# Learning Pruned Structure and Weights Simultaneously from Scratch: an
Attention based Approach

Qisheng He, Weisong Shi, Ming Dong
Computer Science
Wayne State University

###### Abstract

As a deep learning model typically contains millions of trainable weights,
there has been a growing demand for a more efficient network structure with
reduced storage space and improved run-time efficiency. Pruning is one of the
most popular network compression techniques. In this paper, we propose a novel
unstructured pruning pipeline, Attention-based Simultaneous sparse structure
and Weight Learning (ASWL). Unlike traditional channel-wise or weight-wise
attention mechanism, ASWL proposed an efficient algorithm to calculate the
pruning ratio through layer-wise attention for each layer, and both weights
for the dense network and the sparse network are tracked so that the pruned
structure is simultaneously learned from randomly initialized weights. Our
experiments on MNIST, Cifar10, and ImageNet show that ASWL achieves superior
pruning results in terms of accuracy, pruning ratio and operating efficiency
when compared with state-of-the-art network pruning methods.

##  1 Introduction

Deep learning models such as Convolutional Neural Network (CNN) have achieved
great success in various computer vision tasks such as image recognition
Russakovsky et al. (2015); He et al. (2016), object detection Ren et al.
(2015); He et al. (2017), and semantic segmentation Noh et al. (2015). A key
avenue for deploying deep learning models is a mobile device or an edge server
in order to reduce latency and ensure data privacy for users. As a deep
learning model typically contains millions of trainable weights, this practice
has been accompanied by a growing demand for a more efficient network
structure with reduced storage space and improved run-time operating
efficiency.

Recently, there has been a resurgence in neural network compression techniques
(e.g., pruning and quantization) Salehinejad and Valaee (2020). One can
compress a given neural network architecture into an extremely small size
without compromising on the model performance. Even efficient networks Howard
et al. (2017) can be further compressed even though they already have a small
footprint. It was shown, for instance, that SqueezeNet Iandola et al. (2016)
can be compressed to 0.5MB, $510\times$ smaller than AlexNet Krizhevsky et al.
(2012).

Figure 1: Comparison of neural network pruning pipelines. (a) Traditional
network pruning learns pruned structure and fine-tunes from a pre-trained
model. (b) Network fine-pruning learns both structure and weights jointly from
a pre-trained model. (c) Pruning from scratch first generates a pruned
structure and then trains the pruned model from randomly initialized weights.
(d) ASWL learns both sparse structure and weights simultaneously through
layer-wise attention from scratch in a unified procedure.

Pruning, which removes connections or neurons in a network, is one of the most
popular network compression techniques. It reduces both the size and the
computational complexity of a model since pruned weights do not need to be
calculated or stored. Moreover, pruning is size-efficient as deep sparse
models consistently outperform shallow dense models with almost no loss of or
even a greater accuracy Zhu and Gupta (2017). When network pruning was first
introduced for compression, redundant connections were pruned using a three-
step pipeline illustrated in Fig. 1(a): first, the network is trained to learn
which connections are important; next, the unimportant connections are pruned;
finally, the network is retrained to fine-tune the weights of the remaining
connections Han et al. (2015). The same three-step pipeline was later employed
to perform layer-wise pruning in a layer-wise optimal brain surgeon Dong et
al. (2017).

It was shown that structure learning and weights optimization could be
combined to simplify the pruning process as Fig 1(b) shows. Fine-pruning Tung
et al. (2017) proposed a principled method in which the pruning ratio of each
layer is first predicted by the Bayesian optimization with a pre-trained
network, and then the network is jointly fine-tuned and compressed with the
predicted pruning ratios. The joint training was frequently used in
quantization methods known as the Straight-Through Estimator (STE) Yin et al.
(2019) to avoid zero gradients during back-propagation. The advantage of such
a pipeline is that it allows the pruning status to adapt during the change of
weights Tung and Mori (2018). However, pruning with pre-training is a time-
consuming procedure. One needs to repeat pruning and fine-tuning several times
to find a good sparse configuration Dettmers and Zettlemoyer (2019). In the
meantime, using a reward function to find the pruning ratio for each layer is
also an expensive procedure Zhou et al. (2019).

Later work, therefore, shows that the weights of a model and its pruned
structure can be directly learned from randomly initialized weights (Fig.
1(c)). Pruning from Scratch Wang et al. (2020) proposed a novel network
pruning pipeline that first learns the pruned structure directly from randomly
initialized weights and then optimizes the weights of the pruned network. This
kind of pipeline bypasses the time-consuming pre-training procedure.
Nevertheless, the structure learning and weights optimization proceed
separately here so that it only prunes weights based on fixed regularization
before training the weights Liu et al. (2020).

Recently, the attention mechanism was brought into network pruning and
achieved great success Yamamoto and Maeno (2018); Chin et al. (2020). In this
paper, we propose a new pruning pipeline, Attention-based Simultaneous sparse
structure and weight Learning (ASWL). As shown in Fig. 1(d), in ASWL, the
layer-wise sparsity and weights are jointly learned from scratch in a unified
training pipeline. Specifically, we first use the attention mechanism to learn
the importance of each layer in a network and determine the corresponding
pruning ratio. Then, we jointly prune the layer under the guidance of the
pruning ratio and update the unpruned weights. The major contributions of our
work are summarized as follows:

  * •

ASWL provides a unified framework that combines both _layer-wise pruning_ and
weights optimization to learn a pruned network from randomly initialized
weights. During ASWL training, both weights of the dense network and the
sparse network are tracked so that the pruning ratio for each layer is
simultaneously learned as the weights change.

