{2}(0,T;H^{2}(I))}\|y-z\|_{L^{2}(0,T;H^{2}(I))}\\\
=&K\|y-z\|_{L^{2}(0,T;H^{2}(I))}\left(\|y\|_{L^{2}(0,T;H^{2}(I))}+\|z\|_{L^{2}(0,T;H^{2}(I))}\right)\end{split}$

Thus, the mapping $\Theta$ is continuous with respect to the corresponding
topologies. ∎

###  3.3. Well-posedness: The nonlinear problem

The next result ensures the well-posedness of the system (1.2), which is
represented by the problem (2.1).

######  Theorem 3.3.

Let us consider $T>0$ and $a_{0}>0$. If (1.3) is verified, then there exists a
positive constant $C$ such that, for every $U_{0}\in\mathcal{H}$ with

(3.27) $\|U_{0}\|^{2}<\dfrac{1}{16C_{1}^{2}K^{2}}$

where $C_{1}$ is as in (3.21) and $K$ is the positive constant of the Sobolev
embedding $H^{2}\hookrightarrow L^{\infty}(I)$, the problem (2.1) has a unique
global solution $U$ satisfying the regularity (3.6) and consequently, the
problem (1.2) admits a unique global solution $u\in\mathcal{B}.$

###### Proof.

First, consider $U_{0}=(u_{0},\eta^{0})\in\mathcal{H}$ satisfying (3.27).
Next, define the map $\Gamma:\mathcal{B}\rightarrow\mathcal{B}$ by
$\Gamma(z)=u$, where $u$ is a solution of (3.19) with source term
$\varphi(x,t)=-z(x,t)\partial_{x}z(x,t)$ and initial data $U_{0}.$

Claim 1: $\Gamma$ is well-defined.

In fact, take $\alpha>0$ such that

$\|U_{0}\|_{\mathcal{B}}^{2}\leq\alpha<\dfrac{1}{16C_{1}^{2}K^{2}}.$

Theorem 3.2 ensures that for the initial data $U_{0},$ there exists a unique
solution $U=(u,\eta^{t})$ of (3.19) satisfying, thanks to (3.27), the estimate

(3.28) $\|\Gamma(z)\|_{\mathcal{B}}\leq
C_{1}(\alpha+\|z\partial_{x}z\|_{L^{1}(0,T;L^{2}(I))}^{2}).$

Moreover, by using (3.25), we get

(3.29)
$\begin{split}\|\Gamma(z)\|_{\mathcal{B}}^{2}\leq&C_{1}\left(\|(u_{0},\eta^{0})\|_{\mathcal{H}}^{2}+\|z\partial_{x}z\|_{L^{1}(0,T;L^{2}(I))}^{2}\right)\leq
C_{1}(\alpha+K^{2}\|z\|_{L^{2}(0,T;H^{2}(I))}^{4})\\\
\leq&C_{1}(\alpha+K^{2}\|z\|_{\mathcal{B}}^{4}),\end{split}$

for all $z\in\mathcal{B},$ showing the claim 1.

Claim 2: $\Gamma$ is a contraction.

Indeed, we have

(3.30)
$\begin{split}\|\Gamma(y)-\Gamma(z)\|_{\mathcal{B}}^{2}\leq&2K^{2}\|y-z\|_{L^{2}(0,T;H^{2}(I))}^{2}\left(\|y\|_{L^{2}(0,T;H^{2}(I))}^{2}+\|z\|_{L^{2}(0,T;H^{2}(I))}^{2}\right)\\\
\leq&2K^{2}\|y-z\|_{\mathcal{B}}^{2}\left(\|y\|_{\mathcal{B}}^{2}+\|z\|_{\mathcal{B}}^{2}\right).\end{split}$

Then, consider the restriction of $\Gamma$ to the closed ball
$B=\left\\{z\in\mathcal{B};\|z\|_{\mathcal{B}}^{2}\leq r\right\\},$ with
$r=\dfrac{\sqrt{\alpha}}{2K}$. Thus, (3.28) and (3.29) yields that

(3.31) $\|\Gamma(z)\|_{\mathcal{B}}^{2}\leq
C_{1}(\alpha+K^{2}\|z\|_{\mathcal{B}}^{4})\leq
C_{1}(\alpha+K^{2}r^{2})<2C_{1}\alpha<r$

and

(3.32) $\|\Gamma(y)-\Gamma(z)\|_{\mathcal{B}}^{2}\leq
4rK^{2}\|y-z\|_{\mathcal{B}}^{2}\leq\dfrac{1}{2}\|y-z\|_{\mathcal{B}}^{2}.$

Clearly, the mapping $\Gamma$ is well-defined and contractive on the ball $B$
according to the choice (3.27), showing the claim 2.

