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# GUILGET: GUI Layout GEneration with Transformer

###### Abstract.

Sketching out Graphical User Interface (GUI) layout is part of the pipeline of
designing a GUI and a crucial task for the success of a software application.
Arranging all components inside a GUI layout manually is a time-consuming
task. In order to assist designers, we developed a method named GUILGET to
automatically generate GUI layouts from positional constraints represented as
GUI arrangement graphs (GUI-AGs). The goal is to support the initial step of
GUI design by producing realistic and diverse GUI layouts. The existing image
layout generation techniques often cannot incorporate GUI design constraints.
Thus, GUILGET needs to adapt existing techniques to generate GUI layouts that
obey to constraints specific to GUI designs. GUILGET is based on transformers
in order to capture the semantic in relationships between elements from GUI-
AG. Moreover, the model learns constraints through the minimization of losses
responsible for placing each component inside its parent layout, for not
letting components overlap if they are inside the same parent, and for
component alignment. Our experiments, which are conducted on the CLAY dataset,
reveal that our model has the best understanding of relationships from GUI-AG
and has the best performances in most of evaluation metrics. Therefore, our
work contributes to improved GUI layout generation by proposing a novel method
that effectively accounts for the constraints on GUI elements and paves the
road for a more efficient GUI design pipeline.

###### keywords:

Keywords: Graphical User Interface, GUI arrangement graphs, deep learning,
transformer, generative model, GUI layout

Andrey Sobolevsky,*, Guillaume-Alexandre Bilodeau, Jinghui Cheng, Jin L.C. Guo
---
Polytechnique Montréal
McGill University

<EMAIL_ADDRESS>
##  1\. Introduction

The design of Graphical User Interface (GUI) is an important aspect that
affects the success of many software applications. The first step for the GUI
designers is often sketching out the interface layout with wireframes, based
on design constraints such as users’ needs and software requirements
[hartson2018ux]. These GUI layouts define the visual arrangement of elements
in a user interface, such as buttons, text fields, and containers. Creating
these layouts and variations of them manually, however, can be time-consuming.
In this paper, we explore an automated technique that can support designers
creating GUI layouts. In our approach, we capture the design constraints that
the designers have to consider through GUI arrangement graphs (GUI-AGs) and
generate graphical user interface layouts from those constraints. GUI-AGs
specify elements required in the UI design and the relationships among them,
as illustrated in Figure 1. We use GUI-AGs because they can be used to
describe the requirements with an explicit definition of components that are
part of the screen and the definition of the visual relations between those
components. These graph models offer the flexibility to automatically create
layout variations by modifying relations in the graph.

There are several technical challenges for achieving automatically
generatation of GUI layouts from GUI-AGs. One challenge is to accurately
capture the logical and semantic relationships between GUI elements, such as
hierarchical structures and functional dependencies. Another challenge is to
generate visually appealing and functional layouts that adhere to design
principles and constraints [tidwell2010designing]. In addition, the generation
process should be efficient to be used in design workflow. To address these
challenges, we propose a transformer-based approach for generating GUI layouts
from GUI-AGs. Transformer networks have recently achieved state-of-the-art
results in a wide range of natural language processing and computer vision
tasks. They are particularly well-suited for generating GUI layouts from GUI-
AGs, as they can effectively capture the dependencies between elements in the
GUI-AG and generate GUI layouts that reflect these dependencies. Our
transformer-based model takes as input a series of tokens that express the
GUI-AG relationships; it then outputs a realistic GUI layout. We demonstrate
the effectiveness of our approach through a series of experiments on real-
world datasets [li2022learning, deka2017rico] using metrics specific to this
task. Results indicated that our approach produces the most relevant GUI
layouts in regard to specified requirements.

Our contributions can be summarized with the following:

  * •

We propose a new transformer-based method to generate GUI layouts from GUI-AGs
that takes into account the GUI design constraints. Our experiment
demonstrates that it better captures the intended GUI layout compared to
previously proposed methods [johnson2018image, yang2021layouttransformer];

  * •

We introduce new loss functions and metrics for quantitative measurement of
the quality of generated GUI layouts; Those metrics can be used for future
work on similar tasks.

  * •

To encourage reproduction or replication of our study, the source code of our
experiment is publicly available with a pre-trained model at
https://github.com/dysoxor/GUILGET.

Figure 1. GUI layout generation goal is to generate a realistic GUI layout (b)
from a given GUI-AG (a).

##  2\. Related Work

GUI generation is an emerging but yet under-explored computer vision
application, especially for GUI generation based on GUI-AGs. First studies on
generating GUI designs automatically adopt generative adversarial network,
such as GUIGAN [zhao2021guigan] and GANSpiration [mozaffari2022ganspiration].
However, those approaches respectively do not generate new components style
and do not consider any specification of the design – what components should
be included or the relation between them. GUIGAN is based on a sequence of
subtrees, which are hierarchical tree structures made of components, which is
used as input and produces GUI by reusing different components based on their
style without having the ability to produce new components. On the other hand,
GANSpiration produces new design examples from existing screenshots or random
vectors from the latent space representing the screenshots. To the best of our
knowledge, no existing GUI generation methods explicitly consider the
specifications about the content, including the relationship between
components. However, there are several works focusing on image and layout
generation for more general domains that can be inspirational for GUI
generation [li2019pastegan, zhao2022high]. For example, PasteGAN
[li2019pastegan] generates images from scene graphs and image crops by taking
into account the semantic relationships between objects and their visual
appearance. Another method, proposed by Zhao et al. [zhao2022high], uses a
transformer to generate image tokens from a given scene graph and then decodes
those tokens into a plain image using a VQGAN decoder [esser2021taming]. In
comparison, our work aims to produce first GUI layouts, which is usually the
first step in the GUI design process, rather than directly generating the GUI
design. Our work can be used as a part of a GUI design generation pipeline.

Our concept of GUI-AG is inspired by scene graph, a widely used structure in
computer graphics. The task of generating layouts from scene graphs is studied
in computer graphic, but also requires knowledge from natural language
processing (NLP) to process the textual input and understand the interactions
between each element of the scene graph. Scene graphs are commonly used to
represent the structure and relationships between graphical elements in a
scene, and layout generation aims to determine the position and size of these
elements in the final layout.

There have been several approaches proposed for generating layouts from scene
graphs in the literature. For example, SG2IM by Johnson et al.
[johnson2018image] adopted a graph convolutional network (GCN) to process the
scene graph. This approach, however, is unable to capture semantic equivalence
in graphs. The work of Herzig et al. [herzig2020learning] tackle this issue by
learning canonical graph representations from the input scene graph, and it
allows to represent more complex scene graphs. Another way of processing scene
graphs is by using transformer-based model such as the ones presented in
[gupta2021layouttransformer, yang2021layouttransformer]. Transformers have
shown great performances in processing textual components and learning
relations between them. The method presented by Gupta et al.
[gupta2021layouttransformer] uses self-attention to learn contextual
relationships between layout element. In practice, it takes layout elements as
input and produces the next layout elements as output. So it has no
constraints and the model produces random new layouts. On the other hand, in
the work LayoutTransformer by Yang et al. [yang2021layouttransformer], the
transformer-based model takes as input constraints in the form of scene graphs
and predicts the layout based on relationships.

Many works have been done to generate layouts with different type of input
from scene graphs. LayoutVAE [jyothi2019layoutvae], a variational autoencoder
(VAE)-based framework for generating stochastic scene layouts, is a model that
takes as input a set of labels representing the entities that must appear in
the layout. It does not take into account any positional constraints, neither
the amount of each label as with scene graphs. The Neural Design Network (NDN)
[lee2020neural] combines a GCN with a conditional VAE to generate layouts
based on constraints given as input. In NDN, the scene graph is generated
based on the amount of components there are and some desired positional
constraints. Another work uses a generative adversarial network (GAN)
[li2019layoutgan] to generate layouts from an input of randomly-placed 2D
elements. It uses self-attention modules to refine their labels and geometric
parameters jointly to produce a layout.

In contrast to previous works, we propose a method that takes into account the
specifications of a UI design using GUI-AGs that are considering the amount of
each UI component and component position relative to other components.

##  3\. Method

Figure 2. Architecture of our model with three main components. Obj/Rel
Predictor $\mathcal{P}$ takes as input an embedding $e^{P}$ which is a
concatenation of several embeddings that describe different information about
the given GUI-AG, and it produces contextual features $f$. Layout Generator
$\mathcal{G}$ takes as input contextual features $f$, predicted sizes $s$ and
predicted positions $p$ to translate it into a layout-aware representation $c$
and bounding boxes $b$. Layout Refiner $\mathcal{R}$ uses co-attention module
with predicted bounding boxes $b$ and layout-aware representation $c$ to
improve the layout.

Our work is based on the use of transformers [vaswani2017attention] that have
been shown to be state-of-the-art in multiple natural language processing
tasks. A transformer uses the concept of self-attention, which allows to give
different weights depending on the part of the input to make predictions. GUI-
AGs can be then viewed as a natural language input to the transformer since it
is a logical sequence of relationships.

Our proposed method is based on the LayoutTransformer (LT-net)
[yang2021layouttransformer], a transformer-based model that aims to generate
diverse layouts from scene graphs of images. It consists of (1) an
object/relation predictor $\mathcal{P}$ that encodes the input scene graph
into contextual features $f$ using transformer encoder layers, (2) a layout
generator $\mathcal{G}$ that, from contextual features $f$, generates bounding
boxes $b$ with distributions matching a learned Gaussian distribution model
and layout-aware representations $c$ with transformer decoder layers, and (3)
a layout refiner $\mathcal{R}$ made of a co-attention module. Our architecture
is presented in Figure 2. Compared to LT-net, we introduced several
improvements for GUI layout generation that will be presented in the
following.

###  3.1. Building a GUI arrangement graph

Figure 3. Heuristic process of modeling GUI-AG (c) based on given layout (b)
and associated screenshot (a). All nodes represent components and all arrows
represent possible relationships. We will keep only one inside relation among
all children from a component (in blue) randomly, while the others are not
used as input (in gray). The other relations are only possible between
components that are inside the same parent and we keep only one relation
between them.

GUIs are made of two types of components: (1) widgets (e.g. button, pictogram,
text), which are leaf nodes in a GUI-AG representation and do not contain any
other component, and (2) spatial layouts (e.g. container, list item, toolbar),
which are intermediate nodes that allow to organize the structure of widgets
[zhao2021guigan]. This tree structure does not exist in images and it changes
the way GUI-AGs should be modeled and processed compared to scene graphs for
images. In contrast to our work, most of current works that are done in layout
generation for GUI ignore the tree structure and only consider widgets to
remove the complexity of organizing those widgets inside layouts.

GUI-AGs are directed graphs made of relationships that are triplets of
subject-predicate-object [johnson2018image]. A GUI does not require as much
variety of relations and objects as images. However, it has complexity since
GUI layouts have more rules and principles to be learned than in a layout from
an image [tidwell2010designing]. To train a neural network model, GUI-AGs from
ground truth layouts are required. We define five types of possible predicate
specifying the relationship among GUI components: left, right, top, bottom and
inside. We build a GUI-AG from a layout by parsing a layout description.
First, to reduce the size of the GUI-AG in order to have a smaller input in
the transformer, we keep randomly only one inside relation within a group of
components that are children to the same parent. This step is beneficial due
to memory limitations when the input is too large. To illustrate this step, in
Figure 3 we notice that CONTAINER[2] has two children, DATE PICKER[3] and
NAVIGATION[6], but only one inside relationship is kept as input to the model
and the other one is removed. This processing method does not lose information
since all components that do not have an inside relationship are considered at
the same level and are implicitly inside the parent component. We also
randomly choose a sequence of components inside the layout and determine
relationship between each pair of components within the sequence. More
precisely, as shown in Figure 3, DATE PICKER[3] and NAVIGATION[6] are inside
the same parent; in this situation, we randomly choose the sequence in the set
of children [NAVIGATION[6], DATE PICKER[3]] and then add the relation between
each component of the sequence to form triplets ([NAVIGATION[6], below, DATE
PICKER[3]] in this case). By doing so, we get a simplified input that captures
most information from the graph, as we will see in subsection 3.2. It is to
note however, that we may miss some relationships. For example, if there are
three components (a,b,c) and we keep only two relationships (a-b, b-c), the
relation between a-c can be uncertain in some cases. Finally, in GUI-AGs, the
relationships are reversible; e.g., if in the layout from Figure 3 the
NAVIGATION[6] is below DATE PICKER[3], the reverse, DATE PICKER[3] is above
NAVIGATION[6], must also be true. Hence, we can keep only one relationship
between them.

###  3.2. Object/Relation Predictor

To construct the input for our transformer model, we first convert the GUI-AG
into a sequence of relationship triplets $s_{i}$. We refer to this sequence of
relationships as $S=\\{s_{1},s_{2},...s_{T}\\}$ where $T$ is the number of
relationships. Relationships are separated by a special token SEP, and a
special token CLS is used at the beginning of the entire sequence $S$.