  * •

In ASWL, layer-wise pruning decisions are made through a novel attention-based
approach. Attention scales in the attention mechanism are set across each
layer instead of previous channel-wise or weight-wise attention mechanism. A
pruning ratio is then directly computed based on the learned attention value
for each layer.

  * •

ASWL no longer needs the time-consuming pre-training procedure in network
pruning but provides equivalent pruning results. Through extensive experiments
on benchmark datasets, we demonstrate that ASWL leads to superior pruning
results in terms of accuracy, network size, and operating efficiency when
compared with state-of-the-art pruning methods.

Figure 2: ASWL simultaneous training procedure with the attention mechanism.

##  2 Related Work

Neural network pruning techniques were used to avoid the overfitting problem
in early work. For example, optimal brain surgeon Hassibi and Stork (1993) and
dynamic sparsely connected artificial neural networks Ström (1997) pruned
those connections that lead to a limited change of validation errors. The
Dropout was introduced to prune or remove neurons randomly with a given ratio
so that the complexity of the model is reduced. Later, pruning methods were
employed to identify and remove unnecessary weights to reduce the size as well
as the run-time latency of a targeted model. While some of the pruning methods
like ThiNet Luo et al. (2017) and Stripe-wise Pruning Meng et al. (2020) prune
feature maps or filter stripes, most others remove connections/neurons because
sparse structures provided by unstructured pruning typically lead to a higher
accuracy with a lower redundancy of weights Liu et al. (2020).

When pruning methods were first introduced to deep compression, they removed
all connections whose weights were lower than a threshold Han et al. (2015).
It was shown that pruning unimportant weights results in almost no loss of
accuracy as the weights in deep neural networks are highly redundant. Then,
methods were proposed to identify unnecessary weights layer-wisely Yang et al.
(2017); Dong et al. (2017); Yu et al. (2018), which offers flexible pruning
ratios on different layers and thus achieves better results. Automated Machine
Learning (AutoML) was also introduced into pruning He et al. (2018), where the
design space was efficiently sampled using reinforcement learning to find the
best-pruned structure of a pre-trained model. Additionally, the attention
mechanism was recently employed to find unimportant weights Hacene et al.
(2019); Wu et al. (2018); Lin et al. (2019). Most of them introduced attention
scales across each channel or each weight of the targeted layer. For example,
LeGR-based pruning Chin et al. (2020) trained pairs of parameters (scale and
shift) across each channel and then pruned output features accordingly.

More recently, the common three-step pruning pipeline, i.e., pre-training,
pruning and fine-tuning, has been simplified. Fine-pruning Tung et al. (2017)
jointly fine-tunes and compresses the pre-trained network with the pruning
ratio of each layer predicted by the Bayesian optimization. Later, the lottery
ticket hypothesis stated that dense randomly-initialized neural networks
contain subnetworks, known as winning tickets, that reach accuracy equivalent
to its original network with a similar number of training iterations Frankle
and Carbin (2018). The hypothesis was mathematically proved in Malach et al.
(2020). Based on such findings, Pruning from Scratch Wang et al. (2020)
proposed to prune networks from randomly initialized weights. Specifically, it
first learns the channel importance for each layer with a sparsity
regularizer, then searches for a pruned structure with a method adopted from
Network Slimming Liu et al. (2017) but used floating-point operations per
second (FLOPS) as constraints. The pruned structure is then optimized from
random weights. However, sanity-checking Su et al. (2020) showed that initial
tickets are hard to be learned from the training data. Thus, it is necessary
to keep the pruned structure flexible during training. To this end, more
recent work trained sparse networks from scratch by regrowing some connections
after a fixed number of iterations (e.g., per epoch) based on the gradient
momentum, aka, sparse momentum, of the weights in these iterations Dettmers
and Zettlemoyer (2019). Nonetheless, when one of the connections converges
during training, its sparse momentum will also trend to zero, often leading to
a sub-optimal structure and weights during the training.

##  3 Attention-based Simultaneous Learning

The objective of our new training pipeline, Attention-based Simultaneous
sparse structure and weight Learning (ASWL), is to train both pruned structure
and model weights from randomly initialized weights. Different from existing
pruning methods, ASWL obtains layer-wise pruned deep learning models without
pre-training. As shown in Fig. 2, we first convert a model into an attention-
based model by replacing its traditional weighted layers with attention-based
weighted layers. Instead of learning the pruned structure by regrowing
connections after a fixed number of iterations, we pruned the weights layer-
wise based on the pruning ratio calculated from the learned attention of the
target layer in every iteration. In the mean time, the unpruned parameters are
tracked and learned directly from the forward pass of the pruned structure.
Such simultaneous learning tracks both pruned and unpruned weights during
training and thus gives an adaptive pruning decision on the sparse structure.