Therefore, using the Banach Fixed Point Theorem, we deduce that $\Gamma$ has a
unique fixed element $u\in B$, which turns out to be the unique solution to
our problem (1.2). Lastly, the system (1.2) being dissipative as its energy is
decreasing, the solution is global. ∎

##  4\. Proof of the main result

This section is devoted to the proof of the main result, namely, Theorem 1.1.
The main ingredient of the proof is the utilization of the energy method.

###### Proof of Theorem 1.1.

First, multiplying (1.10) by $xu$, integrating by parts several times,
observing that $\partial_{x}(xu)=u+x\partial_{x}u,$ and thanks to the boundary
conditions of (1.2), we have that

(4.1)
$\begin{split}\dfrac{5a_{0}}{2}\|\partial_{x}^{2}u\|^{2}=&-\partial_{t}\left(\dfrac{1}{2}\int_{0}^{L}xu^{2}dx\right)-\dfrac{3}{2}\|\partial_{x}u\|^{2}+\dfrac{1}{3}\int_{0}^{L}u^{3}dx+\dfrac{a_{1}}{2}\|u\|^{2}\\\
&-(-1)^{k}\int_{0}^{L}xu\int_{0}^{\infty}g(s)\partial_{x}^{2k}\eta^{t}(x,s)dsdx.\end{split}$

We are now in position to estimate the terms of the right hand side of (4.1).

Estimate 1: First, the Sobolev embedding yields that

$\left|\int_{0}^{L}u^{3}dx\right|\leq\|u\|_{L^{\infty}(I)}^{2}\int_{0}^{L}|u|dx\leq
M_{S}\|u\|_{H_{1}(I)}^{2}\int_{0}^{L}|u|dx.$

Thus, the previous inequality together with Hölder’s and Poincaré’s
inequalities give us

(4.2)
$\begin{split}\left|\int_{0}^{L}u^{3}dx\right|\leq&M_{S}\|u\|_{H_{1}(I)}^{2}\left(\int_{0}^{L}1^{2}dx\right)^{\frac{1}{2}}\left(\int_{0}^{L}|u|^{2}dx\right)^{\frac{1}{2}}\\\
\leq&M_{S}L^{\frac{1}{2}}\left(\|u\|^{2}+\|\partial_{x}u\|^{2}\right)\|u\|\\\
\leq&M_{S}L^{\frac{1}{2}}\left(M_{P}+1\right)\|\partial_{x}u\|^{2}(2E(t))^{\frac{1}{2}}\\\
\leq&M_{S}L^{\frac{1}{2}}\left(M_{P}+1\right)(2E(0))^{\frac{1}{2}}\|\partial_{x}u\|^{2}.\end{split}$

Estimate 2: We claim that for each $\epsilon>0$, there exists a constant
$C_{\epsilon}>0$ such that

$\left|-(-1)^{k}\int_{0}^{L}xu\int_{0}^{\infty}g(s)\partial_{x}^{2k}\eta^{t}(x,s)dsdx\right|\leq\epsilon\|\partial_{x}^{2}u\|^{2}+C_{\epsilon}\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds.$

We split this estimates in two parts, namely, $k=0$ and $k\in\\{1,2\\}.$

Indeed, for the case $k=0,$ using the Young’s and Poincaré’s inequalities we
have

(4.3)
$\begin{split}\left|\int_{0}^{L}xu\int_{0}^{\infty}g(s)\eta^{t}dsdx\right|\leq&\int_{0}^{\infty}g(s)\int_{0}^{L}\left|xu\right|\left|\eta^{t}\right|dxds\\\
\leq&L\int_{0}^{\infty}g(s)\int_{0}^{L}\left(\delta\left|u\right|^{2}+\dfrac{1}{4\delta}\left|\eta^{t}\right|^{2}\right)dxds\\\
\leq&L\delta\underbrace{\int_{0}^{\infty}g(s)ds}_{=g_{0}}\|u\|^{2}+L\dfrac{1}{4\delta}\int_{0}^{\infty}g(s)\|\eta^{t}\|^{2}ds\\\
\leq&\underbrace{LM_{P}^{2}g_{0}\delta}_{\epsilon}\|\partial_{x}^{2}u\|^{2}+\underbrace{L\dfrac{1}{4\delta}}_{C_{\epsilon}}\int_{0}^{\infty}g(s)\|\eta^{t}\|^{2}ds\\\
\leq&\epsilon\|\partial_{x}^{2}u\|^{2}+C_{\epsilon}\int_{0}^{\infty}g(s)\|\eta^{t}\|^{2}ds,\end{split}$

where $\delta=\dfrac{\epsilon}{Lc_{p}^{2}g_{0}}>0,$ showing the estimate 2.