Instead of using directly the sequence $S$ as input in the Object/Relation
Predictor, we embed $s_{1:T}$ into $e_{1:T}^{P}$, which allows to take into
consideration different features: word embedding $e_{1:T}^{w}$ that allows to
identify the class of the object (e.g. "button" or "container") or the
relation (e.g. "inside" or "right), object ID embedding $e_{1:T}^{o}$ to
differentiate instances of the same object, relationship ID embedding
$e_{1:T}^{s}$ to separate each relationship, type of word embedding
$e_{1:T}^{t}$ to identify parts of a relationship (subject = 1, predicate = 2,
object = 3) for example the part of the the relationship "button inside
container" are "subject predicate object" and instead of writting the plain
text, we rather associate an ID to each part of the relationship to
differentiate them, and parent ID embedding $e_{1:T}^{p}$, which is a feature
that allows for each of the component to know its parent. Those features are
concatenated to form the input embedding for the object/relation predictor. It
is given by

$e_{1:T}^{P}=[e_{1:T}^{w}\bigoplus e_{1:T}^{o}\bigoplus e_{1:T}^{s}\bigoplus
e_{1:T}^{t}\bigoplus e_{1:T}^{p}].$ (1)

The object/relation predictor learns to produce three different outputs: (1)
the contextualized feature vectors $f_{1:T}$, (2) the size vectors $s_{1:T}$,
and (3) the position vectors $p_{1:T}$. The contextualized feature vectors
$f_{1:T}$ describe objects, relations and their context with features from the
input. In order to capture conceptually diverse embedding and exploiting the
co-occurrence among objects, predicates and parents, we follow the technique
used in BERT [devlin2018bert] and mask randomly words from the input that must
be predicted by the object/relation predictor. The size vectors $s_{1:T}$ are
predictions of the bounding boxes size for each object made by the
object/relation predictor. It is used as indicator later in the GUI layout
generator to generate final bounding boxes size. The position vectors
$p_{1:T}$ are predictions of bounding boxes position for each object.

Finally, in order to compute the objective function to train this part of the
network, we need to predict $\hat{e}^{P}_{1:T}$ from the features $f_{1:T}$
using a single linear layer. Indeed, since there is not ground truth for
$f_{1:T}$, we predict $\hat{e}^{P}_{1:T}$ from $f_{1:T}$ and match it with
$e^{P}_{1:T}$ to minimize the reconstruction error. The objective function for
training the module is composed of cross-entropy losses
$\mathcal{L}_{predSem}$, given by

$\mathcal{L}_{predSem}=CrossEntropy(e^{P}_{t},\hat{e}^{P}_{t}),$ (2)

for the matching word, object ID, type of word, parent ID which are all
extracted from the input GUI-AG, and regression losses given by

$\mathcal{L}_{predBox}=Regression(s_{t},\hat{s}_{t})+Regression(p_{t},\hat{p}_{t}),$
(3)

which are computed on predicted positions $\hat{p}_{1:T}$ and sizes
$\hat{s}_{1:T}$ with their corresponding ground truth positions and sizes. The
total loss of the predictor $\mathcal{L}_{pred}$ is a combination of the two
losses and given by

$\mathcal{L}_{pred}=\mathcal{L}_{predSem}+\mathcal{L}_{predBox}.$ (4)

###  3.3. Layout generator

The goal of the layout generator module is to produce layout-aware
representations $c_{1:T}$ and bounding boxes $\hat{b}_{1:T}$. This module is
made of transformer decoder layers and interprets jointly and sequentially
contextual features $f_{1:T}$, predicted bounding box sizes $s_{1:T}$ and
predicted bounding box positions $p_{1:T}$ that are computed by the
object/relation predictor module. The three given inputs are concatenated and
expressed as $e^{G}_{1:T}$. After that it is translated into diverse bounding
box output $\hat{b}_{1:T}$. A bounding box is described by its top-left corner
position in a 2D Cartesian coordinate system and its size in terms of width
and height, i.e., $\hat{b}_{t}=(x_{t},y_{t},w_{t},h_{t})$, and there is a
bounding box produced for each subject, predicate and object. The bounding box
of the predicate is the difference between the position of the object and the
one of the subject, i.e., $\hat{b}_{t}=(x_{t+1}-x_{t-1},y_{t+1}-y_{t-1})$.

To produce sequentially the bounding boxes $\hat{b}_{t}$, the features from
the input $e^{G}_{t}$ are also concatenated with the previously produced
bounding box $\hat{b}_{t-1}$. This input is not directly translated into the
bounding box but to a layout-aware representation $c_{t}$ that is used to
model a distribution in order to use Gaussian Mixture Models (GMM)
[reynolds2009gaussian]. We use this instead of directly predicting the
bounding box from layout-aware representation in order to have a generative
ability. Given a bounding box distribution, the bounding box $\hat{b}_{t}$
will be sampled from the posterior distribution
$p_{\theta_{t}}(\hat{b}_{t}|c_{t})$ knowing $c_{t}$. It can be described as
follows:

$p_{\theta_{t}}(\hat{b}_{t}|c_{t})=\sum^{K}_{i=1}\pi_{i}\mathcal{N}(\hat{b}_{t};\theta_{t,i}),$
(5)

where $i$ indicates the $i$-th distribution out of $K$ multivariate normal
distributions, $\theta_{t,i}$ are the parameters of each distribution defined
by
$(\mu_{t,i}^{x},\mu_{t,i}^{y},\sigma_{t,i}^{x},\sigma_{t,i}^{y},\rho_{t,i}^{xy})$
where $\mu$, $\sigma$ and $\rho$ denote respectively the mean, standard
deviation and the correlation coefficient, $\pi_{i}$ is a magnitude factor,
and $\mathcal{N}$ is the multivariate normal distribution.

To define the objective function of the generator $\mathcal{L}_{gen}$, we
start by defining the box reconstruction loss $\mathcal{L}_{box}$, which
maximizes the log-likelihood of the generated GMM to fit the training data
where the ground-truth bounding boxes are denoted as
$b_{t}=(x_{t},y_{t},w_{t},h_{t})$.

$\mathcal{L}_{box}=-\frac{1}{K}\log(\sum^{K}_{i=1}\pi_{i}\mathcal{N}(b_{t};\theta_{t,i})).$
(6)

To avoid the over-fitting with this loss function, the GMM distributions are
fitted to a multivariate normal distribution $Q$ using a Kullback-Leibler (KL)
divergence loss:

$\mathcal{L}_{KL}=\sum^{K}_{i=1}D_{KL}(P_{i}||Q_{i}).$ (7)

Finally, a relation consistency loss $\mathcal{L}_{rel}$ is also used since
the two previous losses focuses only on the bounding boxes. It is given by:

$\mathcal{L}_{rel}=\frac{1}{N}\sum(\Delta\hat{b}_{t}-\hat{b}^{rel}_{t})^{2},$
(8)

where $N$ denotes the number of relationships in S. It calculates the Mean
Square Error (MSE) between the box disparity of the relation we get, i.e.
$\hat{b}_{t}^{rel}$ which is the predicted bounding box for the predicate, and
the corresponding box disparity we calculate from the object and the subject
$\Delta\hat{b}_{t}=(x_{t+1}-x_{t-1},y_{t+1}-y_{t-1})$. The layout generator is
trained by minimizing the weighted sum of losses using

$\mathcal{L}_{gen}=\lambda_{box}\mathcal{L}_{box}+\lambda_{KL}\mathcal{L}_{KL}+\lambda_{rel}\mathcal{L}_{rel},$
(9)

where $\lambda_{box}$, $\lambda_{KL}$ and $\lambda_{rel}$ are weighting
factors for each corresponding loss.

###  3.4. Layout refiner

Since the bounding boxes are generated sequentially, they require refinement
in the layout in order to consider the semantic $c_{1:T}$ and the bounding box
$\hat{b}_{1:T}$. This is done in the layout refiner using the Visual-Textual
Co-Attention (VT-CAtt) [yang2021layouttransformer], which predicts the
residual $\Delta\hat{b}_{1:T}$ for updating the bounding boxes. The objective
function of this module $\mathcal{L}_{ref}$ is defined with multiple losses.
The first one is a regression loss $\mathcal{L}_{reg}$ between predicted
bounding boxes $b^{\prime}_{1:T}$ by the layout refiner and the ground truth
bounding boxes $b_{1:T}$. Another new and task-specific loss that we
implemented is the overlap between children loss $\mathcal{L}_{CC}$, which
aims to minimize the overlap of components that share the same parent in the
GUI interface. This is specific to design principles in GUI since we do not
want the components to overlap because it will hide some components on the
final interface. This loss is given by

$\mathcal{L}_{CC}=\frac{C_{1}\cap C_{2}}{\min(C_{1},C_{2})},$ (10)

where $C$ designate the area of a children. Another principle to follow is
that a children must be inside its parent. To enforce this principle, we
define the overlap between children and parent loss $\mathcal{L}_{CP}$ and we
express it as

$\mathcal{L}_{CP}=1-\frac{C\cap P}{C},$ (11)

where $C$ is the area of the children and $P$ is the area of its parent that
is defined in the input GUI-AG. The objective function on which the layout
refiner is trained is a weighted sum of those losses, that is

$\mathcal{L}_{ref}=\lambda_{reg}\mathcal{L}_{reg}+\lambda_{CC}\mathcal{L}_{CC}+\lambda_{CP}\mathcal{L}_{CP}.$
(12)

where $\lambda_{reg}$, $\lambda_{CC}$ and $\lambda_{CP}$ are weighting factors
for each corresponding loss.

##  4\. Experiments

In our experiment, we validated our proposed method by generating layouts from
GUI-AGs. We aim to evaluate how close the generated layouts are to the ground
truth layouts based on the graphs associated to them.

###  4.1. Dataset

We tested our method on the CLAY dataset [li2022learning], which is a UI
design dataset. UI layouts in RICO dataset [deka2017rico] are often noisy and
have visual mismatches hence CLAY is a dataset that improves RICO by denoising
UI layouts. It contains 59,555 human-annotated screen layouts, based on
screenshots and layouts from RICO. A total of 24 component categories (e.g.
Image, button, text) are available in layouts and 5 predicate categories
(above, below, right, left, inside) are considered in GUI-AGs.

By observing data from the CLAY dataset, we decided to remove several
irrelevant GUIs and their layouts. Firstly, we removed GUIs that contain two
or less types of components. Those screenshots are usually not representing an
application GUI but rather GUIs with, for instance, full screen image and
video screenshot that do not contain useful information for our task, as there
is a lack of component and interactions (predicates) between components. Then,
we also removed screens that contain only a navigation bar or popup for
example, for the same reason. Also usually in the dataset there are several
screenshots associated to the same application but some contain only the
navigation bar and others have the navigation bar and also some content. So we
removed the former to avoid overfitting. In practice, we achieved that by
removing GUIs in which components cover less than 25% of the total area of the
screen.

###  4.2. Evaluation metrics

To evaluate the generated layouts with respect to the ground truth, we used
the following metrics.

CP Inclusion (CPI) is a metric that captures the overlapping of children with
its parent. The metric is computed as $1-\mathcal{L}_{CP}$ (see Equation 11).
Hence, the goal of this metric is to indicate if the generated GUI layouts
tend to satisfy the UI design principle that states that children must be
fully inside its parent as we see in Table 1 with ground truth data.

CC Separation (CCS) is another metric that aims to evaluate if the UI design
principles tend to be satisfied. It is computed by $1-\mathcal{L}_{CC}$ (see
Equation 10), so the metric measure the ratio of components that does not
overlap between each other and in the same time share a common parent.

Alignment [lee2020neural] metric evaluates an important design principle that
is components must be either in center alignment or in edge alignment (i.e.
left-, right-, bottom- or top-aligned). We computed alignment with the
following:

$1-\frac{1}{N_{C}}\sum_{d}\sum_{i}\min_{j,i\neq
j}\\{\min(l(c_{i}^{d},c_{j}^{d}),m(c_{i}^{d},c_{j}^{d}),r(c_{i}^{d},c_{j}^{d}),t(c_{i}^{d},c_{j}^{d}),v(c_{i}^{d},c_{j}^{d}),b(c_{i}^{d},c_{j}^{d}))\\},$
(13)

where $N_{C}$ is the number of components, $c_{k}^{d}$ is the $k_{th}$
component of the $d_{th}$ layout and $l$, $m$, $r$, $t$, $v$ and $b$ are
alignment functions where the distance between the left, horizontal center,
right, top, vertical center and bottom are measured, respectively.

W bbox is the similarity between bounding box properties
$(x_{left},y_{top},w,h)$ distribution of the generated GUI layouts and the
ground truth GUI layouts. This is computed using Wasserstein distance and
inverting it to be a similarity, between 0 and 1, by subtracting the maximum
possible distance by the actual distance and normalizing the value. It is a
way to measure if the generated bounding boxes are as diversified as in the
ground truth data.

GUI-AG Correctness (GUI-AGC) computes the average number of correct
relationships that appears in the generated GUI layout. In practice, we
compare the input GUI-AG with the corresponding generated GUI layout and count
the number of satisfied relationships divided by the total number of
relationships.