###  3.1 Attention-based Neural Networks

The sparsity of a pruned network is weakly related to its output and hence is
hardly to be learned directly Lin et al. (2020). Recent research shows that
the attention mechanism provides a promising solution Yamamoto and Maeno
(2018); Hacene et al. (2019). Unlike the traditional methods which set
attention scalars channel-wisely or weight-wisely, we introduced a much
simpler layer-wise attention mechanism, where an attention scalar value was
defined across all $L$ layers in our attention-based neural network. Such
model is denoted as $f(x;\mathbf{w},\mathbf{A})$ where $x$ is the input of the
model, $\mathbf{w}$ is the traditional trainable weights, and
$\mathbf{A}=\\{a_{1},a_{2},...,a_{L}\\}$ is the layer-wise attention values.
Given a scalar gate value $a_{l}\in(0,1]$ for the $l^{th}$ layer as the
attention, it is multiplied with the output of the layer. That is, assuming
the output of the $l^{th}$ original layer is $f_{l}(x_{l-1};\mathbf{w}_{l})$,
the attended output is modified as
$\hat{f}_{l}(x_{l-1};\mathbf{w}_{l},a_{l})=a_{l}\cdot
f_{l}(x_{l-1};\mathbf{w}_{l})$. Each attention value is initialized to 0.5 at
the beginning. During training, both weights and attentions are updated. With
pruning applied, the optimization objective of an attention-based neural
network with compressed weights $\mathbf{\hat{w}}$ is:

$\underset{\mathbf{\hat{w}},\mathbf{A}}{min}\sum_{i}^{N}{\mathfrak{L}(f(x_{i};\mathbf{\hat{w}},\mathbf{A}))}+\gamma\Psi(\mathbf{A})+\lambda\sum_{j}\mathbf{\hat{w}}_{j}^{2}$
(1)

where $\mathfrak{L}(\cdot)$ denotes the cross-entropy loss, $\Psi(\cdot)$ is
the sparsity regularizer for structure learning (discussed in the next
section), L2 regularizer encourages all weights to be small Krogh and Hertz
(1992), and $\gamma$ and $\lambda$ are the coefficients for the sparsity
regularizer and L2 regularizer, respectively.

###  3.2 Pruning Ratios and Sparsity Regularizer

Layer-wise pruning methods typically require a pre-trained model as the
starting point to search for a pruned structure. Recent work in Tung et al.
(2017) and He et al. (2018) use naive Bayesian to optimize the pruning ratio
layer-wisely in a given network. However, employing this kind of optimization
in every training iteration is prohibitively time-consuming. In order to
facilitate simultaneous optimization of structure and weights in each
iteration, we propose a more efficient algorithm to calculate the pruning
ratios directly from the attentions in our attention-based model.

Algorithm 1 Training Algorithm for Attention-based Simultaneous sparse
structure and Weight Learning

1:function Train($iterations$, $\alpha$)

2: $\mathbf{w},\mathbf{A}\leftarrow$ initialize()

3:

4: for $i$ from 1 to $iterations$ do

5: $\mathbf{p}\leftarrow(1-\mathbf{A})^{\alpha}$

6: $\mathbf{\hat{w}}\leftarrow prune(\mathbf{w},\mathbf{p})$

7:

8: $\mathbf{x}\leftarrow getNextBatch()$

9:
$\mathfrak{l}\leftarrow\sum_{i}^{N}{\mathfrak{L}(f(x_{i};\mathbf{\hat{w}},\mathbf{A}))}+\gamma\Psi(\mathbf{A})+\lambda\sum_{i}\mathbf{\hat{w}}_{i}^{2}$

10:

11: $\mathbf{G}\leftarrow
calculateGradients(\mathfrak{l};\mathbf{\hat{w}},\mathbf{A})$

12: $\mathbf{w},\mathbf{A}\leftarrow
updateWeights(\mathbf{G};\mathbf{w},\mathbf{A})$

13: end for

14:end function

In Wang et al. (2020); Hacene et al. (2019), it was shown that scaling the
network weights will suppress the unimportant ones, resulting in a pruning
effect. As a result, the learned attentions can be used to represent the
importance of each layer. That is, if a layer has a larger attention, it is
considered to be more important so that it needs to be pruned less, and vice
versa. Here, we introduce a positive hyper-parameter, the pruning factor
$\alpha$, to gain more control when computing the pruning ratio $p_{l}$ based
on the given attention $a_{l}$ of the $l^{th}$ layer:

$p_{l}(a_{l})=(1-a_{l})^{\alpha},$ (2)

where $a_{l}$ is the attention for the $l^{th}$ layer and $\alpha$ is the
pruning factor. To ensure not all weights are pruned, we limited the maximum
pruning ratio to be $99\%$. Since our attentions are applied along the layer
dimension when weights are pruned, the overall sparsity $S$ of our attention-
based model is computed as:

$S(\mathbf{A})=\frac{\sum_{l}(1-p_{l}(a_{l}))n_{\mathbf{w}_{l}}}{\sum_{l}n_{\mathbf{w}_{l}}},$
(3)

where $p_{l}$ is the pruning ratio and $n_{\mathbf{w}_{l}}$ is the total
number of unpruned weights in the $l^{th}$ layer. Note that
$n_{\mathbf{w}_{l}}$ is a constant as it is pre-determined for each layer.

Regularizers like L1 or L2 encourage the network weights $\mathbf{w}$ to be
small, but not necessarily zeros. Thus, an additional regularizer is required
in ASWL to encourage the pruning procedure to remove unimportant weights in
each layer. Moreover, the L1/L2 regularizer encourages attentions to be zeroes
without giving any consideration on the sparsity of the entire model,
especially when the pruning factor is not set to 1. As a result, we adopted
the sparsity regularizer proposed in Wang et al. (2020) as follows and
combined it with L2 in ASWL:

$\Psi(\mathbf{A})=S(\mathbf{A})^{2}$ (4)

The square sparsity regularizer is differentiable Luo and Wu (2020) and will
help minimize the layer-wise sparsity during optimization.