Now, we turn to the case $k=1$. Applying once again Young’s and Poincaré’s
inequalities gives the following

(4.4)
$\begin{split}\left|-\int_{0}^{\infty}g(s)\int_{0}^{L}u\partial_{x}\eta^{t}(x,s)dxds\right|\leq&\int_{0}^{\infty}\int_{0}^{L}\left|g(s)u\partial_{x}\eta^{t}(x,s)\right|dxds\\\
\leq&\delta\int_{0}^{\infty}g(s)\|u\|^{2}ds+\dfrac{1}{4\delta}\int_{0}^{\infty}g(s)\|\partial_{x}\eta^{t}\|^{2}ds\\\
\leq&\delta\int_{0}^{\infty}g(s)\|u\|^{2}ds+\dfrac{1}{4\delta}\|\eta^{t}\|^{2}_{L_{g}}\\\
\leq&\delta
g_{0}M_{P}^{2}\|\partial_{x}^{2}u\|^{2}+\dfrac{1}{4\delta}\|\eta^{t}\|^{2}_{L_{g}}.\end{split}$

In a similar way, we also have the following estimate

(4.5)
$\begin{split}\left|-\int_{0}^{\infty}g(s)\int_{0}^{L}x\partial_{x}u\partial_{x}\eta^{t}(x,s)dxds\right|\leq&\delta\int_{0}^{\infty}g(s)\|x\partial_{x}u\|^{2}ds+\dfrac{1}{4\delta}\int_{0}^{\infty}g(s)\|\partial_{x}\eta^{t}\|^{2}ds\\\
\leq&\delta
L^{2}\int_{0}^{\infty}g(s)\|\partial_{x}u\|^{2}ds+\dfrac{1}{4\delta}\|\eta^{t}\|^{2}_{L_{g}}\\\
=&\delta
g_{0}L^{2}\|\partial_{x}u\|^{2}+\dfrac{1}{4\delta}\|\eta^{t}\|^{2}_{L_{g}}\\\
\leq&\delta
g_{0}M_{P}L^{2}\|\partial_{x}^{2}u\|^{2}+\dfrac{1}{4\delta}\|\eta^{t}\|^{2}_{L_{g}}.\end{split}$

Now, observe that

$\begin{split}\left|(-1)^{k}\int_{0}^{L}xu\int_{0}^{\infty}g(s)\partial_{x}^{2}\eta^{t}dsdx\right|\leq&\left|-\int_{0}^{\infty}g(s)\int_{0}^{L}u\partial_{x}\eta^{t}(x,s)dxds\right|\\\
&+\left|-\int_{0}^{\infty}g(s)\int_{0}^{L}x\partial_{x}u\partial_{x}\eta^{t}(x,s)dxds\right|.\end{split}$

Thus, thanks to the inequalities (4.4) and (4.5) applied on the right-hand
side of the previous inequality we have

$\begin{split}\left|(-1)^{k}\int_{0}^{L}xu\int_{0}^{\infty}g(s)\partial_{x}^{2}\eta^{t}dsdx\right|\leq&\delta
g_{0}M_{P}^{2}\|\partial_{x}^{2}u\|^{2}+\dfrac{1}{4\delta}\|\eta^{t}\|^{2}_{L_{g}}+\delta
g_{0}M_{P}L^{2}\|\partial_{x}^{2}u\|^{2}+\dfrac{1}{4\delta}\|\eta^{t}\|^{2}_{L_{g}}\\\
=&\underbrace{\delta
g_{0}(M_{P}+M_{P}^{2})}_{\epsilon}\|\partial_{x}^{2}u\|^{2}+\underbrace{\dfrac{1}{2\delta}}_{C_{\epsilon}}\|\eta^{t}\|^{2}_{L_{g}},\end{split}$

where $\delta=\dfrac{\epsilon}{g_{0}(M_{P}+M_{P}^{2})}>0,$ showing the
estimate 2.