###  4.3. Quantitative results

| Metrics
---|---
| CPI $\uparrow$ | CCS $\uparrow$ | Alignment $\uparrow$ | W bbox $\uparrow$ | GUI-AGC $\uparrow$
GT data | 1.0 | 0.987 | 1.0 | - | -
SG2IM [johnson2018image] | 0.191 | 0.974 | 0.997 | 0.81 | 0.369
LayoutTransformer [yang2021layouttransformer] | 0.392 $\pm$ 0.001 | 0.805 $\pm$ 0.002 | 0.998 $\pm$ 2E-5 |  0.834 $\pm$ 1E-4 | 0.797 $\pm$ 0.001
GUILGET (ours) |  0.592 $\pm 0.001$ | $0.623\pm 0.002$ |  0.9983 $\pm$ 2E-5 | 0.811 $\pm$ 8E-5 |  0.868 $\pm 0.001$
Table 1. Quantitative evaluation on CLAY dataset [li2022learning]. 3000
layouts are generated with each method and are compared using metrics
presented in subsection 4.2. For SG2IM [johnson2018image] as well as for
LayoutTransformer [yang2021layouttransformer], we only consider the parts of
the architectures responsible to generate the layout.

Table 1 gives the results of our method compared to SG2IM [johnson2018image]
and to LayoutTransformer [yang2021layouttransformer]. As we can observe, there
are several metrics where our model gives the best results but not with all
metrics. If we compare our model to SG2IM, we can see that in terms of CCS we
do not get as good results. However, we notice that CPI and GUI-AGC are the
worst for SG2IM. In particular, GUI-AGC shows that only $36.9\%$ of the
relations given as input to SG2IM are satisfied in the output, which means
that the design constraints are not met with this method. Moreover, we can see
with this model that the CCS is high while CPI is low, which means that SG2IM
does not organize components inside the layout but uses the whole screen, as
it is also shown in the results from Figure 5, which makes it easier to get a
high CCS since it does not learn the inside constraint. It is easy not to have
overlap between child components in that case. In other words, it is not
meaningful to have a high CCS if CPI and GUI-AGC are not also high. The
LayoutTransformer model has more understanding of predicates but it is still
worse than with our model. We can also see that there is a negative
correlation between CCS and CPI – if the model learns to place components
inside its parent, there are more possible overlaps between components inside
a layout. We want both of these metrics to be similarly high to respect both
of those GUI design constraints as we can observe it in the GT data, where all
metrics related to GUI design constraint are close to 1. W bbox metric is
similar for all of the models which is understandable since our model and
LayoutTransformer model generate bounding boxes based on distribution of
bounding boxes from the training GUI layouts. On the other hand, SG2IM is not
a generative model and aims to predict bounding boxes which leads to learn the
most common sizes and positions for different types of component.

####  4.3.1. Influence of UI category

Figure 4. Quantitative evaluation on different screen categories (a) from CLAY
dataset [li2022learning]. Evaluation metrics are applied on all 27 screen
categories separately. (b) shows the influence of number of unique type
components on evaluation metrics.

In order to see if the screen category has an impact on the performance, we
conducted an experiment where we compute each evaluation metric for each
category separately. Figure 4 (a) summarizes the results. Overall, our model
yields similar performances among different app categories. This is a
conclusive result which shows that our model is not biased toward certain
types of screen categories and is able to produce equally good GUI layout for
any of the category.

####  4.3.2. Influence of UI complexity

Similarly as with the previous experiment, we want to understand the influence
of the UI complexity (indicated by the number of unique component types in the
UI [mozaffari2022ganspiration]) on the performance. Figure 4 (b) shows that
with smaller number of unique component types most of the evaluation metrics
are better; in other words, our model achieved better performances when the UI
is less complex. Particularly, the GUI-AGC metric is inversely proportional to
the number of unique component types in the UI. For both CCS and CPI metrics,
these results are expected due to the fact that with more diverse components
that have various sizes and position standards, it becomes harder to organize
all elements inside spatial layouts. There are however two metrics performing
equally well over all number of unique component types, which are the
Alignment and W bbox metrics. This shows that our model succeeds to always
align components whether the complexity is low or high, and generated bounding
boxes have almost the same similarity in distribution with the ground truth
distribution.

###  4.4. Qualitative results

Figure 5. Qualitative comparison between our model, LayoutTransformer and
SG2IM. The same input is given for the three models. The input from the first
row has a low complexity with 3 unique component types while the second input
has a larger complexity with 5 unique component types.

Figure 5 shows two examples produced by our model, LayoutTransformer, and
SG2IM. Examples were chosen manually based on number of unique component types
inside the GUI layout. We show results for low complexity (3-4 unique
component types) and medium complexity (5-7 unique component types). We do not
show GUI With large complexity (8 or more unique component types) as it is
harder to analyze visually because of the larger number of components. The
results of Figure 5 are aligned with the quantitative results from subsection
4.3. Indeed, we can see that SG2IM has a poor semantic understanding of
relations for both cases in Figure 5; it also struggles to place components
inside their parent as we can observe in the second row from Figure 5 where
all components that are supposed to be inside the CONTAINER[3] are not and the
container which is supposed to be below the MAP[2] is actually entirely inside
it instead. We can note however that the sizes of bounding boxes and their
alignments are realistic. The LayoutTransformer shows a better understanding
of relations but is not able in the second case to place components inside its
parent as exemplified by BUTTON[4] and BUTTON[5] that are outside the
CONTAINER[3] in the second row from Figure 5. In contrast, our model respects
all the given constraints and placed correctly buttons inside the container.
Also, GUILGET generates plausible bounding boxes even though the generated
layouts are not aligned in the way it is in the ground truth. However,
information from GUI-AG is not complete enough to reproduce the same
alignment. Adding global positioning constraints on components to the GUI-AG
could be an interesting avenue to investigate.

##  5\. Conclusion

This work propose a transformer-based model that generates a GUI layout from a
given GUI-AG. Our approach is the state-of-the-art in quantitative performance
across several metrics and in visual quality. We saw that using attention
provides a higher performance than using graph convolution network in
capturing semantic of the GUI-AG. The new components from our model compared
to LayoutTransformer bring also more understanding in GUI layout constraints.
This work also introduce new loss functions and evaluation metrics specific to
this task of GUI layout generation. Future work is to generate GUI from the
layouts to complete the GUI design pipeline.


# Knowledge Graph Guided Semantic Evaluation of Language Models For User Trust

Kaushik Roy Artificial Intelligence Institute
University of South Carolina
Columbia, SC, USA
<EMAIL_ADDRESS>Tarun Garg Vedant Palit
Birla Institute of Technology
Pilani
<EMAIL_ADDRESS>Indian Institute of Technology
Kharagpur
<EMAIL_ADDRESS>Yuxin Zi Vignesh Narayanan
Artificial Intelligence Institute
University of South Carolina
Columbia, SC, USA
<EMAIL_ADDRESS>Artificial Intelligence Institute
University of South Carolina
Columbia, SC, USA
<EMAIL_ADDRESS>Amit Sheth Artificial Intelligence Institute
University of South Carolina
Columbia, SC, USA
<EMAIL_ADDRESS>
###### Abstract

A fundamental question in natural language processing is - what kind of
language structure and semantics is the language model capturing? Graph
formats such as knowledge graphs are easy to evaluate as they explicitly
express language semantics and structure. This study evaluates the semantics
encoded in the self-attention transformers by leveraging explicit knowledge
graph structures. We propose novel metrics to measure the reconstruction error
when providing graph path sequences from a knowledge graph and trying to
reproduce/reconstruct the same from the outputs of the self-attention
transformer models. The opacity of language models has an immense bearing on
societal issues of trust and explainable decision outcomes. Our findings
suggest that language models are models of stochastic control processes for
plausible language pattern generation. However, they do not ascribe object and
concept-level meaning and semantics to the learned stochastic patterns such as
those described in knowledge graphs. Furthermore, to enable robust evaluation
of concept understanding by language models, we construct and make public an
augmented language understanding benchmark built on the General Language
Understanding Evaluation (GLUE) benchmark. This has significant application-
level user trust implications as stochastic patterns without a strong sense of
meaning cannot be trusted in high-stakes applications.

###### Index Terms:

Knowledge Graph, Graph Neural Networks, Transformers

##  I Introduction

Recent studies have studied self-attention models such as transformers for
their ability to encode underlying graph structures by drawing parallels with
graph neural networks (GNNs)[1] Intuitively, there is a correspondence between
the self-attention map in the transformer and the normalized adjacency matrix
in GNNs. Also, there is a correspondence between GNN node representations and
the output value vectors from a transformer. The multiple routings of the
transformer output through layers of the transformer are similar to multiple
graph convolution aggregations in a GNN. Thus, both transformers may be an
effective way to learn graph contexts between language tokens. In this study,
we aim to test this perceived equivalence rigorously.

Do transformers encode semantic graphs between input sequence tokens? We
perform simple experiments that feed various graph path sequence inputs to
transformers (we test with multiple knowledge graphs (KGs) and language models
(LMs)) and try reconstructing the input graph from transformer outputs. In our
experiments, we find that in doing so, a high reconstruction error is observed
for certain types of graph paths, paths that require strongly typed real-world
concept level knowledge (e.g., Volvo is typically a type of high-performance
car which is, in turn, a type of car). Several previous works have performed
similar knowledge graph-based reconstruction experiments. However, they have
measured link prediction performance alone and not path predictions [2]. Link
prediction is a weak evaluation of knowledge graph semantics as the richness
of concepts in a knowledge graph comes from graph paths connecting concepts
comprising multiple relationships. Furthermore, they have not qualitatively
analyzed the results of successful and failed outcomes. In this study, we
quantitatively measure the ability of transformers to predict relationships
and concepts in knowledge graph paths. We also qualitatively inspect the paths
on which the model makes errors to evaluate their conceptual understanding
capabilities.

##  II Methdology

First, we extract masked graph paths from the knowledge graphs for processing
by the language model. Figure 1 illustrates the masked graph path extraction
process from the knowledge graph.

Figure 1: Steps 1, 2, and 3 show the process of converting the knowledge graph
links to paths. Step 4 shows the masked inputs to the language model that will
predict the masked tokens. The links are connected two make longer paths
through the use of inverse relationships, e.g., has-1.

Next, we predict the masked tokens and calculate the percentage of times the
language models assign the correct token top five prediction ranks (measured
using softmax over logits). Figure 2 illustrates this process. The final
softmax logits obtained can be ranked in order of probability values. For our
evaluation metric, we calculate the percentage of times the correct answer is
within the top five probabilities. We call this metric %Top@5.

Figure 2: The figure shows how the masked graph path inputs are processed
through the self-attention transformer models to obtain softmax logit outputs.

##  III Experiments

We extract approximately 300K knowledge graph links from the knowledge graphs
DBPedia, ConceptNet, Wiktionary, WordNet, and OpenCyc Ontology [3]. The
relationships we find are Antonym, DistinctFrom, EtymologicallyRelatedTo,
LocatedNear, RelatedTo, SimilarTo, Synonym, AtLocation, CapableOf, Causes,
CausesDesire, CreatedBy, DefinedAs, DerivedFrom, Desires, Entails,
ExternalURL, FormOf, HasA, HasContext, HasFirstSubevent, HasLastSubevent,
HasPrerequisite, HasProperty, InstanceOf, IsA, MadeOf, MannerOf,
MotivatedByGoal, ObstructedBy, PartOf, ReceivesAction, SenseOf, SymbolOf, and
UsedFor. The data can be found at this link. For the language models, we use
bert-base-uncased, bert-large, GPT-Neo small, medium, and large with 0.1B,
0.3B, 1B, 2.7B, and 6B parameters, respectively (B stands for billion) [4].

###  III-A Quantitative Results

Figure 3 shows the quantitative results. We explain the results in the figure
caption due to space limitations.

Figure 3: The X-axis denotes the number of parameters in billions, and the
Y-axis measures the %Top@5. The performance measured using %Top@5 increases
steadily with the number of model parameters. However, after a certain amount
of parameters is reached ($\sim$ 1 billion), the performance starts to flat-
line. The variance across different runs remains significant ($\sim\pm$ 5),
although it also shows a decreasing trend with increased model parameters.

###  III-B Qualitative Results

We manually inspect the knowledge graph paths at which the language models
fail, which we will call false paths. Interestingly, the false paths almost
exclusively involve knowledge of strongly typed objects and their properties
as seen in the real world. Some examples include “volvo IsA car CapableOf
slow_down”, “retrograde_motion HasContext astronomy IsA physics”, “handicapped
SimilarTo unfit RelatedTo unhealthy”, and “ultimate_frisbee IsA field_game IsA
outdoor_game”. The remaining examples are at this link. This finding is
particularly interesting as it supports third-party observations about
language models’ fundamental lack of a conceptual world model when asked about
physics-related questions (e.g., block-stacking)[5].

##  IV The GLUE Benchmark

The General Language Understanding Evaluation (GLUE) benchmark revolutionized
the evaluation of LMs with several challenging language understanding
phenomena, specifically, Textual Entailment (TE), Textual Similarity (TS), and
Natural Language Inference (NLI) [6]. The success of the GLUE benchmark
directly led to rapid advances in language understanding evaluation benchmarks
such as SuperGLUE, KILT, and BIG-BENCH, to name a few [7, 8, 9]. We identify
defining characteristics of these benchmarks and explain why such
characteristics are meaningless without further qualification by aligning with
conceptual semantics such as those found in knowledge graphs. We first
describe TE, TS, and NLI phenomena as they are currently established and
defined. Then, we report shortcomings resulting from their incomplete
definitions due to non-alignment with knowledge graph semantics.