###  3.3 Simultaneous Sparse Structure Learning and Weight Optimization

Traditional pruning methods learn the pruned structure first, and then
optimize the weights based on the pruned structure. The pruned structure can
be easily found if a pre-trained model is available. However, with the weights
training from random initialized values, efficient weights might change
gradually from shallow to deep layers Dettmers and Zettlemoyer (2019).
Meanwhile, a dense neural network contains a subnetwork that has the same
accuracy even without fine-tuning Malach et al. (2020). These results
motivated us to perform simultaneous learning on both the sparse structure and
weights in ASWL by tracking both the dense network (through the backward
propagation) and the sparse subnetwork (through the forward pass).

In ASWL training, both weights for the dense network and the sparse network
are tracked so that the structure is simultaneously learned as the weights
change. The parameters of the sparse network are pruned by removing at least
the bottom $p_{i}$ percent dense weights in the $i_{th}$ layer based on the
absolute weight values $|w_{i}|$. After training, the attention values are
applied to the weights and discarded together with the weights in the dense
network. Only the weights in the sparse network are stored.

Each training iteration in ASWL contains four steps: (1) the network
classification loss is calculated through a forward pass based on the
compressed weights $\mathbf{\hat{w}}$ and attentions $\mathbf{A}$, (2) both
uncompressed weights $\mathbf{w}$ and attentions $\mathbf{A}$ are updated
through back-propagation, (3) the pruning ratio $\mathbf{p}$ is computed by
the new attentions $\mathbf{A}$ and the pruning factor $\alpha$, and (4) the
model is compressed layer-wise using the pruning ratio $\mathbf{p}$, and the
compressed weights $\mathbf{\hat{w}}$ are updated. The detailed training
procedure is summarized in Algorithm 1. In ASWL, through simultaneous
optimization, weights that have been pruned at first may be recovered later,
and the weights defined as important at first can be pruned, all depending on
the evolution of the network structure.

##  4 Experimental Results

In this section, we perform extensive experiments with ASWL on VGG-16 Simonyan
and Zisserman (2014), ResNet50 He et al. (2016), and MobileNetV2 Sandler et
al. (2018). For VGG-16 and ResNet50, we simply replace the traditional
convolutional layer and dense layer with our attention-based convolutional
layer and dense layer. For MobileNetV2, we replace the $1\times 1$ point-wise
convolutional layer with an attention-based convolutional layer but left the
depth-wise layer uncompressed since 99% of the parameters and calculations are
contained in point-wise convolutional layers. There are totally three
hyperparameters in ASWL: the coefficients of sparsity and L2 regularizer
($\gamma$ and $\lambda$), and the pruning factor $\alpha$. They are specified
later in different experiments.

Our ASWL training pipeline was implemented in TensorFlow Abadi et al. (2016).
All models are trained on a computer with Intel i7 8700K CPU, 16GB RAM, and
two NVIDIA RTX 2080 Ti graphic cards, each of which has 11GB of GDDR SDRAM.
The source code of this work is available in the supplemental materials with
detailed comments and will be made publicly available after the review period
of AAAI 2022.

(a) Pruning details for VGG-16 with $\alpha=1.5$

(b) Pruning details for MobileNetV2 with $\alpha=1$

(c) Pruning details for ResNet56 with $\alpha=1$

Figure 3: Layer-wise pruning details on MNIST for (a) VGG-16 with
$\alpha=1.5$, where the last two layers are attention-based dense layers, and
all others are attention-based convolutional layers, (b) MobileNetV2 with
$\alpha=1$, where the first layer is an attention-based convolutional layer
with a $3\times 3$ filter, and others are point-wise ($1\times 1$ filter)
attention-based convolutional layers, (c) ResNet56 with $\alpha=1$, where the
last layer is an attention-based dense layer, and others are attention-based
convolutional layer.

(a) Training graph for VGG16 with $\alpha=1.5$

(b) Training graph for MobileNetV2 with $\alpha=1$

(c) Training graph for ResNet56 with $\alpha=1$

Figure 4: MNIST Training graph: training accuracy (blue line), validation
accuracy (green line), and pruning ratio (red line) for (a) VGG-16 with
$\alpha=1.5$, (b) MobileNetV2 with $\alpha=1$, and (c) ResNet56 with
$\alpha=1$.

Specifically, we conducted our experiments on the standard MNIST LeCun et al.
(1998), Cifar-10 Krizhevsky and Hinton (2009) and ImageNet (ILSVRC-2012 in
version 2.0.1) Russakovsky et al. (2015) datasets and compared with the
following pruning methods: Learning both Weights and Connections for Efficient
Neural Networks (ENN) Han et al. (2015), Rethingking Network Pruning (RNP) Liu
et al. (2018), Discrimination-aware Channel Pruning (DCP) Zhuang et al.
(2018), Convolutional neural network pruning with structural redundancy
reduction (SRR) Wang et al. (2021), Network Pruning via Performance
Maximization (NPPM) Gao et al. (2021), Learned Threshold Pruning (LTP) Azarian
et al. (2020), Pruning from Scratch (PFS) Wang et al. (2020), Dynamic Sparse
Reparameterization (DS) Mostafa and Wang (2019), Pruning Filter in Filter
(PFF) Meng et al. (2020), WoodFisher (WF) Singh and Alistarh (2020), Sparse
Networks from Scratch (SNS) Dettmers and Zettlemoyer (2019), and Dynamic
Pruning with Feedback (DPF) Lin et al. (2020).