Now, we turn to the case $k=2$. Thanks to Young’s and Poincaré’s inequalities,
we have

(4.6)
$\begin{split}\left|2\int_{0}^{\infty}g(s)\int_{0}^{L}\partial_{x}u\partial_{x}^{2}\eta^{t}(x,s)dxds\right|\leq&\delta
g_{0}M_{P}\|\partial_{x}^{2}u\|^{2}+\dfrac{1}{\delta}\|\eta^{t}\|^{2}_{L_{g}}.\end{split}$

Using the same arguments as for the case $k=1$, we have

(4.7)
$\begin{split}\left|-\int_{0}^{\infty}g(s)\int_{0}^{L}x\partial_{x}^{2}u\partial_{x}^{2}\eta^{t}(x,s)dxds\right|\leq&\delta
g_{0}L^{2}\|\partial_{x}^{2}u\|^{2}+\dfrac{1}{\delta}\|\eta^{t}\|^{2}_{L_{g}}.\end{split}$

Moreover, since

$\begin{split}\left|-(-1)^{k}\int_{0}^{L}xu\int_{0}^{\infty}g(s)\partial_{x}^{4}\eta^{t}dsdx\right|\leq&\left|2\int_{0}^{\infty}g(s)\int_{0}^{L}\partial_{x}u\partial_{x}^{2}\eta^{t}(x,s)dxds\right|\\\
&+\left|\int_{0}^{\infty}g(s)\int_{0}^{L}x\partial_{x}^{2}u\partial_{x}^{2}\eta^{t}(x,s)dxds\right|,\end{split}$

and thanks to (4.6) and (4.7), we reach

$\begin{split}\left|-(-1)^{k}\int_{0}^{L}xu\int_{0}^{\infty}g(s)\partial_{x}^{4}\eta^{t}dsdx\right|\leq&\delta
g_{0}M_{P}\|\partial_{x}^{2}u\|^{2}+\delta
g_{0}L^{2}\|\partial_{x}^{2}u\|^{2}+\dfrac{2}{\delta}\|\eta^{t}\|^{2}_{L_{g}}\\\
=&\underbrace{\delta
g_{0}(M_{P}+L^{2})}_{\epsilon}\|\partial_{x}^{2}u\|^{2}+\underbrace{\dfrac{2}{\delta}}_{C_{\epsilon}}\|\eta^{t}\|^{2}_{L_{g}},\end{split}$

where $\delta=\dfrac{\epsilon}{g_{0}(M_{P}+L^{2})}>0,$ showing the estimate 2.

Estimate 3: There are two constants $C_{1}>0$ and $C_{2}>0$ such that

(4.8)
$\begin{array}[]{rcl}{\displaystyle\|\partial_{x}^{2}u\|^{2}}&\leq&{\displaystyle-
C_{1}\partial_{t}\left(\int_{0}^{L}xu^{2}dx\right)+C_{2}\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds.}\end{array}$

Indeed, estimates 1 and 2 together with the Poincaré inequality yield

$\begin{array}[]{rcl}{\displaystyle\dfrac{5a_{0}}{2}\|\partial_{x}^{2}u\|^{2}}&\leq&{\displaystyle-\partial_{t}\left(\dfrac{1}{2}\int_{0}^{L}xu^{2}dx\right)+\dfrac{1}{3}\left|\int_{0}^{L}u^{3}dx\right|}\\\
&&\\\
&&{\displaystyle+\dfrac{a_{1}}{2}\|u\|^{2}+\left|-(-1)^{k}\int_{0}^{L}xu\int_{0}^{\infty}g(s)\partial_{x}^{2k}\eta^{t}(x,s)dsdx\right|}\\\
&&\\\
&\leq&{\displaystyle-\partial_{t}\left(\dfrac{1}{2}\int_{0}^{L}xu^{2}dx\right)+\dfrac{1}{3}M_{S}L^{\frac{1}{2}}(M_{P}+1)(E(0))^{\frac{1}{2}}\|\partial_{x}u\|^{2}}\\\
&&\\\
&&{\displaystyle+\dfrac{a_{1}}{2}\|u\|^{2}+\epsilon\|\partial_{x}^{2}u\|^{2}+C_{\epsilon}\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds}\\\
&&\\\
&\leq&{\displaystyle-\partial_{t}\left(\dfrac{1}{2}\int_{0}^{L}xu^{2}dx\right)+\dfrac{1}{3}M_{S}L^{\frac{1}{2}}M_{P}(M_{P}+1)(E(0))^{\frac{1}{2}}\|\partial_{x}^{2}u\|^{2}}\\\
&&\\\
&&{\displaystyle+\dfrac{a_{1}M_{P}^{2}}{2}\|\partial_{x}^{2}u\|^{2}+\epsilon\|\partial_{x}^{2}u\|^{2}+C_{\epsilon}\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds}\end{array}$