#### Natural Language Inference (NLI)

The GLUE tasks - MNLI (Multi-genre Natural Language Inference), QNLI (Question
Answering Natural Language Inference), and WNLI (Winograd Natural Language
Inference) test NLI capabilities from varying angles. MNLI tests whether the
model can appropriately judge if a sentence logically follows from another,
i.e., logical entailment. QNLI tests similar logical entailment between
question and statement pairs - does it make logical sense to ask a follow-up
question? WNLI tests logical entailment in the presence of pronouns and the
nouns they reference.

#### Textual Entailment (TE)

The GLUE task RTE (Recognizing Textual Entailment) tests for logical
entailment similar to the NLI task MNLI. However, RTE emphasizes the meaning -
given two text fragments, whether the meaning of one can be entailed (or can
be inferred) from the other.

#### Textual Similarity (TS)

The GLUE task QQP (Quora Question Pairs) tests for the ability to assess the
semantic equivalence, measured as the similarity between a pair of questions
that appear on the social media forum Quora.

Together, the TE, TS, and NLI phenomena constitute the fundamental
“characteristics” of language understanding. Many other characteristics define
human-like language understanding, such as abstraction, analogy, and implicit
mentions. We posit that if the TE, TS, and NLI characteristics are
satisfactorily incorporated into an LM’s basic skill set, those other
characteristics can arise through compositions of this basic skill set.

##  V Constructing the Modified GLUE Benchmark

###  V-A Knowledge Graphs Augmented GLUE

We now augment the GLUE benchmark using the Knowledge Graphs (KGs) DBPedia,
ConceptNet, Wiktionary, WordNet, and the OpenCyc Ontology which consist of
semantic associations as networks of connected objects and their relationships
[10, 3, 11]. These KGs are human-curated and carry ontological commitments (a
good majority of humans that curated the knowledge agree on a common semantic
interpretation of the KGs). For the first step of building the new benchmark,
we augment the task datasets described in Section IV, using order one and
order two (one-hop and two-hop paths) associations from the aforementioned
knowledge sources. Tokenization of the data points is essential when querying
knowledge sources for paths. Therefore, we define tokenization tailored to
specific sources from which we extract knowledge paths based on manual
inspection and determination of optimal practices and heuristics. For example,
one of the tokenization techniques we use is using a sliding window on the
input data and extracting tokens of span length 1-5 as we found entity
mentions of span length 5 (e.g., Standard and Poor Index 500). We limit the
augmentation to order two paths as we find higher-order paths introduce noisy
concepts and paths to the dataset and can, therefore, hinder the semantic
correctness of the dataset.

The relationships we find are the same as described in Section III. We make
our augmented dataset available at this link.

#### Concept Understanding Evaluation Metrics

Our augmented dataset supports the evaluation metric for knowledge graph path
alignment, i.e., %Top@5, introduced in Section II as well as standard
performance scores of accuracy, F1 score, precision, and recall, both being
necessary to conclusively determine a language model’s conceptual
understanding capabilities.

##  VI Conclusion

This paper opens the black-box language models’ ability to model knowledge
graph semantics by proposing masked prediction tasks on graph paths. We do
this to understand a language model’s conceptual understanding and its
bearings on application-level user trust issues. We introduce metrics for the
evaluation of the results and also manually inspect the outcomes.

Our findings suggest that language models are models of stochastic control
processes for plausible language pattern generation. However, they do not
ascribe object and concept-level meaning and semantics to the learned
stochastic patterns such as those described in knowledge graphs. This has
significant application-level user trust implications for applications
requiring concept-level understanding (e.g., healthcare) and physical
simulations (e.g., war-time strategies). Our findings suggest that using
language models alone, which are stochastic control models, to drive high-
stake application-level decisions would be highly unsafe and irresponsible.
Finally, the paper constructs and makes public a knowledge-augmented GLUE
benchmark that can foster the development of trustworthy language models
through concept grounding in human-curated knowledge sources such as knowledge
graphs. We also touch upon how novel evaluation metrics combined with such
augmented datasets can be used to further increase the reliability and
robustness of language model evaluations with regard to their ability to
accurately encode conceptual understanding.

## Acknowledgements

This work was supported in part by the National Science Foundation under Grant
2133842, “EAGER: Advancing Neuro-symbolic AI with Deep Knowledge-infused
Learning” and was carried out under the advisement of Prof. Amit Sheth [12,
13].

## References

  * [1] E. Choi, Z. Xu, Y. Li, M. Dusenberry, G. Flores, E. Xue, and A. Dai, “Learning the graphical structure of electronic health records with graph convolutional transformer,” in Proceedings of the AAAI conference on artificial intelligence, vol. 34, pp. 606–613, 2020.
  * [2] H. Ma and D. Z. Wang, “A survey on few-shot knowledge graph completion with structural and commonsense knowledge,” arXiv preprint arXiv:2301.01172, 2023.
  * [3] R. Speer, J. Chin, and C. Havasi, “Conceptnet 5.5: An open multilingual graph of general knowledge,” in Proceedings of the AAAI conference on artificial intelligence, vol. 31, 2017.
  * [4] R. Gozalo-Brizuela and E. C. Garrido-Merchan, “Chatgpt is not all you need. a state of the art review of large generative ai models,” arXiv preprint arXiv:2301.04655, 2023.
  * [5] E. M. Bender, T. Gebru, A. McMillan-Major, and S. Shmitchell, “On the dangers of stochastic parrots: Can language models be too big?,” in Proceedings of the 2021 ACM conference on fairness, accountability, and transparency, pp. 610–623, 2021.
  * [6] A. Wang, A. Singh, J. Michael, F. Hill, O. Levy, and S. R. Bowman, “Glue: A multi-task benchmark and analysis platform for natural language understanding,” arXiv preprint arXiv:1804.07461, 2018.
  * [7] A. Wang, Y. Pruksachatkun, N. Nangia, A. Singh, J. Michael, F. Hill, O. Levy, and S. Bowman, “Superglue: A stickier benchmark for general-purpose language understanding systems,” Advances in neural information processing systems, vol. 32, 2019.
  * [8] F. Petroni, A. Piktus, A. Fan, P. Lewis, M. Yazdani, N. De Cao, J. Thorne, Y. Jernite, V. Karpukhin, J. Maillard, et al., “Kilt: a benchmark for knowledge intensive language tasks,” arXiv preprint arXiv:2009.02252, 2020\.
  * [9] A. Srivastava, A. Rastogi, A. Rao, A. A. M. Shoeb, A. Abid, A. Fisch, A. R. Brown, A. Santoro, A. Gupta, A. Garriga-Alonso, et al., “Beyond the imitation game: Quantifying and extrapolating the capabilities of language models,” arXiv preprint arXiv:2206.04615, 2022.
  * [10] S. Auer, C. Bizer, G. Kobilarov, J. Lehmann, R. Cyganiak, and Z. Ives, “Dbpedia: A nucleus for a web of open data,” in The semantic web, pp. 722–735, Springer, 2007.
  * [11] C. Matuszek, M. Witbrock, J. Cabral, and J. DeOliveira, “An introduction to the syntax and content of cyc,” UMBC Computer Science and Electrical Engineering Department Collection, 2006.
  * [12] A. Sheth, K. Roy, and M. Gaur, “Neurosymbolic ai-why, what, and how,” arXiv preprint arXiv:2305.00813, 2023.
  * [13] A. Sheth, M. Gaur, K. Roy, and K. Faldu, “Knowledge-intensive language understanding for explainable ai,” IEEE Internet Computing, vol. 25, no. 5, pp. 19–24, 2021.


#  Global solutions for a 2D BENNEY-LIN type EQUATION posed ON RECTANGLES and
on a half-strip

Nikolai A. Larkin† Departamento de Matemática, Universidade Estadual de
Maringá, Av. Colombo 5790: Agência UEM, 87020-900, Maringá, PR, Brazil
<EMAIL_ADDRESS>
###### Abstract.

We formulate on rectangles and on the right horizontal half-strip initial-
boundary value problems for a two-dimensional Benney-Lin type equation.
Existence and uniqueness of a regular solution as well as the exponential
decay rate for the norm of a regular solution have been established.

###### Key words and phrases:

Benney-Lin equation , Dispersive equations, Exponential Decay

MSC 2010:35Q53;35B35

##  1\. Introduction

We are concerned with initial-boundary value problems (IBVP) posed on
rectangles and the right horizontal half-strip for the 2D Benney-Lin-Kawahara
equation (BLK)

$u_{t}+\Delta^{2}u+\gamma\Delta u+\Delta u_{x}+uu_{x}+\eta\partial^{5}_{x}u=0$
$None$

which is a two-dimensional analog of the well-known Benney-Lin equation, [3,
4, 7, 13, 14, 16, 20],

$u_{t}+\eta D^{5}_{x}u+\beta D^{4}_{x}u+\alpha D^{3}_{x}u+\gamma
D^{2}_{x}u+uu_{x}=0.$ $None$

In turn, (1.1) includes as special cases the KdV, Kuramoto-Sivashinsky,
Kawahara equations., [5, 6, 10, 15, 31]. The theory of the Cauchy problem for
(1.2) and other dispersive equations has been extensively studied and is
considerably advanced today [7, 11, 17, 30]. In recent years, results on IBVPs
for dispersive equations both in bounded and unbounded domains have appeared
[8, 9, 12, 15, 20, 21, 22, 23, 26, 28]. It was discovered in [21, 25] that the
KdV and Kawahara equations have an implicit internal dissipation. This allowed
to prove exponential decay of small solutions in bounded domains without
adding any artificial damping term. Later, this effect was proved for a wide
class of dispersive equations of any odd order with one space variable [15].
Publications on dispersive multy-dimensional equations of a higher order (such
as the KZK equation) appeared quite recently and were concerned with the
existence of weak solutions, [14], and physical motivation [13].

Our work here was motivated by [31] where stability of solutions due to
presense of higher order terms has been studied. Coefficient $\gamma$ in (1.1)
may be positive or negative depending on wavenumbers of long waves considered.
The condition $\gamma>0$ is considered to be instability condition, hence to
guarantee stability of the system, we must take into account the fourth-order
terms. On the other hand, the condition $\gamma<0$ guarantees stability while
the fourth-order terms give an additional damping effect. Differently from
[31], we added the fifth-order Kawahara term putting $\eta=-1$.

We study (1.1) on rectangles

$D=\left\\{(x,y)\in\mathbb{R}^{2}:\quad x\in(0,L),\quad
y\in(0,B),\quad\partial D\right\\},$

where $\partial D$ is a boundary of $D$. For $\gamma>0$, we establish the
existence and uniqueness of global regular solutions for $t\in(0,T),$ where
$T$ is an arbitrary positive number, without smallness conditions for the
initial data. More precisely, we formulate in Section 2 IBVP (2.1)-(2.3). In
order to demonstrate existence of global regular solutions, we exploit the
Faedo-Galerkin method. Estimates, independent of the parameter of
approximations $N$, permit us to establish the existence of regular solutions
for the original problem (2.1)-(2.3). We prove these estimates in Section 3.

In Section 4, we pass to the limit as $N\to\infty$ and obtain a global regular
solution of (2.1)-(2.3). In Section 5, we prove uniqueness of a regular
solution. Finally, in Section 6, we establish the exponential decay rate of
small solutions both for $\gamma>0$ and for $\gamma\leq 0.$ Section 7 contains
Conclusions.

##  2\. Notations and Auxiliary Facts

Let $B,L$ be positive numbers and
$x\in(0,L);y\in(0,B);D=D(x,y)\in\mathbf{R^{2}}.$ We use the standard notations
of Sobolev spaces $W^{k,p}$, $L^{p}$ and $H^{k}$ for functions and the
following notations for the norms [1, 2]:

$\mathbf{R}^{+}=\\{t\in\mathbf{R}^{1};\;t>0\\},\;\;\|f\|^{2}=(f,f)=\int_{0}^{L}\int_{0}^{B}|f|^{2}dxdy,$
$\|f\|^{p}_{L^{p}}=\int_{0}^{L}\int_{0}^{B}|f|^{p}dxdy,\;\;\|f\|_{W^{k,p}}=\sum_{0\leq\alpha\leq
k}\|D^{\alpha}_{x}f\|_{L^{p}},\;$
$D^{\alpha}_{x}=\frac{d^{\alpha}}{dx^{\alpha}},\;D^{\alpha}_{y}=\frac{d^{\alpha}}{dy^{\alpha}},\;\;D_{x}=D^{1}_{x},\;D^{0}_{x}u=u.$

We will use also the standad notations:

$D_{x}f=f_{x},\;D_{x}^{2}f=f_{xx},\;\frac{\partial}{\partial
t}f=f_{t},\;D_{y}f=f_{y}.$

When $p=2$, $W^{k,p}=H^{k}$ is a Hilbert space with the scalar product

$((u,v))_{H^{k}}=\sum_{|j|\leq
k}(D^{j}u,D^{j}v),\;\|u\|_{\infty}=\|u\|_{L^{\infty}(D)}=ess\sup_{(D)}|u(x)|.$

We use the notation $H_{0}^{k}(D)$ to represent the closure of
$C_{0}^{\infty}(D)$, the set of all $C^{\infty}$ functions with compact
support in $(D)$, with respect to the norm of $H^{k}(D)$.