Model | Pruning Factor | Baseline |  $\Delta$ Accuracy | Pruning Ratio
---|---|---|---|---
$\alpha$ | $\%$ | $\%$ | $\%$
VGG16-1 | 1 | 99.51% | +0.04% | 90.69%
VGG16-1.5 | 1.5 | +0.09% | 88.56%
VGG16-2 | 2 | +0.03% | 87.19%
MobileNetV2-1 | 1 | 99.56% | +0.00% | 90.10%
MobileNetV2-1.5 | 1.5 | -0.07% | 86.32%
MobileNetV2-2 | 2 | -0.09% | 82.03%
ResNet56-1 | 1 | 99.50% | +0.07% | 87.74%
ResNet56-1.5 | 1.5 | +0.00% | 83.70%
ResNet56-2 | 2 | +0.05% | 75.61%
Table 1: ASWL training results with different pruning factors on VGG-16 (top row), MobileNetV2 (middle row), and ResNet50 (bottom row) for MNIST. Model | Pruning Factor | Baseline |  $\Delta$ Accuracy | Pruning Ratio
---|---|---|---|---
$\alpha$ | $\%$ | $\%$ | $\%$
VGG16-1 | 1 | 92.64 | +0.50% | 93.80%
VGG16-1.5 | 1.5 | +0.76% | 91.96%
VGG16-2 | 2 | +0.18% | 78.11%
MobileNetV2-1 | 1 | 93.14 | +0.27% | 95.86%
MobileNetV2-1.5 | 1.5 | +0.41% | 94.61%
MobileNetV2-2 | 2 | +0.11% | 78.89%
ResNet56-1 | 1 | 93.44 | +0.28% | 95.97%
ResNet56-1.5 | 1.5 | +0.10% | 94.86%
ResNet56-2 | 2 | +0.26% | 96.19%
Table 2: ASWL training results with different pruning factors on VGG-16 (top
row), MobileNetV2 (middle row), and ResNet56 (bottom row) for Cifar10.

Model | Method | Simu. | From | Unstructured | Baseline |  $\Delta$ Acc. | Pr. Ratio
---|---|---|---|---|---|---|---
Training | Scratch | Pruning | % | % | %
VGG16 | DCP | $\times$ | $\times$ | $\times$ | 93.80% | +0.17% | 52.1%
RNP | $\times$ | $\times$ | $\checkmark$ | 93.76% | -0.04% | 80.0%
PFS | $\times$ | $\checkmark$ | $\times$ | 93.44% | +0.19% | 93.6%
PFF | $\times$ | $\checkmark$ | $\times$ | 93.25 | -0.40% | 92.7%
SNS | $\circ$ | $\checkmark$ | $\checkmark$ | 93.51% | -0.50% | 95.0%
ASWL (Ours) | $\boldsymbol{\checkmark}$ | $\boldsymbol{\checkmark}$ | $\boldsymbol{\checkmark}$ | 92.81% | +0.76% | 92.0%
ResNet56 | DCP | $\times$ | $\times$ | $\times$ | 93.80% | -0.31% | 92.8%
RNP | $\times$ | $\times$ | $\checkmark$ | 93.80% | -0.31% | 93.1%
PFS | $\times$ | $\checkmark$ | $\times$ | 93.23% | -0.18% | 93.1%
PFF | $\times$ | $\checkmark$ | $\times$ | 93.10% | +0.12% | 77.7%
DPF-90 | $\checkmark$ | $\checkmark$ | $\checkmark$ | 94.51% | -0.56% | 90.0%
DPF-95 | $\checkmark$ | $\checkmark$ | $\checkmark$ | 94.51% | -1.26% | 95.0%
ASWL (Ours) | $\boldsymbol{\checkmark}$ | $\boldsymbol{\checkmark}$ | $\boldsymbol{\checkmark}$ | 93.44% | +0.28% | 96.0%

Table 3: ASWL training results on Cifar10, and the comparison with state-of-
the-art network pruning methods. In the “simultaneous training” column,
$\times$ indicates static pruned structure during training, $\circ$ indicates
that the network connections are iteratively regrown after a fixed number of
iterations (e.g., every epoch), and $\checkmark$ indicates fully simultaneous
structure and weight training. In the “from scratch” column, $\times$
indicates pruning from pre-trained, and $\checkmark$ indicates pruning from
scratch. In the “unstructured pruning” column, $\times$ indicates structured
pruning, and $\checkmark$ indicates unstructured. For each method, accuracy,
and pruning ratio are reported, and the best results are highlighted in bold.
The same applies to Table 4

.

###  4.1 Results on MNIST

We first trained selected models with ASWL on MNIST, which contains 10
different handwriting digits with 60,000 training images and 10,000 testing
images. Each of the models was trained with various pruning factors of 1, 1.5,
and 2 for 100 epochs by the Adam optimizer at a learning rate of 0.001 and
0.98 decay for each epoch. The attentions of each layer are initialized at
0.5. The hyper-parameter $\gamma$ (the sparsity regularizer coefficient) is
used mainly to achieve a desired pruning ratio and set at 0.5 for all models.
Following Simonyan and Zisserman (2014), He et al. (2016), and Sandler et al.
(2018), the other hyper-parameter $\lambda$ (the L2 regularizer coefficient)
is set at $5\cdot 10^{-4}$, $0.0001$, and $0.00004$ for VGG16, ResNet 56,
MobileNetV2, respectively. The models with the best results are selected.