which ensures

(4.9)
$\begin{split}\underbrace{\left(5a_{0}-\dfrac{2}{3}M_{S}L^{\frac{1}{2}}M_{P}(M_{P}+1)(E(0))^{\frac{1}{2}}-a_{1}M_{P}^{2}-2\epsilon\right)}_{:=D}\|\partial_{x}^{2}u\|^{2}\leq&-\partial_{t}\left(\int_{0}^{L}xu^{2}dx\right)\\\
&+2C_{\epsilon}\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds.\end{split}$

Now on, taking $\epsilon>0$ small enough, it follows from (1.16) that $D>0$
and hence (4.8) holds true for $C_{1}=\dfrac{1}{D}>0$ and
$C_{2}=\dfrac{2C_{\epsilon}}{D}>0.$ Thus, the estimate 3 is achieved.

To conclude the proof, consider the following function

$F(t)=\mu E(t)+C_{1}\xi(t)\left(\int_{0}^{L}xu^{2}dx\right),$

where $\mu=2\left(C_{2}+\dfrac{1}{M_{P}^{2}}\right)$. As $\xi^{\prime}\leq 0$,
we have

$0\leq\xi(t)\left(\int_{0}^{L}xu^{2}dx\right)\leq\xi(0)\left(\int_{0}^{L}xu^{2}dx\right)\leq\xi(0)L\|u\|^{2}\leq
2L\xi(0)2E(t).$

Consequently, owing to the previous inequality, we get

(4.10) $\begin{split}\mu E(t)\leq&F(t)\leq\mu
E(t)+C_{1}\xi(t)\left(\int_{0}^{L}xu^{2}dx\right)\\\ \leq&\mu
E(t)+2LC_{1}\xi(0)E(t)=\left(\mu+2LC_{1}\xi(0)\right)E(t).\end{split}$

Observe that $\xi^{\prime}\leq 0$ ensures that

(4.11) $\begin{split}F^{\prime}(t)=&\mu
E^{\prime}(t)+C_{1}\xi^{\prime}(t)\left(\int_{0}^{L}xu^{2}dx\right)+C_{1}\xi(t)\partial_{t}\left(\int_{0}^{L}xu^{2}dx\right)\\\
\leq&\mu
E^{\prime}(t)+C_{1}\xi(t)\partial_{t}\left(\int_{0}^{L}xu^{2}dx\right).\end{split}$

Now, putting (4.8) into (4.11) and using the Poincaré’s inequality, we get

(4.12) $\begin{split}F^{\prime}(t)\leq&\mu
E^{\prime}(t)+\xi(t)\left(C_{2}\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds-\|\partial_{x}^{2}u\|^{2}\right)\\\
\leq&\mu
E^{\prime}(t)+\xi(t)\left(C_{2}\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds-\dfrac{1}{M_{P}^{2}}\|u\|^{2}\right)\\\
=&\mu
E^{\prime}(t)+\xi(t)\left(C_{2}+\dfrac{1}{M_{P}^{2}}\right)\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds-\dfrac{2}{M_{P}^{2}}\xi(t)E(t)\\\
\leq&\mu
E^{\prime}(t)+\xi(t)\left(C_{2}+\dfrac{1}{M_{P}^{2}}\right)\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds-\lambda_{0}\xi(t)F(t),\end{split}$

where in the last inequality we have used (4.10). Here,

$\lambda_{0}=\dfrac{2}{M_{P}^{2}\left[\mu+2LC_{1}\xi(0)\right]}.$

Now on, we shall distinguish two cases.

Case 1: $\xi$ is a constant function.