######  Lemma 2.1 (Steklov’s Inequality [29]).

Let $v\in H^{1}_{0}(0,L).$ Then

$\frac{\pi^{2}}{L^{2}}\|v\|^{2}(t)\leq\|v_{x}\|^{2}(t).$

######  Lemma 2.2 (Differential form of the Gronwall Inequality).

Let $I=[t_{0},t_{1}]$. Suppose that functions $A,B:I\to\mathbf{R}$ are
integrable and a function $A(t)$ may be of any sign. Let $u:I\to\mathbf{R}$ be
a differentiable function satisfying

$u_{t}(t)\leq A(t)u(t)+B(t),\text{ for }t\in I\text{ and }\,\,u(t_{0})=u_{0},$

then

$u(t)\leq
u_{0}e^{\int_{t_{0}}^{t}A(\tau)\,d\tau}+\int_{t_{0}}^{t}e^{\int_{t_{0}}^{s}A(r)\,dr}B(s)\,ds.$

.

###### Proof.

Multiply the last inequality by the integrating factor
$e^{\int_{t_{0}}^{s}A(r)\,dr}$ and integrate from $t_{0}$ to $t$. ∎

The next Lemmas will be used in estimates:

######  Lemma 2.3.

(See [27], p. 125.) Suppose $u$ and $D^{m}u$, $m\in\mathbb{N},$ belong to
$L^{2}(0,L)$. Then for the derivatives $D^{i}u$, $0\leq i<m$, the following
inequality holds:

$\|D^{i}u\|\leq
A_{1}\|D^{m}u\|^{\frac{i}{m}}\|u\|^{1-\frac{i}{m}}+A_{2}\|u\|,$

where $A_{1}$, $A_{2}$ are constants depending only on $L$, $m$, $i$.

######  Lemma 2.4 (See: [19] ).

i) For all $u\in H^{1}_{0}(D)$

$\|u\|^{2}_{L^{4}(D)}\leq 2\|u\|_{L^{2}(D)}\|\nabla u\|_{L^{2}(D)}.$

ii) For all $u\in H^{1}(D)$

${\|u\|}_{L^{4}(D)}^{2}\leq C_{D}{\|u\|}_{L^{2}(D)}{\|u\|}_{H^{1}(D)},$

where the constant $C_{D}$ depends on a way of continuation of $u\in H^{1}(D)$
as $\tilde{u}(\mathbb{R}^{2})$ such that $\tilde{u}(D)=u(D).$

######  Lemma 2.5 (see [20], Lemma 5.).

Let $f(t)$ be a continuous positive function such that $f^{\prime}(t)$ is a
measurable, integrable function and

$f^{\prime}(t)+(\alpha-kf^{n}(t))f(t)\leq 0,\;t>0,\;n\in\mathbf{N};$ $None$
$\alpha-kf^{n}(0)>0,\;k>0.$ $None$

Then

$f(t)<f(0)$

for all $t>0$.

##  3\. Formulation of the problem

Let $T,L,B$ be arbitrary real positive numbers;

$\displaystyle D=\left\\{(x,y)\in\mathbb{R}^{2}:\quad x\in(0,L),\quad
y\in(0,B)\right\\};$ $\displaystyle Q_{t}=D\times(0,t),\quad t\in(0,T).$

Consider in $Q_{T}$ the following IBVP:

$\displaystyle Lu\equiv u_{t}+\Delta^{2}u+\Delta u+\Delta
u_{x}+uu_{x}-\partial^{5}_{x}u=0\quad\textrm{in}\quad Q_{t};$ (3.1)
$\displaystyle u|_{\partial D}=u_{yy}(x,0)=u_{yy}(x,B)=u_{x}(0,y)$
$\displaystyle=u_{x}(L,y)=u_{xx}(L,y)=0;$ (3.2) $\displaystyle
u(x,y,0)=u_{0}(x,y),\quad(x,y)\in D.$ (3.3)

Here ${\Delta}=\partial_{x}^{2}+\partial_{y}^{2}$. We adopt the usual
notations $H^{k}$ for $L^{2}$-based Sobolev spaces; $\|\cdot\|$ and
$(\cdot,\cdot)$ denote the norm and the scalar product in $L^{2}(D)$,
${|\nabla u|}^{2}=u_{x}^{2}+u_{y}^{2}$.

##  4\. Existence Theorem

######  Theorem 4.1.

Let $T,B,L$ be arbitrary real positive numbers. Given $u_{0}(x,y)$ such that

$\displaystyle u_{0}\in H^{4}(D),\quad\partial_{x}^{5}u_{0}\;\in L^{2}(D),$
$\displaystyle u_{0}(0,y)=u_{0x}(0,y)=u_{0}(L,y)=u_{0x}(L,y)=u_{0xx}(L,y)$
$\displaystyle=u_{0}(x,0)=u_{0}(x,B)=u_{0yy}(x,0)=u_{0yy}(x,B)=0,$
$\displaystyle J_{w}=\int_{D}\big{[}|\Delta^{2}u_{0}|^{2}+|\Delta
u_{0}|^{2}+u^{2}_{0}u^{2}_{0x}+|D^{5}_{0x}|^{2}\big{]}dxdy<\infty,$

then there exists a unique regular solution of (3.1)-(3.3):

$\displaystyle u\in L^{\infty}(0,T;H^{2}(D))\cap L^{2}(0,T;H^{4}(D)),$
$\displaystyle\partial_{x}^{5}u\;\in L^{2}(0,T;L^{2}(D)),u_{t}\in
L^{\infty}(0,T;L^{2}(D))\cap L^{2}(0,T;H^{2}(D)).$

###### Proof.

Approximate Solutions.

To prove the existence part of this theorem, we use the Faedo-Galerkin Method
as follows:
for all $N$ natural, we define an approximate solution of (3.1)-(3.3) in the
form

$u^{N}(x,y,t)=\sum^{N}_{j=1}\omega_{j}(y)g_{j}(x,t),$ (4.1)

where $\omega_{j}(y)$ are orthonormal in $L^{2}(0,L)$ eigenfunctions of the
following problem:

$\displaystyle-D^{2}_{y}\omega=\lambda_{j}\omega_{j}(y),\;y\in(0,B);\;\omega_{j}(0)=\omega_{j}(B)=0$
(4.2)

and $g_{j}(x,t)$ are solutions to the following initial- boundary value
problem for the system of N generalized KdV equations:

$\displaystyle\frac{\partial}{\partial
t}g_{j}(x,t)+\sum_{l,k=1}^{N}a_{lkj}g_{l}(x,t)g_{kx}(x,t)+(\Delta
u^{N},\omega_{j})$
$\displaystyle+(\Delta^{2}u^{N},\omega_{j})-\partial^{5}_{x}g_{j}(x,t)+(\Delta
u^{N}_{x},\omega_{j})=0,$ $\displaystyle
g_{j}(0,t)=g_{jx}(0,t)=0,\;g_{j}(x,0)=u_{0j}(x),$ (4.3)

where

$\displaystyle
a_{klj}=\int_{0}^{L}\omega_{k}(y)\omega_{l}(y)\omega_{j}(y)\,dy,\;j,k,l=1,...,N;$
$\displaystyle u_{0j}(x)=\int_{0}^{L}u_{0}(x,y)\omega_{j}(y)dy.$

Solvability of (4.3) (at least local in t) follows from [18]. Hence, our goal
is to prove necessary a priori estimates, uniform in $N$, which will permit us
to pass to the limit in (4.1) as $N\to\infty$ and to establish the existence
result. We assume first that a function $u_{0}$ is sufficiently smooth to
ensure calculations. Exact conditions for $u_{0}$ will follow from a priori
estimates for $u^{N}$ independent of $N$ and usual compactness arguments.

ESTIMATE I. Multiplying the $j$-equation of (4.3) by $g_{j}(x,t)$, summing
over $j=1,..,N$ and integrating the result with respect to $x$ over $R^{+}$,
we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\|u^{N}\|^{2}(t)+(|u^{N}|^{2},u_{x}^{N})(t)+(u^{N},\partial^{3}_{x}u^{N})(t)+(\Delta
u^{N},u^{N})(t)$
$\displaystyle+(\Delta^{2}u^{N},u^{N})(t)-(u^{N},\partial^{5}_{x}u^{N})(t)+(u^{N},\partial^{2}_{y}u_{x}^{N})(t)=0.$

In our calculations we will drop the index $N$ where this is not ambiguous.
Integrating by parts the last equality, we get

$\frac{d}{dt}\|u\|^{2}(t)+\|\Delta
u\|^{2}(t)+\int^{L}_{0}u_{xx}^{2}(0,y,t)\,dy\leq\|u\|^{2}(t).$ (4.4)

Applying Lemma 2.2 to the inequality

$\frac{d}{dt}\|u\|^{2}(t)\leq\|u\|^{2}(t),$

we get

$\|u^{N}\|^{2}(t)\leq e^{T}\|u_{0}\|^{2}.$ (4.5)

Returning to (4.4), we find that for $N$ sufficiently large and $\forall
t\in(0,T)$

$\|u^{N}\|^{2}(t)+\int_{0}^{t}[\|\Delta
u^{N}\|^{2}(\tau)+\int^{L}_{0}|u^{N}_{xx}|^{2}(0,y,\tau)\,dy]d\tau\leq
e^{T}\|u_{0}\|^{2}.$ (4.6)

######  Lemma 4.2 ([22] Lemma 3.1.).

Let $f\in H^{4}(D)\cap H^{1}_{0}(D)$ and $f|_{\partial D}$ satisfies (4.2).
Then

$\displaystyle\|\nabla f\|^{2}\geq a\|f\|^{2},\;\;a^{2}\|f\|^{2}\leq\|\Delta
f\|^{2},\;\;a\|\nabla f\|^{2}\leq\|\Delta f\|^{2},$ (4.7) $\displaystyle
a^{2}\|\Delta
f\|^{2}\leq\|\Delta^{2}f\|^{2},\;\;\text{where}\;a=\frac{\pi^{2}}{L^{2}}+\frac{\pi^{2}}{B^{2}}=\frac{\pi^{2}(B^{2}+L^{2})}{L^{2}B^{2}}.$
(4.8)

######  Lemma 4.3.

Let $f\in H^{2}(D)\cap H^{1}_{0}(D).$ Then

$\sup_{D}u^{2}(x,y)\leq C_{s}\|\Delta
u\|^{2},\;\text{where}\;C_{s}=1+\frac{1}{a}+\frac{1}{a^{2}}.$ (4.9)

###### Proof.

Dropping variable $t$ and making use of Lemma 4.2, we can write

$\displaystyle\sup_{D}u^{2}(x,y)\leq\sup_{x\in(0,L)}\sup_{y\in(0,B)}|\int_{0}^{y}\frac{\partial}{\partial
s}u^{2}(s,x)ds|$
$\displaystyle\leq\sup_{x\in(0,L)}\int_{0}^{B}\\{u_{y}^{2}(x,y)+u^{2}(x,y)\\}dy$
$\displaystyle\leq\int_{0}^{L}\int_{0}^{B}\\{|2u_{xy}(x,y)u_{y}(x,y)+2u_{x}(x,y)u(x,y)|\\}dxdy$
$\displaystyle\leq\|u_{xy}\|^{2}+\|\nabla
u\|^{2}+\|u\|^{2}\leq(1+\frac{1}{a}+\frac{1}{a^{2}})\|\Delta u\|^{2}.$

∎

The proof of Lemma 4.3 is complete.

ESTIMATE III. Taking into account the structure of $u^{N}(x,y,t)$, consider
the scalar product

$-2(\partial^{2}_{y}u^{N},[u^{N}_{t}+\Delta^{2}u+\Delta
u^{N}+u^{N}u_{x}^{N}+\Delta u^{N}_{x}-\partial^{5}_{x}u^{N}])(t)=0.$

Acting as by proving Estimate II and dropping the index $N$, we come to the
following equality:

$\displaystyle\frac{d}{dt}\|u^{2}_{y}\|(t)+\int^{L}_{0}u_{xxy}^{2}(0,y,t)\,dy-2(\Delta^{2}u,D^{2}_{y}u)(t)$
$\displaystyle=2(u_{yy},uu_{x})(t).$ (4.10)

We estimate

$I_{1}=-2(\Delta^{2}u,D^{2}_{y}u)(t)=2([|D^{3}_{y}u|^{2}+2u^{2}_{xyy}+u^{2}_{xxy}])(t),$

$I_{2}=2(u_{yy},uu_{x})(t)=(u_{yy},(u^{2})_{x})(t)\leq+2|(u_{xy},uu_{y})|(t)$
$\leq+\epsilon\|u_{xy}\|^{2}(t)+\frac{sup_{D}u^{2}(x,y)}{\epsilon}\|u_{y}\|^{2}(t)$
$\leq\epsilon\|u_{xy}\|^{2}(t)+\Big{[}2kC^{1/2}_{s}\|\Delta
u\|(t)+\frac{C_{s}}{\epsilon}\Big{]}\|u_{y}\|^{2}(t)$
$\leq\epsilon\|u_{xy}\|^{2}(t)+\frac{C}{\epsilon}[1+\|\Delta
u\|^{2}(t)]\|u_{y}\|^{2}(t).$

Substituting $I_{1}-I_{2}$ into (4.10), taking $\epsilon>0$ sufficiently small
and using (4.5)-(4.6), we come to the inequality

$\displaystyle\dfrac{d}{dt}{\|u_{y}\|}^{2}(t)+\displaystyle\int_{0}^{L}u^{2}_{yxx}(0,y,t)\,dy+2(|D^{3}_{y}u|^{2}+2u^{2}_{xyy}+u^{2}_{xxy}])(t)$
$\displaystyle\leq\epsilon\|u_{xy}\|^{2}(t)+\frac{C}{\epsilon}[1+\|\Delta
u\|^{2}(t)]\|u_{y}\|^{2}(t).$ (4.11)