Table 1 shows the ASWL training results on MNIST with VGG16 (Top), MobileNetV2
(Middle), and ResNet56 (Bottom). In most situations, the pruning ratio
progressively increases when we reduce the pruning factor. There is no obvious
relation found between model accuracy and pruning factors, while a smaller
pruning factor offers a greater pruning ratio. With about $10\%$ to $25\%$ of
weights in the original dense models, our ASWL provides similar or many times
higher accuracy.

Fig. 3 (a) shows the pruning details of each layer in VGG-16 with a pruning
factor of $\alpha=1.5$. Deeper attention-based convolutional layers are pruned
more than the shallower ones, while the last two attention-based dense layers
have pruning ratios much less than the convolutional layers. Fig. 3 (b)
demonstrates the pruning ratio of each regular (non depth-wise) attention-
based convolutional layer in MobileNetV2 with a pruning factor of $\alpha=1$.
Its first convolutional layer with a filter size of $3\times 3$ has the least
pruning ratio, just above $50\%$. Meanwhile, pruning ratios appear to be
greater in deeper convolutional layers, similar to the VGG-16 case. The layer-
wise pruning details of ResNet56 are shown in Fig. 3 (c), where the last layer
is an attention-based dense layer. ResNet56 has much more layers compared with
VGG-16 and MobileNetV2 but still has an obvious trend of a higher pruning
ratio for a deeper layer. Fig. 4 (a) shows the training graph of VGG-16 with a
pruning factor of $\alpha=1.5$. Overall, the pruning ratio continues
increasing as the model is being trained, and the validation accuracy
stabilizes after 30 epochs. We observe the same in the training graphs of
MobileNetV2 with $\alpha=1$ and ResNetV2 with $\alpha=1$ (Fig. 4 (b) and (c)).

###  4.2 Results on Cifar10

Cifar10 is a dataset that contains 10 different classes with 50,000 training
images and 10,000 testing images. Similar to our experiment on MNIST, we
trained the selected models with various pruning ratios of 1, 1.5, and 2. For
VGG16, we used an initial learning rate of 0.1 and multiplied 0.5 for every 20
epochs with the SGD optimizer (momentum 0.9), and trained for 250 epochs at a
batch size of 128. For ResNet 56, we followed the same settings in He et al.
(2016). For MobileNetV2, we trained for 350 epochs on SGD optimizer with an
initial learning rate of 0.1, which was divided by 10 after 150 and 250
epochs. Again, the hyper-parameter $\gamma$ (sparsity regularizer coefficient)
is used to help us achieve a desired pruning ratio and was set at 2.5, 5 and
0.5 for VGG16, ResNet 56, MobileNetV2, respectively. Following Simonyan and
Zisserman (2014), He et al. (2016), Sandler et al. (2018), the other hyper-
parameter $\lambda$ is set at $5\cdot 10^{-4}$, $0.0001$, and $0.00004$ for
VGG16, ResNet 56, and MobileNetV2, respectively. The models with the best
results are selected.

Table 2 shows the ASWL training results on Cifar10 with VGG16 (Top),
MobileNetV2 (Middle), and ResNet56 (Bottom). Similar to MNIST, the results
show that small pruning factors typically lead to higher pruning ratios.
Additionally, we observe that ASWL offers an improvement on model accuracy
with reduced network size and higher run-time efficiency over uncompressed
baselines in all cases. The pruning details of the three models the three
models are shown in Fig. 1 in supplemental materials. Similar to the case of
MNIST, deeper layers are pruned more than the shallower ones except for the
last dense layer for classification. Similar training graphs (Fig. 2 in
supplemental materials) of the three models were observed as in MNIST that the
model converges after about 30 epochs and reaches the highest validation
accuracy at around 130 epochs, at which time the overall pruning ratio stops
increasing as well.

Table 3 compares ASWL with state-of-the-art pruning methods on Cifar10 with
ResNet56 (top) and VGG16 (bottom). Clearly, ASWL achieved a higher accuracy
than the baseline on both models. Overall, our ASWL model achieved an
outstanding pruning ratio with the highest increase of accuracy when compared
to the baseline. These results clearly demonstrate the advantages of
simultaneous training and layer-wise attention-based pruning in ASWL.

Method | Simu. | From | Unstructured | Baseline | T1 Acc. | Pr. Ratio
---|---|---|---|---|---|---
Training | Scratch | Pruning | % | % | %
ENN | $\times$ | $\times$ | $\checkmark$ | 76.1% | 76.1% | 60.0%
CP 0.5$\times$ | $\times$ | $\times$ | $\times$ | 75.4% | 48.5%
SRR | $\times$ | $\times$ | $\times$ | 75.1% | 55.1% (FLOPS)
PFS 0.75$\times$ | $\times$ | $\checkmark$ | $\times$ | 75.6% | 63.9%
DS-20 | $\circ$ | $\checkmark$ | $\checkmark$ | 73.3% | 80.0%
SNS-20 | $\circ$ | $\checkmark$ | $\checkmark$ | 73.8% | 80.0%
NPPM | $\checkmark$ | $\checkmark$ | $\times$ | 76.0% | 56.0% (FLOPS)
LTP | $\checkmark$ | $\times$ | $\checkmark$ | 73.3% | 85.6%
WF-90 | $\checkmark$ | $\times$ | $\checkmark$ | 75.2% | 90.0%
DPF-80 | $\checkmark$ | $\checkmark$ | $\checkmark$ | 75.5% | 73.5%
DPF-90 | $\checkmark$ | $\checkmark$ | $\checkmark$ | 74.6% | 82.6%
ASWL (Ours) | $\boldsymbol{\checkmark}$ | $\boldsymbol{\checkmark}$ | $\boldsymbol{\checkmark}$ |  | 76.5% | 86.1%
Table 4: ASWL training result for ResNet-50 on ImageNet, and the comparison
with state-of-the-art network pruning methods. Pruning ratio with FLOPS
indicates the pruning ratio of the target method is calculated by FLOPS
instead of number of weights.