In this case, taking into account (1.14) and (2.2), we have

$\xi\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds\leq-\int_{0}^{\infty}g^{\prime}(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds\leq-2E^{\prime}(t)$

that substituting in (4.12) gives us

$F^{\prime}(t)\leq\mu
E^{\prime}(t)-2\left(C_{2}+\dfrac{1}{M_{P}^{2}}\right)E^{\prime}(t)-\lambda_{0}\xi
F(t)=-\lambda_{0}\xi F(t)$

implying $F(t)=e^{-ct}F(0),$ with $c=\lambda_{0}\xi$. Finally, (4.10) yields

$E(t)\leq\dfrac{1}{\mu}F(t)=\dfrac{F(0)}{\mu}e^{-ct}\leq\dfrac{2\xi(0)E(0)}{\mu}e^{-ct},$

which ensures item (i) of the Theorem 1.1.

Case 2: $\xi$ is not a constant function.

First, observe that integrating (4.8) over $[0,t]$ and using the definition of
$E(t)$, since $E$ is decreasing, we get

(4.13)
$\begin{split}\int_{0}^{t}\|\partial^{2}_{x}u\|^{2}ds\leq&-C_{1}\int_{0}^{t}\partial_{\tau}\left(\int_{0}^{L}xu^{2}dx\right)d\tau+C_{2}\int_{0}^{t}\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}dsd\tau\\\
\leq&C_{1}\left(\int_{0}^{L}xu_{0}^{2}dx\right)+C_{2}\int_{0}^{t}\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}dsd\tau\\\
\leq&C_{1}\left(\int_{0}^{L}xu_{0}^{2}dx\right)+C_{2}\int_{0}^{t}2E(\tau)d\tau\\\
\leq&C_{1}\left(\int_{0}^{L}xu_{0}^{2}dx\right)+C_{2}\int_{0}^{t}2E(0)d\tau:=C_{3}(1+t),\end{split}$

where
$C_{3}=\max\left\\{C_{1}\left(\displaystyle\int_{0}^{L}xu_{0}^{2}dx\right),2C_{2}E(0)\right\\}$.
Now, Young’s and Hölder’s inequalities together with (4.13) , ensures that

(4.14)
$\begin{split}\left\|\int_{t-s}^{t}\partial_{x}^{k}u(\cdot,\tau)d\tau\right\|^{2}\leq&2\left\|\int_{t-s}^{0}\partial_{x}^{k}u(\cdot,\tau)d\tau\right\|^{2}+2\left\|\int_{0}^{t}\partial_{x}^{k}u(\cdot,\tau)d\tau\right\|^{2}\\\
\leq&2\left\|\int^{t-s}_{0}\partial_{x}^{k}u_{0}(\cdot,\tau)d\tau\right\|^{2}+2t\int_{0}^{L}\int_{0}^{t}(\partial_{x}^{k}u)^{2}(\cdot,\tau)d\tau
dx\\\
\leq&2\left\|\int^{t-s}_{0}\partial_{x}^{k}u_{0}(\cdot,\tau)d\tau\right\|^{2}+2t\int_{0}^{t}\|\partial_{x}^{k}u\|^{2}d\tau:=c_{1}h(t,s),\end{split}$

for $0\leq t\leq s.$ Here $c_{1}=2\max\\{1,M_{P}^{2-k}C_{3}\\}$ and
${\displaystyle
h(t,s)=t^{2}+t+\left\|\int^{t-s}_{0}\partial_{x}^{k}u_{0}(\cdot,\tau)d\tau\right\|^{2}.}$
On the other hand, thanks to (2.2)

(4.15)
$\begin{split}\xi(t)\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds=&\xi(t)\int_{0}^{t}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds+\xi(t)\int_{t}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds\\\
\leq&-\int_{0}^{t}g^{\prime}(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds+\xi(t)\int_{t}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds\\\
\leq&-\int_{0}^{t}g^{\prime}(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds+c_{1}\xi(t)\int_{t}^{\infty}g(s)h(t,s)ds\\\
\leq&-\int_{0}^{\infty}g^{\prime}(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds+c_{1}\xi(t)\int_{t}^{\infty}g(s)h(t,s)ds\\\
\leq&-2E^{\prime}(t)+c_{1}\xi(t)\int_{t}^{\infty}g(s)h(t,s)ds.\end{split}$