Taking $\epsilon$ sufficiently small and making use of the Gronwall lemma and
Estimates I, II, we find

$\displaystyle\|u_{y}\|^{2}(t)$ $\displaystyle\leq$
$\displaystyle\|u_{0y}\|^{2}e^{C(\|u_{0}\|,T)\displaystyle\int_{0}^{t}[1+\|\Delta
u\|^{2}(\tau)]d\tau.}$ $\displaystyle\leq$
$\displaystyle\|u_{0y}\|^{2}e^{C(\|u_{0}\|,T)}\leq
C(T,\|u_{0}\|_{H^{1}_{0}}).$

Integrating (4.11) over $(0,t)$ gives

$\displaystyle\|u^{N}_{y}\|^{2}(t)+\displaystyle\int_{0}^{t}2([\|D^{3}_{y}u^{N}\|^{2}+2\|u^{N}_{xyy}\|^{2}+\|u^{N}_{xxy}\|^{2}])(\tau)$
$\displaystyle+[\|u^{N}_{xy}\|^{2}(\tau)+\|u^{N}_{yy}\|^{2}(\tau)]\,d\tau$
$\displaystyle+\displaystyle\int_{0}^{t}\int^{L}_{0}|u^{N}_{xxy}(0,y,\tau)|^{2}\,dyd\tau\leq
C(T,\|u_{0}\|)\|u_{0y}\|^{2}).$ (4.12)

ESTIMATE IV. Dropping the index $N$, transform the scalar product

$2(\partial^{4}_{y}u^{N},[u^{N}_{t}+\Delta^{2}u^{N}+\Delta
u^{N}+u^{N}u^{N}_{x}+\Delta u^{N}_{x}-\partial^{5}_{x}u^{N}])(t)=0$

into the following equality:

$\displaystyle\dfrac{d}{dt}\|u_{yy}\|^{2}(t)$
$\displaystyle+\displaystyle\int^{L}_{0}|\partial^{2}_{y}u_{xx}(0,y,t)|^{2}\,dy++2(D^{4}_{y}u,\Delta^{2}u)(t)$
(4.13) $\displaystyle=-2(\Delta u,D^{4}_{y}u)(t)-2(D^{4}_{y}u,uu_{x})(t).$

We estimate

$I=2(D^{4}_{y}u,\Delta^{2}u)(t)=2(D^{4}_{y}u,[D^{4}_{y}+2D^{2}_{y}D^{2}_{x}+D^{4}_{x}]u)(t)=I_{1}+I_{2}+I_{3},$

where

$\displaystyle I_{1}=2(D^{4}_{y}u,D^{4}_{y}u)(t)=2\|D^{4}_{y}u\|^{2})(t),$
$\displaystyle
I_{2}=(D^{4}_{y}u,2D^{2}_{y}D^{2}_{x}u)(t)=4\|D^{3}_{y}u_{x}\|^{2})(t),$
$\displaystyle
I_{3}=2(D^{4}_{y}u,D^{4}_{x}u)(t)=2\|D^{2}_{y}D^{2}_{x}u\|^{2})(t).$

Hence

$\displaystyle
I=2[\|D^{4}_{y}\|^{2}+2\|D^{3}u_{x}\|^{2}+\|D^{2}_{y}D^{2}_{x}u\|^{2}](t).$

For any positive $\epsilon$, integrating by parts, we obtain

$\displaystyle
I_{4}=2(D^{4}_{y}u,uu_{x})=-2(D^{3}_{y}u_{x},uu_{y})\leq\epsilon\|D^{3}_{y}u_{x}\|^{2})$
$\displaystyle+\frac{1}{\epsilon}(u^{2},u^{2}_{y})\leq\epsilon\|D^{3}_{y}u_{x}\|^{2})+\frac{1}{\epsilon}\sup_{D}u^{2}(x,y)\|u_{y}\|^{2}(t)$
$\displaystyle\leq\epsilon\|D^{3}_{y}u_{x}\|^{2})+\frac{1}{\epsilon}C_{s}\|\Delta
u\|^{2}(t)\|u_{y}\|^{2}(t).$

Taking $\epsilon>0$ sufficiently small and substituting $I_{1}-I_{4}$ into
(4.13), we obtain

$\displaystyle\dfrac{d}{dt}\|u_{yy}\|^{2}(t)+2([\|D^{4}_{y}\|^{2}+2\|D^{3}u_{x}\|^{2}+\|D^{2}_{y}D^{2}_{x}u\|^{2}])(t)$
$\displaystyle+\displaystyle\int_{0}^{L}{|D_{y}^{2}u_{xx}(0,y,t)|}^{2}dy\leq
C(k)\left[\|\Delta u\|^{2}(t)+1\right]\|u_{y}\|^{2}(t).$ (4.14)

The previous estimates and the Gronwall lemma yield

$\displaystyle\|u^{N}_{yy}\|^{2})(t)+\displaystyle\int_{0}^{t}\\{2([\|D^{4}_{y}u^{N}\|^{2}+2\|D^{3}_{y}u^{N}_{x}\|^{2}+\|D^{2}_{y}D^{2}_{x}u^{N}\|^{2}])(\tau)\,d\tau$
$\displaystyle+\displaystyle\int_{0}^{t}\displaystyle\int_{0}^{L}{|D_{y}^{2}u^{N}_{xx}(0,y,\tau)|}^{2}dyd\tau\leq
C(\|u_{0y}\|^{2}+\|u_{0yy}\|^{2}),$ (4.15)

where the constant $C$ does not depend on $N,L.$

ESTIMATE V. To estimate $u^{N}_{t}$, we differentiate (4.3) with respect to
$t$, multiply the $j$-equation of the resulting system by $g_{jt}$, sum up
over $j=1,...,N$ and integrate over $D$. Calculations, similar to those
exploited in Estimate II, imply

$\displaystyle\frac{d}{dt}\|u_{t}\|^{2}(t)+\int_{0}^{L}u_{txx}^{2}(0,y,t)\,dy+2\|\Delta
u_{t}\|^{2}(t)$ $\displaystyle+2(u_{t},\Delta
u_{t})(t)=2([uu_{x}]_{t},u_{t})(t)=-(u_{xt},[u^{2}]_{t})(t)-2(u_{xt},uu_{t})(t)$
$\displaystyle\leq\epsilon\|u^{2}_{xt}\|(t)+[2ksup_{D}|u(x,t)|+\frac{1}{\epsilon}]\|u_{t}\|^{2}(t).$
(4.16)

Taking $\epsilon>0$ sufficiently small and making use of Estimates I-IV and
the Gronwall lemma, we find

$\displaystyle\|u_{t}\|^{2}(t)$ $\displaystyle\leq$ $\displaystyle
C(T,\|u_{0}\|)J_{w}.$

Returning to (4.16), we deduce

$\displaystyle\|u^{N}_{t}\|^{2}(t)+\int_{0}^{t}\int_{0}^{L}|u^{N}_{xxs}(0,y,s)|^{2}\,dyds$
$\displaystyle+\int_{0}^{t}\|\Delta u^{N}_{s}\|^{2}])(s)\,ds\leq C(T)J_{w},$
(4.17)

where

$J_{w}=\int_{D}[|\Delta^{2}u_{0}|^{2}+|\Delta
u_{0}|^{2}+u^{2}_{0}u^{2}_{0x}+|D^{5}_{0x}|^{2}]dxdy<\infty.$ (4.18)

ESTIMATE VI.

From (4.5),(4.6),(4.12),(4.15),(4.17), it follows that

$\displaystyle u^{N},\;u^{N}_{t}\in L^{\infty}(0,T;L^{2}(D))\cap
L^{2}(0,T;H^{2}_{0}(D));$ (4.19) $\displaystyle D^{2}_{y}u^{N},\;\nabla
D^{2}_{y}u^{N},\;\nabla D^{3}_{y}u^{N},\;D^{2}_{y}D^{2}_{x}u^{n}\in
L^{2}(0,T;(D))$ (4.20)

and these inclusions do not depend on $N.$ Making use of (4.19)-(4.20), we
find that

$u^{N}\in L^{\infty}(0,T;H^{2}_{0}(D))$ (4.21)

uniformly on $N$.
Estimates (4.19)-(4.21) make it possible to pass to the limit as $N\to\infty$
in (4.1) and to prove thee existence of a weak solution of (3.1)-(3.3) in the
following sense:

$\displaystyle\int_{0}^{T}\\{([u_{t}+D^{4}_{y}u+2D^{2}_{x}D^{2}_{y}u+\Delta
u-D^{2}_{y}u],\phi)(t)$
$\displaystyle+(D^{2}_{x}u,D^{3}_{x}\phi)(t)+(D^{2}_{x}u,D^{2}_{x}\phi)(t)-(D^{2}_{x}u,\phi_{x})(t)\\}dt.$
(4.22)

Here, $u(x,y,t)$ satisfies the same inclusions (4.19)-(4.21) as $u^{N}$ and
$\phi(x,y,t)$ is an arbitrary function from the class: $\phi\in
L^{\infty}(0,T;L^{2}(D))\cap L^{2}(0,T;H^{3}_{0}(D)).$ Taking into account
(4.19)-(4.21), for any $t_{0}\in(0,T)$ nonsingular, we can write

$\displaystyle([u_{t}+D^{4}_{y}u+2D^{2}_{x}D^{2}_{y}u+\Delta
u-D^{2}_{y}u],\phi)(t_{0})$
$\displaystyle+(D^{2}_{x}u,D^{3}_{x}\phi)(t_{0})+(D^{2}_{x}u,D^{2}_{x}\phi)(t_{0})-(D^{2}_{x}u,\phi_{x})(t_{0})=0.$
(4.23)

In turn, for any $t\in(0,T)$ nonsingular, (4.23) may be rewritten as a
distribution in $D$:

$\displaystyle D^{5}_{x}u-D^{4}_{x}u-D^{3}_{x}u$
$\displaystyle=u_{t}+D^{4}_{y}u+2D^{2}_{x}D^{2}_{y}u+D^{2}_{y}u_{x}+\Delta
u+uu_{x}\equiv f(x,y).$ (4.24)

Since $u$ is a limit of $u^{N}$ and satisfies (4.19)-(4.21), then $f\in
L^{2}(0,T;L^{2}(D)).$ By Lemma 2.3,

$\displaystyle\|D^{4}_{x}u\|\leq
C\|D^{5}_{x}u\|^{4/5}\|u\|^{1/5}\leq\delta\|D^{5}_{x}u\|+C_{4}(\delta)\|u\|,$
$\displaystyle\|D^{3}_{x}u\|\leq
C\|D^{5}_{x}u\|^{3/5}\|u\|^{2/5}\leq\delta\|D^{5}_{x}u\|+C_{3}(\delta)\|u\|,$
(4.25)

where $\delta$ is an arbitrary positive number. Substituting (4.25) into
(4.24), we get

$\|D^{5}_{x}u\|(1-2\delta)\leq f+C(\delta)\|u\|.$

Taking $\delta=\frac{1}{4},$ by (4.24),

$\|D^{5}_{x}u\|,\|D^{4}_{x}u\|,\|D^{3}_{x}u\|\in L^{2}(0,T;L^{2}(D)).$ (4.26)

Returning to (4.24), we find that a weak soltuion of(3.1)-(3.3) is such that

$u\in L^{\infty}(0,T;H^{2}_{0}(D)),\;\Delta^{2}u,D^{5}_{x}u\in
L^{2}(0,T;L^{2}(D)).$ (4.27)

Hence, (4.22) can be rewritten as

$\int_{0}^{t}([u_{\tau}+\Delta^{2}u+\Delta u_{x}+\Delta
u+uu_{x}],\phi)(\tau)d\tau=0,\;t\in(0,T),$ (4.28)

where $\phi(x,y,t)$ is an arbitrary function from $L^{2}(Q_{t}).$

##  5\. Uniqueness

Let $u_{1}$ and $u_{2}$ be distinct solutions of (3.1)-(3.3) and
$z=u_{1}-u_{2}$. Then $z(x,y,t)$ satisfies the following initial-boundary
value problem:

$\displaystyle Lz=z_{t}+\dfrac{1}{2}(u_{1}^{2}-u_{2}^{2})_{x}+\Delta
z_{x}-D_{x}^{5}z$ $\displaystyle+\Delta z+\Delta^{2}z=0\ \textrm{in}\ Q_{t};$
(5.1) $\displaystyle z(0,y,t)=z_{x}(0,y,t)=z_{x}(L,y,t)=z_{xx}(L,y,t)$
$\displaystyle=z(x,0,t)=z(x,B,t)=0,;\;\;y\in(0,L),\;x>0,\;t>0;$ (5.2)
$\displaystyle z(x,y,0)=0,\quad(x,y)\in D.$ (5.3)

From the scalar product

$2(Lz,z)(t)=0,$ (5.4)

acting in the same manner as by the proof of Estimate II, we obtain

$\displaystyle\frac{d}{dt}\|z\|^{2}(t)+2\|\Delta
z\|^{2}(t)+\int_{0}^{B}z^{2}{xx}(0,y,\tau)dy+2(\Delta z,z)(t)$
$\displaystyle=-([u_{1}^{2}-u_{2}^{2}]_{x},z)(t).$ (5.5)