###  4.3 Results on ImageNet

Compared with MNIST and Cifar10, ImageNet is a much larger dataset that
contains 1000 classes with 1.2M training images and 50K testing images. We
trained ResNet-50 using ASWL following the same training setting in He et al.
(2016) for ResNet-50. The hyper-parameter $\gamma$ was set at 0.5.
Additionally, the pruning factor was set to 1. The models with the best
results are selected.

Table 4 compares ASWL with state-of-the-art pruning methods on ResNet-50.
Through simultaneous training with layer-wise pruning from randomly
initialized weights, ASWL achieves the top-1 accuracy of $76.5\%$ with a great
pruning ratio ($86.1\%$). Note that this accuracy is higher than the
uncompressed ResNet-50 baseline model ($76.1\%$). Considering the balance
between the top-1 accuracy and the pruning ratio, ASWL provides a superior
pruning result when compared to all existing pruning methods. If we consider
accuracy alone, it is the second best.

##  5 Conclusion

In this paper, we proposed a novel pruning pipeline, Attention-based
Simultaneous sparse structure and weight Learning (ASWL). In ASWL, we first
use the attention mechanism to learn the importance of each layer in a network
and determine the corresponding pruning ratio. Then, the layer-wise sparsity
and weights are jointly learned from scratch in a unified training procedure.
Our extensive experiments on benchmark datasets show that ASWL achieves
outstanding pruning results in terms of accuracy, pruning ratio, and operating
efficiency.

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# Persistence properties and asymptotic analysis for a family of hyperbolic
equations including the Camassa-Holm equation

Igor Leite Freire
1Centro de Matemática<EMAIL_ADDRESS>or<EMAIL_ADDRESS>Computação e Cognição
Universidade Federal do ABC
Santo André  Brazil

###### Abstract

Persistence and asymptotic properties for solutions of a family of hyperbolic
equations having the Camassa-Holm equation as a particular member are
revisited and investigated from a different perspective. We show that if the
initial data has a certain asymptotic exponential decaying for each time
fixed, then such property persists for any time as long as the solution
exists. Moreover, depending on the behaviour of the initial data for large
values of the spatial variable and if for some further time the solution
shares the same behaviour, then it necessarily vanishes identically. The
approach used does not permit us to infer whether members of the family has
non-trivial solutions compactly supported, except for the Camassa-Holm
equation. For this reason we use recent developments about unique continuation
of solutions to study this sort of solutions. We show that if a given solution
of the equation vanishes on an open set of its domain, then the only possible
solution is the trivial one. Therefore, as long as the initial data of the
equation is non-trivial the corresponding solution does not vanish identically
on each compact set of the space variable for each time fixed. As a
consequence we have that the solution cannot be compactly supported.

MSC classification 2010: 35A01, 74G25, 37K40, 35Q51.

_K_ eywords Camassa-Holm type equations $\cdot$ Conserved quantities $\cdot$
Persistence of decay rates $\cdot$ Unique continuation of solutions $\cdot$
Compactly supported solutions

###### Contents

  1. 1 Introduction
  2. 2 Main results of the paper and its outline
    1. 2.1 Notation and conventions
    2. 2.2 Main results and outline of the paper
  3. 3 Preliminaries and technical results
  4. 4 Proof of the main results
  5. 5 Discussion
  6. 6 Conclusion
  7. A Proof of Claim 1
  8. B Proof of Claim 2
  9. C Proof of Claim 3
  10. D Proof of Claim 4
  11. E Proof of estimates (4.0.9)

##  1 Introduction

In [11, 12] Hakkaev and Kirchev considered the following equation

$u_{t}-u_{txx}+\frac{(n+1)(n+2)}{2}u^{n}u_{x}=\frac{n(n-1)}{2}u^{n-2}u_{x}^{3}+2nu^{n-1}u_{x}u_{xx}+u^{n}u_{xxx},$
(1.0.1)

where $n$ is a constant, that henceforth will be assumed to be a positive
integer, and $u=u(t,x)$. We observe that whenever $n=1$ we recover the
celebrated Camassa-Holm (CH) equation [2], which is nowadays a by far famous
equation not only for its relevance in hydrodynamics [2, 5], but also for its
rich mathematical properties, see [6, 7] and references therein about several
properties of the CH equation. For the CH equation the variables $t$ and $x$
are regarded as time and space, respectively, so that we maintain this
terminology for (1.0.1).

In regard to (1.0.1), it is locally well-posed in Sobolev spaces
$H^{s}(\mathbb{R})$, with $s>3/2$, see [17, Theorem 2] and [11, Theorem 3.4]
(see also [12, Theorem 2.2]), and for $s\in(1,3/2]$ the existence of weak
solutions is granted by [17, Theorem 1].