Recall that

$F(t)=\mu E(t)+C_{1}\xi(t)\left(\int_{0}^{L}xu^{2}dx\right).$

Analogously to the previous case, we have

(4.16) $\begin{split}F^{\prime}(t)\leq&-\lambda_{0}\xi(t)F(t)+\mu
E^{\prime}(t)+\xi(t)\dfrac{\mu}{2}\int_{0}^{\infty}g(s)\|\partial_{x}^{k}\eta^{t}\|^{2}ds\\\
\leq&-\lambda_{0}\xi(t)F(t)+\mu E^{\prime}(t)-\mu
E^{\prime}(t)+c_{1}\dfrac{\mu}{2}\xi(t)\int_{t}^{\infty}g(s)h(t,s)ds.\end{split}$

Setting $c:=\lambda_{0}$, (4.16) ensures that

$F^{\prime}(t)+c\xi(t)F(t)\leq\mu E^{\prime}(t)-\mu
E^{\prime}(t)+\dfrac{c_{1}\mu}{2}\xi(t)\int_{t}^{\infty}g(s)h(t,s)ds=\dfrac{c_{1}\mu}{2}\xi(t)\int_{t}^{\infty}g(s)h(t,s)ds,$

or equivalently,

$\left(e^{c\int_{0}^{t}\xi(\tau)d\tau}F(t)\right)^{\prime}\leq\dfrac{c_{1}\mu}{2}e^{c\int_{0}^{t}\xi(\tau)d\tau}\xi(t)\int_{t}^{\infty}g(s)h(t,s)ds.$

Finally, the previous inequality and $E(t)\leq\dfrac{1}{\mu}F(t),$ gives us

(4.17)
$E(t)\leq\tilde{c}e^{-c\int_{0}^{t}\xi(\tau)d\tau}\left(1+\int_{0}^{t}e^{c\int_{0}^{\sigma}\xi(\tau)d\tau}\xi(\sigma)\int_{\sigma}^{\infty}g(s)h(\sigma,s)dsd\sigma\right),$

where $\tilde{c}=\dfrac{\max\left\\{F(0),\dfrac{c_{1}\mu}{2}\right\\}}{\mu}.$
Thereby, the proof of the second part (ii) of Theorem 1.1 is complete. ∎

##  5\. Conclusion

In this paper, we considered the well-known Kawahara equation under the
presence of only an internal infinite memory term. Then, it is shown that the
energy of the system decays under some assumptions of the memory kernel.
Moreover, an estimate of the energy decay is provided depending on the
property of the kernel. The main ingredient of the proof is the utilization of
the Fixed Point Theorem and the energy method. Based on this outcome, one can
conclude that the distributed memory term creates enough dissipation for the
energy of the system so that the exponential stability holds. On the other
hand, we believe that our results remain valid if the memory term occurs in a
boundary condition. Of course, this could be the subject of a future work to
ascertain the claim.

### Acknowledgments

Capistrano–Filho was supported by CAPES grant 88881.311964/2018-01 and
88881.520205/2020-01, CNPq grant 307808/2021-1 and 401003/2022-1, MATHAMSUD
grant 21-MATH-03 and Propesqi (UFPE). This work is part of the Ph.D. thesis of
de Jesus at the Department of Mathematics of the Universidade Federal de
Pernambuco.

### Data availability statement

Data sharing does not apply to the current paper as no data were generated or
analyzed during this study.

### Conflict of Interest

The authors declare that they have no known conflict of interest.