For an arbitrary positive $\delta$, we estimate

$I_{1}=2(\Delta z,z)\leq\delta\|\Delta z\|^{2}+\frac{1}{\delta}\|z\|^{2},$
$I_{2}=-([u_{1}^{2}-u_{2}^{2}]_{x},z)=(z[u_{1}+u_{2}],z_{x})\leq\frac{1}{2}\delta\|z_{x}\|^{2}+\frac{1}{2\delta}\|[u_{1}+u_{2}]z\|^{2}.$

By Steklov‘s Lemma,

$I_{3}=\|z_{x}\|^{2}\leq a\|z_{xx}\|^{2}\leq a\|\Delta z\|^{2}$

and

$I_{4}=\frac{1}{2\delta}\|[u_{1}+u_{2}]z\|^{2}\leq\frac{1}{\delta}sup_{D}|u_{1}^{2}+u_{2}^{2}|\|z\|^{2}.$

Making use of Lemma 4.3, we find that

$I_{5}=\frac{1}{\delta}sup_{D}|u_{1}^{2}+u_{2}^{2}|\|z\|^{2}\leq
C(\delta)[\|\Delta u_{1}\|^{2}+\|\Delta u_{2}\|^{2}]\|z\|^{2}.$

Substituting $I_{1}-I_{5}$ into (5.5) and taking $\delta>0$ sufficiently
small, we come to the inequality

$\dfrac{d}{dt}\|z\|^{2}(t)\leq C(k)\sum_{i=1}^{2}[1+\|\Delta
u_{i}\|^{2}(t)]\|z\|^{2}(t).$

Taking into account that by (4.27) $[1+\|\Delta u_{i}\|^{2}(t)]\in L^{1}(0,T)$
$(i=1,2),$ we get $\|z\|(t)\equiv 0$ for $a.e.\>t\in(0,T).$ This proves
uniqueness of a regular solution of (3.1)-(3.3) and completes the proof of
Theorem 3.1.

∎

######  Remark 1.

Assertions of Theorem 3.1 stay true if equation (3.1) is changed by

$Lu\equiv u_{t}+\Delta^{2}u+\gamma\Delta u+\Delta
u_{x}+uu_{x}-\partial^{5}_{x}u=0\quad\textrm{in}\quad Q_{t},$

where $\gamma$ is a positive real number.

##  6\. Decay of Solutions

In Theorem 3.1, we have proved the existence and uniqueness of global
solutions for $t\in(0,T),$ where $T$ is an arbitrary positive number, without
smallness restrictions on initial data and value of an interval $(0,L)$.
Nevertheless, we could not prove decay of solutions because our estimates
depended on $T$. Here, we give sufficient conditions which guarantee decay of
global solutions as $t\to+\infty.$

1\. Decay of solutions to the initial-boundary value problem posed on
rectangles.

First, we prove exponential decay with destabilizing effect of the term
$\gamma\Delta u,\;\gamma>0$. Consider the following problem:

$\displaystyle Lu\equiv u_{t}+\Delta^{2}u+\gamma\Delta u+\Delta
u_{x}+uu_{x}-\partial^{5}_{x}u=0\quad\textrm{in}\quad Q_{t};$ (6.1)
$\displaystyle u|_{\partial D}=u_{yy}(x,0)=u_{yy}(x,B)=u_{x}(0,y)$
$\displaystyle=u_{x}(L,y)=u_{xx}(L,y)=0;$ (6.2) $\displaystyle
u(x,y,0)=u_{0}(x,y),\quad(x,y)\in D.$ (6.3)

######  Theorem 6.1.

Let $B,L$ satisfy the following condition:
$\gamma>0;\;b=\gamma\left\\{\frac{\pi^{2}}{B^{2}}+\frac{\pi^{2}}{L^{2}}\right\\}^{-1}<1,$
$\theta=1-b>0$. Given $u_{0}(x,y)$ subjected to conditions ofTheorem 3.1. Then
for regular solutions of (6.1)-(6.3) the following inequality is true:

$\displaystyle{\|u\|}^{2}(t)\leq\|u_{0}\|^{2}e^{-\chi t},$

where $\chi=2a^{2}\theta$ and $a=\frac{\pi^{2}}{B^{2}}+\frac{\pi^{2}}{L^{2}}.$

###### Proof.

Multiplying (6.1) by $2u$ and integrating the result by parts, we obtain

$\displaystyle\frac{d}{dt}\|u\|^{2}(t)-2\gamma\|\nabla u\|^{2}(t)+2\|\Delta
u\|^{2}(t)$ $\displaystyle+\int_{0}^{B}|D^{2}_{x}(0,y,t)^{2}dy=0.$ (6.4)

By Lemma 4.2, $a\|\nabla u\|^{2}\leq\|\Delta u\|^{2}$ that transform (6.4)
into the following inequality:

$\frac{d}{dt}\|u\|^{2}(t)+2\theta\|\Delta u\|^{2}(t)\leq 0.$ (6.5)

Again by Lemma 4.2, $a^{2}\|u\|^{2}\leq\|\Delta u\|^{2},$ hence (6.5) becomes

$\frac{d}{dt}\|u\|^{2}(t)+2a^{2}\theta\|u\|^{2}(t)\leq 0.$ (6.6)

Integrating (6.6), we get

${\|u\|}^{2}(t)\leq\|u_{0}\|^{2}e^{-2a^{2}\theta t}.$

This completes the proof of Theorem 6.1. ∎

######  Remark 2.

We have proved Theorem 6.1 in the ”bad case” $\gamma>0$ imposing restrictions
on dimensions of $D,$ but in the case $\gamma\leq 0$ we will not have any
restrictions on dimensions of $D$ or initial data.

######  Theorem 6.2.

Let $\gamma$ be nonpositive real number; $B,L$ are arbitrary positive numbers
and $u_{0}(x,y)$ is a given function subjected to conditions of Theorem 3.1.
Then global soluions to the problem (6.1)-(6.3) satisfy the following
inequality:

$\|u\|^{2}(t)\leq\|u_{0}\|^{2}e^{-\chi t},$

where $\chi=2\\{\frac{|\gamma|}{a}+1\\}a^{2}.$

###### Proof.

Multiplying (6.1) by $2u$ and integrating the result by parts, we obtain

$\displaystyle\frac{d}{dt}\|u\|^{2}(t)+2|\gamma|\|\nabla u\|^{2}(t)+2\|\Delta
u\|^{2}(t)$ $\displaystyle+\int_{0}^{B}|D^{2}_{x}(0,y,t)^{2}dy=0.$ (6.7)

Since, by Lemma 4.2, $a\|\nabla u\|^{2}\geq\|u\|^{2}$ and
$a^{2}\|u\|^{2}\leq\|\Delta u\|^{2},$ then (6.7) becomes

$\displaystyle\frac{d}{dt}\|u\|^{2}(t)+2\\{\frac{|\gamma|}{a}+1\\}a^{2}\|u\|^{2}(t)\leq
0.$ (6.8)

This implies

$\|u\|^{2}(t)\leq\|u_{0}\|^{2}e^{-\chi t},$

where $\chi=2\\{\frac{|\gamma|}{a}+1\\}a^{2}.$ This proves Theorem 6.2. ∎

2\. Decay of solutions to the initial-boundary value problem posed on the
right half-strip.

Define the right half-strip as follows:
$D=(x,y)\in\mathbf{R}^{2};x\in(0,+\infty),$
$y\in(0,B);\;Q_{t}=D\times(0,t).$

In $Q_{t}$ consider the following problem:

$\displaystyle Lu\equiv u_{t}+\Delta^{2}u+\gamma\Delta u+\Delta
u_{x}+uu_{x}-\partial^{5}_{x}u=0\quad\textrm{in}\quad Q_{t};$ (6.9)
$\displaystyle u|_{\partial D}=u_{yy}(x,0)=u_{yy}(x,B)=u_{x}(0,y)=0;$ (6.10)
$\displaystyle u(x,y,0)=u_{0}(x,y),\quad(x,y)\in D.$ (6.11)
$\displaystyle\lim_{x\to+\infty}D^{i}_{x}u_{0}=0,\;i=0,1,2,3.$ (6.12)

######  Theorem 6.3.

Let $\gamma\in(0,\frac{1}{8});\;k\in(0,\frac{1}{4})$ and
$a^{2}=\frac{\pi^{2}}{B^{2}}>1.$ Given $u_{0}(x,y)$ such that

$\displaystyle u_{0}\in H^{4}(D),\quad\partial_{x}^{5}u_{0}\;\in
L^{2}(D),\;u_{0}(0,y)=u_{0x}(0,y)$
$\displaystyle=u_{0}(x,0)=u_{0}(x,B)=u_{0yy}(x,0)=u_{0yy}(x,B)=0,$
$\displaystyle J_{w}=\int_{D}e^{kx}\big{[}|\Delta^{2}u_{0}|^{2}+|\Delta
u_{0}|^{2}+u^{2}_{0}u^{2}_{0x}+|D^{5}_{0x}|^{2}\big{]}dxdy<\infty,$

$(e^{kx},u_{0}^{2})<\frac{9}{8k}a^{2}.$ (6.13)

. Then for regular solutions of (6.9)-(6.12) the following inequality is true:

$\displaystyle{\|u\|}^{2}(t)\leq\|u_{0}\|^{2}e^{-\chi t},$

where $\chi=\frac{1}{2}a^{2}.$

###### Proof.

Makng use of (4.1), multiplying the $j$-equation of (4.3) by
$e^{kx}g_{j}(x,t)$, summing over $j=1,..,N$ , we obtain

$([u^{N}_{t}+\Delta^{2}u^{N}+\gamma\Delta
u^{N}+\partial^{3}_{y}u^{N}_{x}+u^{N}u^{N}_{x}-\partial^{5}_{x}u^{N}],e^{kx}u^{N})(t)=0.$
(6.14)

Dropping the index $N$ and integrating by parts, we transform (6.14) to the
form

$\displaystyle\frac{d}{dt}(e^{kx},u^{2})(t)+(3k-5k^{3})(e^{kx},u^{2}_{x})(t)+5k(e^{kx},u_{xx}^{2})(t)$
$\displaystyle\int_{0}^{B}u_{xx}^{2}(0,y,t)(t)\,dy+k(e^{kx},u^{2}_{y})(t)+(k^{5}-k^{3})(e^{kx},u^{2})(t)$
$\displaystyle+2(e^{kx}u,\Delta^{2}u)(t)+2\gamma(e^{kx}u,\Delta
u)(t)=\frac{2k}{3}(e^{kx},u^{3})(t).$ (6.15)

We estimate

$I_{1}=2(\Delta^{2}u,e^{kx}u)=2([D^{4}_{y}+2D^{2}_{y}D^{2}_{x}+D^{4}_{x}]u,e^{kx}u)=$
$2(e^{kx},|D^{2}_{y}u|^{2})+2(e^{kx},|D_{x}D_{y}u|^{2})+(e^{kx},|D^{2}_{x}u|^{2})-2k^{2}(e^{kx},u_{y}^{2})-k^{3}(e^{kx},u^{2}_{x})$
$=2(e^{kx},|\Delta u|^{2})-2k^{2}(e^{kx},u_{y}^{2})-k^{3}(e^{kx},u^{2}_{x}).$

Making use of Lemma 4.2 and Steklov‘s inequality, we obtain
$a^{2}\|u\|^{2}\leq
a^{2}\|e^{\frac{kx}{2}}u\|^{2}\leq(e^{kx},|D^{2}_{y}u|^{2}),$ and $I_{1}$
becomes

$I_{1}=2(\Delta^{2}u,e^{kx}u)=2([D^{4}_{y}+2D^{2}_{y}D^{2}_{x}+D^{4}_{x}]u,e^{kx}u)\geq$
$2(e^{kx},|D^{2}_{y}u|^{2})-2k^{2}(e^{kx},u_{y}^{2})-k^{3}(e^{kx},u^{2}_{x})$
$\geq 2a^{2}(e^{kx},u^{2})-2k^{2}(e^{kx},u_{y}^{2})-k^{3}(e^{kx},u^{2}_{x}).$

Similarly,

$I_{2}=2\gamma(e^{kx}u,\Delta u)\leq\gamma[(e^{kx},|\Delta
u|^{2})+(e^{kx},u^{2})].$

Substituting first $I_{2}$ and then $I_{1}$ into (6.15) and taking into
account that $k\in(0,1/4),$ and $a>1,$we get

$\displaystyle\frac{d}{dt}(e^{kx},u^{2})(t)+(3k/2)(e^{kx},u^{2}_{x})(t)+5k(e^{kx},u_{xx}^{2})(t)$
$\displaystyle\int_{0}^{B}u_{xx}^{2}(0,y,t)(t)\,dy+(k/2)(e^{kx},u^{2}_{y})(t)+(k^{5}-k^{3})(e^{kx},u^{2})(t)$
$\displaystyle+(2-2\gamma)a^{2}(e^{kx},u^{2})\leq\frac{2k}{3}(e^{kx},u^{3})(t).$
(6.16)

For arbitrary $\epsilon>0$, extending $u$ by zero into the exterior of $D$ and
making use of Lemma 2.4 and (4.5), we estimate

$\displaystyle
I_{3}=\frac{2k}{3}(e^{kx},u^{3})(t)\leq\dfrac{4k}{3}{\|u\|(t)\|e^{\frac{kx}{2}}u\|}(t){\|\nabla(e^{\frac{kx}{2}}u)\|}(t)$
$\displaystyle\leq
2\epsilon(e^{kx},[(u_{x}^{2}+u^{2}_{y}])(t)+\frac{k^{2}\epsilon}{2}(e^{kx},u^{2})(t)+\frac{k^{2}}{9\epsilon}(e^{kx},u^{2})(t).$

Substituting $I_{3}$ into (6.16), we find

$\displaystyle\frac{d}{dt}(e^{kx},u^{2})(t)+(3k/2-2\epsilon)(e^{kx},u^{2}_{x})(t)+5k(e^{kx},u_{xx}^{2})(t)$
$\displaystyle\int_{0}^{B}u_{xx}^{2}(0,y,t)(t)\,dy+(k/2-2\epsilon)(e^{kx},u^{2}_{y})(t)$
$\displaystyle+(2-2\gamma-k^{3}-k^{2}\epsilon)a^{2}(e^{kx},u^{2})\leq\frac{k^{2}}{9\epsilon}(e^{kx},u^{2})^{2}(t).$
(6.17)

Putting $\gamma=1/8,\;\epsilon=k/8$ e $k=1/4,$ we reduce (6.17) as follows:

$\displaystyle\frac{d}{dt}(e^{kx},u^{2})(t)+\frac{1}{2}a^{2}(e^{kx},u^{2})$
$\displaystyle+([a^{2}-\frac{8k}{9}(e^{kx},u^{2})(t)])(t)(e^{kx},u^{2})(t)\leq
0$ (6.18)

Making use of (6.13) and Lemma 2.5, we come to the inequality

$\frac{d}{dt}(e^{kx},u^{2})(t)+\frac{1}{2}a^{2}(e^{kx},u^{2})\leq 0.$

Integrating this inequality, we get

$\displaystyle{\|u\|}^{2}(t)\leq\|u_{0}\|^{2}e^{-\chi t},$

where $\chi=\frac{1}{2}a^{2}$ and $a=\frac{\pi^{2}}{B^{2}}.$

This proves Theorem 6.3. ∎

The following step is to consider problema (6.9)-(6.12) for $\gamma\geq 0.$

######  Theorem 6.4.

Let $\gamma$ be an arbitrary nonpositive real number; $k\in(0,\frac{1}{4})$;
$B$ is an arbitrary positive number and $u_{0}(x,y)$ is a given function
subjected to conditions of Theorem 6.3. Then global soluions to the problem
(6.9)-(6.12) satisfy the following inequality:

$\|u\|^{2}(t)\leq\|u_{0}\|^{2}e^{-\chi t},$

where $\chi=2\\{\frac{|\gamma|}{a}+1\\}a^{2}.$

###### Proof.

Acting as by proving (6.17), we obtain

$\displaystyle\frac{d}{dt}(e^{kx},u^{2})(t)+(3k-5k^{3})(e^{kx},u^{2}_{x})(t)+5k(e^{kx},u_{xx}^{2})(t)$
$\displaystyle\int_{0}^{B}u_{xx}^{2}(0,y,t)(t)\,dy+k(e^{kx},u^{2}_{y})(t)+(k^{5}-k^{3})(e^{kx},u^{2})(t)$
$\displaystyle+2(e^{kx}u,\Delta^{2}u)(t)+2|\gamma|\|\nabla u\|^{2}(t)\leq
2\epsilon(e^{kx},[(u_{x}^{2}+u^{2}_{y}])(t)$
$\displaystyle+\frac{k^{2}\epsilon}{2}(e^{kx},u^{2})(t)+\frac{k^{2}}{9\epsilon}(e^{kx},u^{2})(t)$
(6.19)

where $\epsilon$ is an arbitrary positive number. Acting similar to the proof
of Theorem 6.3, we get

$I_{1}=2(\Delta^{2}u,e^{kx}u)\geq
2a^{2}(e^{kx},u^{2})-2k^{2}(e^{kx},u_{y}^{2})-k^{3}(e^{kx},u^{2}_{x}).$

Substituting $I_{1}$ into (6.19) and taking into account that $k\in(0,1/4),$
and $a>1,$we get

$\displaystyle\frac{d}{dt}(e^{kx},u^{2})(t)+(3k/2)(e^{kx},u^{2}_{x})(t)+5k(e^{kx},u_{xx}^{2})(t)$
$\displaystyle\int_{0}^{B}u_{xx}^{2}(0,y,t)(t)\,dy+(k/2)(e^{kx},u^{2}_{y})(t)+(k^{5}-k^{3})(e^{kx},u^{2})(t)$
$\displaystyle+2(1+\frac{|\gamma|}{a})a^{2}(e^{kx},u^{2})\leq
2\epsilon(e^{kx},[(u_{x}^{2}+u^{2}_{y}])(t)$
$\displaystyle+\frac{k^{2}\epsilon}{2}(e^{kx},u^{2})(t)+\frac{k^{2}}{9\epsilon}(e^{kx},u^{2})(t).$
(6.20)

Putting $\epsilon,k$ sufficiently small, we find that

$\displaystyle\frac{d}{dt}(e^{kx},u^{2})(t)+2(1+\frac{|\gamma|}{a})a^{2}(e^{kx},u^{2})\leq
0.$ (6.21)

Integrating (6.21), we prove Theorem 6.4. ∎

######  Remark 3.

We have not proved the existence of regular global solutions in Theorems 6.3,
6.4. It can be done exploiting the approach used in [23].

##  7\. Conclusions

In this work, motivated by [31], we study an initial-boundary value problem
for the 2D Benney-Lin equation adding to their model the Kawahara term
$-D_{x}^{5}u$, see [16]. We studied stability of the system considering an
influense of the term $\gamma\Delta u$. As has been observed in [31], this
term caused instability while $\gamma$ was positive and implied stability
effect while it was negative.

Taking this into account, first we studied in Section 4 the case $\gamma>0$
and proved the existence and uniqueness of global regular solutions on the
time interval $(0,T),$ where $T$ was an arbitrary positive number, without
restrictions on the size of initial data or on dimensions of an arbitrary
bounded domain $D.$ Unfortunately, we could not prove decay of solutions as
$t\to+\infty$ because our estimates of solutions depended on $T.$ On the other
hand, in Section 6, we have proved in Theorem 6.1 exponential decay of
solutionsfor sufficiently small $\gamma>0$ assuming some restrictions on sizes
of a bounded domain $D.$ In this section, we studied also the case $\gamma\leq
0$ and proved in Theorem 6.2 exponential decay of global solutions without
restrictions on sizes of a bounded $D$ and initial data. In Theorem 6.3, we
have proved exponential decay of solutions defined on the right half-strip
with $\gamma>0$ sufficiently small and some restrictions on $u_{0}$ and $B.$
In Theorem 6.4, exponential decay have been established for global regular
solutions defined on the same half-strip while $\gamma\leq 0$ without
restrictions on $u_{0},B.$ It has been observed that for $\gamma<0$ the term
$\gamma\Delta u$ itself supplied stability of the system while the term
$\Delta^{2}u$ served as additional damping. On the other hand, in the case
$\gamma=0$ only the term $\Delta^{2}u$ guaranteed stability .

## References

  * [1] Adams, R.A.; Fournier, J.F.  _Sobolev Spaces_ ;  Elsevier Sciuence Ltd.: Oxford, UK, 2003.
  * [2] Brezis, H.  _Functional Analysis, Sobolev Spaces and Partial Differential Equations_ ;  Springer: New York, NY, USA, 2011.
  * [3] Benney, D.J.  Long waves on liquid films.  _J. Math. Phys._ 1966, _45_ , 150–155.
  * [4] Lin, S.P.  Finite amplitude side band stability of a viscous film.  _J. Fluid Mech._ 1974, _63_ , 417–429.
  * [5] Kuramoto, Y.; Tsuzuki, T.  On the formation of dissipative structures in reaction-diffusion systems.  _Progr. Theor. Phys._ 1975, _54_ , 687–699.
  * [6] Sivashinsky, G.I.  Nonlinear analysis of hydrodinamic instability in laminar flames-1. Derivation of basic equations.  _Acta Astronauica_ 1977, _4_ , 1177–1206
  * [7] Biagioni H.A., Linares F: On the Benney - Lin and Kawahara equations. J. Math. Anal. Appl. 211, 131–152 (1997)
  * [8] Bubnov B.A.: Solvability in the large of nonlinear boundary-value problems for the Korteweg-de Vries equation in a bounded domain. (Russian) Differentsial’nye uravneniya. 16, No 1, 34–41 (1980). Engl. transl. in: Differ. Equations. 16, 24–30 (1980)
  * [9] Ceballos J., Sepulveda M., Villagran O.: The Korteweg-de Vries- Kawahara equation in a bounded domain and some numerical results. Appl. Math. Comput. 190, 912–936 (2007)
  * [10] Cousin, A.T.; Larkin, N.A.  Kuramoto-Sivashinsky equation in domains with moving boundaries.  _Port. Math._ 2002, _59_ , 335–349.
  * [11] Cui S.B., Deng D.G., Tao S.P.: Global existence of solutions for the Cauchy problem of the Kawahara equation with $L_{2}$ initial data. Acta Math. Sin. (Engl. Ser.) 22, 1457–1466 (2006)
  * [12] Doronin G.G., Larkin N.A.: Kawahara equation in a bounded domain. Discrete and Continuous Dynamical Systems. Serie B, 10, 783–799 (2008)
  * [13] Elwakil S.A., El-Shewy E.K. and Abdelwahed H.G.: Solution of the Perturbed Zakharov-Kuznetsov (ZK) Equation describing Electron-Acoustic Solitary Waves in a Magnetized Plasma. Chinese Journal of Physics. 49, N3, 732–744 (2011)
  * [14] Faminskii A.V.: Weak solutions to initial-boundary value problems for quasilinear evolution equations of any odd order. Adv. Differential Equations. vol.17 N 5-6, 421–470 (2012)
  * [15] Faminskii A.V., Larkin N.A.: Initial-boundary value problems for quasilinear dispersive equations posed on a bounded interval. Electron. J. Differ. Equations. 1–20 (2010)
  * [16] Kawahara T.: Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan. 33, 260–264 (1972)
  * [17] Kato T.: On the Cauchy problem for the (generalized) Korteweg-de Vries equations. Advances in Mathematics Suplementary Studies, Stud. Appl. Math. 8, 93–128 (1983)
  * [18] Kuvshinov R.V., Faminskii A.V.: A mixed problem in a half-strip for the Kawahara equation. (Russian) Differ. Uravn. 45, N. 3, 391-402 (2009); translation in Differ. Equ. 45 N. 3, 404–415 (2009)
  * [19] Ladyzhenskaya O.A., Solonnikov V.A. and Uraltseva N.N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society. Providence. Rhode Island, 1968.
  * [20] Larkin N.A.: Regularity and decay of global solutions for the generalized Benney-Lin equation posed on bounded intervals and on a half-line. Axioms 2022, MDPI; 1-14.
  * [21] Larkin N.A.: Korteweg-de Vries and Kuramoto-Sivashinsky Equations in Bounded Domains. J. Math. Anal. Appl. 297, 169–185 (2004)
  * [22] Larkin N.A.: The 2D Kawahara equation on a half-strip. Applied Mathematics and Optimization. DOI 10.1007/s00245-014-9246-4
  * [23] Larkin N.A.: Existence and decay of global solutions to the three-dimensional Kuramoto-Sivashinsky-Zakharov-Kuznetsov equation. J. Math. Anal. Appl. 514, 1-18 (2022) 126046.
  * [24] Larkin N. A., Luchesi J.: General mixed problems for the KdV equations on bounded intervals. EJDE, vol. 2010(2010), No 168, 1–17 (2010)
  * [25] N.A. Larkin, M.H. Simoes, General boundary conditions for the Kawahara equation on bounded intervals. Electron. J. Differential Equations. 2013 No 159, 1–21 (2013)
  * [26] Linares F., Pastor A. and Saut J.-C.: Well-posedness for the ZK equation in a cylinder and on the background of a KdV Soliton. Commun. Partial Differential Equations. 35 No 9, 1674–1689 (2010)
  * [27] Nirenberg, L.  On elliptic partial differential equations.  In _Annali Della Scuola Normale Superiore di Pisa_ ; Classe di Scienze 3ª série; Springer: Berlin/Heidelberg, Germany,
  * [28] Saut J.-C., Temam R. and Wang C.: An initial and boundary value problem for the Zakharov-Kuznetsov equation. J. Math. Phys. 53 115612 (2012) doi:10.1063/1.4752102.
  * [29] Steklov, A.V.  The problem of cooling of an heterogeneous rigid rod.  _Commun. Kharkov Math. Soc. Ser._ 1896, _2_ , 136–181. (In Russian)
  * [30] Temam R.: Sur un probléme non linéaire. J. Math. Pures Appl. 48, 159–172 (1969)
  * [31] Topper, J., Kawahara T.  Approximate equations for long nonlinear waves on a viscous fluid.  Journal of Phys. Soc. Japan, vol. 44, No 2, pages 663-666, 1978.
  * [32] Zakharov V.E. and Kuznetsov E.A., On three-dimensional solitons. Sov. Phys. JETP 39, 285–286 (1974)