## References

  * [1] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, _Invent. Math._ , 171 (2008), 485–541.
  * [2] G. Amendola, M. Fabrizio, J.M. Golden, Thermodynamics of Materials with Memory: Theory and Applications, Springer, 2012.
  * [3] F. D. Araruna, R. A. Capistrano-Filho and G. G. Doronin, Energy decay for the modified Kawahara equation posed in a bounded domain, _J. Math. Anal. Appl._ , 385 (2012), 743–756.
  * [4] J. L. Bona, D. Lannes and J.-C. Saut, Asymptotic models for internal waves, _J. Math. Pures Appl._ , 9 (2008), 538–566.
  * [5] R. A. Capistrano-Filho, B. Chentouf, L. de Sousa and V. H. Gonzalez Martinez, Two stability results for the Kawahara equation with a time-delayed boundary control, Zeitschrift für angewandte Mathematik und Physik, to appear.
  * [6] R. A. Capistrano-Filho and M. M. de S. Gomes, Well-posedness and controllability of Kawahara equation in weighted Sobolev spaces, _Nonlinear Analysis_ , 207 (2011), 1–24.
  * [7] R. A. Capistrano-Filho and V. H. Gonzalez Martinez, Stabilization results for delayed fifth order KdV-type equation in a bounded domain, arXiv:2112.14854 [math.AP].
  * [8] R. A. Capistrano-Filho and L. S. de Sousa, Control results with overdetermination condition for higher order dispersive system, _Journal of Mathematical Analysis and Applications_ , 506 (2022), 1–22.
  * [9] M. Chen, Internal controllability of the Kawahara equation on a bounded domain, _Nonlinear Analysis_ , 185 (2019), 356–373.
  * [10] B. Chentouf, Well-posedness and exponential stability of the Kawahara equation with a time-delayed localized damping, _Mathematical Methods in the Applied Sciences,_ 45 (2022), 10312–10330.
  * [11] B. Chentouf and A. Guesmia, Well-posedness and stability results for the Korteweg–de Vries–Burgers and Kuramoto–Sivashinsky equations with infinite memory: A history approach, _Nonlinear Analysis: Real World Applications_ , 65 (2022), 103508.
  * [12] G. M. Coclite and L. di Ruvo, Wellposedness of the classical solutions for a Kawahara–Korteweg–de Vries type equation., _J. Evolution Equations_ , 21 (2021), 625–651.
  * [13] G. M. Coclite and L. di Ruvo, Convergence results related to the modified Kawahara equation, _Boll. Unione Mat. Ital._ , 8 (2016), 265–286.
  * [14] S. Cui, S. Tao, Strichartz estimates for dispersive equations and solvability of the Kawahara equation, _J. Math. Anal. Appl._ , 304 (2005), 683–702.
  * [15] C.M. Dafermos, Asymptotic stability in viscoelasticity, _Arch. Ration. Mech. Anal._ , 37 (1970) 297–308.
  * [16] G. G. Doronin, N. A. Larkin, Kawahara equation in a quarter-plane and in a finite domain, _Bol. Soc. Parana. Mat._ , 25 (2007) 9–16.
  * [17] G. G. Doronin and N. A. Larkin, Boundary value problems for the stationary Kawahara equation, _Nonlinear Analysis_ , 69 (2008), 1655–1665.
  * [18] G. G. Doronin and N. A. Larkin, Kawahara equation in a bounded domain, _Discrete Contin. Dyn. Syst. Ser. B_ , 10 (2008) 783–799.
  * [19] H. Hirayama, Local well-posedness for the periodic higher order KdV type equations, _NoDEA Nonlinear Differential Equations Appl._ , 19 (2012) 677–693.
  * [20] H. Hasimoto, Water waves, Kagaku, 40 (1970), 401–408.
  * [21] P. Isaza, F. Linares, G. Ponce, Decay properties for solutions of fifth order nonlinear dispersive equations, _J. Differential Equations_ , 258 (2015), 764–795.
  * [22] T. Kato, Low regularity well-posedness for the periodic Kawahara equation, _Differential Integral Equations_ , 25 (2012), 1011–1036.
  * [23] T. Kawahara, Oscillatory solitary waves in dispersive media, _J. Phys. Soc. Japan_ , 33 (1972), 260–264.
  * [24] D. Lannes, The water waves problem. Mathematical analysis and asymptotics. Mathematical Surveys and Monographs, 188. American Mathematical Society, Providence, RI, 2013. xx+321 pp.
  * [25] N. A. Larkin, M.H. Simoes, The Kawahara equation on bounded intervals and on a half-line, _Nonlinear Anal. TMA_ , 127 (2015), 397–412.
  * [26] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed memory, Diff. Integral Equations, 21 (2008), 935–958.
  * [27] L. Pandolfi, _Distributed Systems with Persistent Memory. Control and Moment Problems_ , Springer-Verlag, New York, 1983.
  * [28] A. Pazy, _Semigroups of Linear Operators and Applications to Partial Differential Equations_ , Springer-Verlag, New York, 1983.
  * [29] C. F. Vasconcellos and P. N. Silva, Stabilization of the linear Kawahara equation with localized damping, _Asymptotic Analysis_ , 58 (2008), 229–252.
  * [30] C. F. Vasconcellos and P. N. Silva, Stabilization of the linear Kawahara equation with localized damping, _Asymptotic Analysis_ , 66 (2010), 119–124.
  * [31] C. F. Vasconcellos and P. N. Silva, Stabilization of the Kawahara equation with localized damping, _ESAIM Control Optim. Calc. Var._ , 17 (2011), 102–116.