In addition to the results in [11, 17], in [19] was shown the local well-
posedness of solutions of (1.0.1) in Besov spaces (see Theorem 1.1 in the
mentioned reference) and under certain circumstances, the solutions are also
analytic on a certain open set of $\mathbb{R}^{2}$, see [19, Theorem 1.2].

In [21] some persistence properties for (1.0.1) were considered. In
particular, among the results proved there, one is concerned with asymptotic
properties of (1.0.1): in case $n$ is odd and for each $t$ fixed, then
$u(t,x)\sim u_{0}(x)+c(x)e^{-x}$, where $c(x)\rightarrow 0$ as
$x\rightarrow\infty$, see [21, Theorem 1.3].

On the other hand, it is well known that the Camassa-Holm equation does not
have compactly supported solutions in $H^{s}(\mathbb{R})$, for a suitable
choice of $s$, see [4, 13]. In particular, in [16, Theorem 1.4] it was shown
that compactly supported initial data for the CH equation gives rise to
solutions behaving like $e^{-|x|}$, for each fixed $t>0$. Moreover, if a
solution of the CH equation vanishes on an open set of its domain, then the
solution itself is identically zero, see [18, 9].

We would like to note that (1.0.1) can be rewritten as

$m_{t}+2nu^{n-1}u_{x}m+u^{n}m_{x}=\frac{n(1-n)}{2}(u^{n}u_{x}-u^{n-2}u_{x}^{3}),$
(1.0.2)

and observe that the ideas used in [13, 16] to prove the non-existence of non-
trivial compactly supported solutions for the CH equation (note that these
works use different approaches) cannot be applied to (1.0.2) because the right
hand side of (1.0.2) does not vanish identically if $n\neq 1$ and $u\not\equiv
0$ (this is better discussed in Section 5). Therefore, the available machinery
seems to be unsuitable for addressing the same question for (1.0.1) when
$n\neq 1$.

The purpose of this paper is to shed light on the topics mentioned above and
revisit the results proved in [21] from a different perspective influenced by
recent developments considered in [18, 9, 8, 10] and extending to (1.0.1) some
results proved in [18, 9, 10]. To do this, we consider families of functions
obtained from a known solution $u(t,x)$ of (1.0.1). More precisely, we
consider the families

${\cal F}_{1}=(f_{t})_{t\in[0,T)}\quad\text{and}\quad{\cal
F}_{2}=(F_{t})_{t\in[0,T)},$

where

$f_{t}(x):=\frac{n}{2}u(t,x)^{n-1}u_{x}(t,x)^{2}+\frac{n(n+3)}{2(n+1)}u(t,x)^{n+1}\quad\text{and}\quad
F_{t}(x):=\partial_{x}\Lambda^{-2}\ast f_{t}(x).$

The meaning of $\partial_{x}\Lambda^{-2}$ will be given in Section 2, where we
present the notations and conventions used throughout the paper, as well as
our main results and outline of the work.

##  2 Main results of the paper and its outline

Let us firstly introduce the notation and conventions used throughout the
paper and then, its main results and outline.

###  2.1 Notation and conventions

Throughout this paper we denote by $\|\cdot\|_{p}$ and
$\|\cdot\|_{H^{s}(\mathbb{R})}$ the norms of the Banach spaces
$L^{p}(\mathbb{R})$, $1\leq p\leq\infty$, and the Sobole space
$H^{s}(\mathbb{R})$, respectively. Given two functions $f,g\in
L^{1}(\mathbb{R})$, their convolution is denoted by $f\ast g$. If $u=u(t,x)$,
we denote by $u_{0}(x)$ the function $x\mapsto u(0,x)$. Given a function $u$
depending on two variables, $u_{t}$ and $u_{x}$ denotes the derivatives of $u$
with respect to the first and the second arguments, respectively. The
Helmholtz operator $1-\partial_{x}^{2}$ and its inverse
$(1-\partial_{x}^{2})^{-1}$ are denoted by $\Lambda^{2}$ and $\Lambda^{-2}$,
respectively. In particular, $\Lambda^{-2}(f)=g\ast f$, where $g$ is the Green
function

$g(x)=\frac{e^{-|x|}}{2}.$ (2.1.1)

We say that $|f(x)|\sim O(e^{ax})$ as $x\nnearrow\infty$ if

$\lim_{x\rightarrow\infty}\frac{|f(x)|}{e^{ax}}=L,$

whereas $|f(x)|\sim o(e^{ax})$ as $x\nnearrow\infty$ if

$\lim_{x\rightarrow\infty}\frac{|f(x)|}{e^{ax}}=0.$

The first definition does not exclude the case $L=0$. If this happens, both
definitions coincide, meaning that if $|f(x)|\sim o(e^{ax})$ as
$x\nnearrow\infty$, then $|f(x)|\sim O(e^{ax})$ as $x\nnearrow\infty$, but the
converse may not be true.

Finally, we recall the Gronwall inequality, e.g., see [20, page 29], which
says that if $M(t)$ is non-negative and $t\geq 0$ and

$u(t)\leq A+\int_{0}^{t}M(s)u(s)ds,$

then $u(t)\leq Ae^{\int_{0}^{t}M(s)ds}$. Note that sometimes the original
inequality is replaced by $u^{\prime}(t)\leq M(t)u(t)$.

We recall that we can rewrite equation (1.0.1) as a hyperbolic evolution
equation

$u_{t}+u^{n}u_{x}+\partial_{x}(1-\partial_{x}^{2})^{-1}\left(\frac{n}{2}u^{n-1}u_{x}^{2}+\frac{n(n+3)}{2(n+1)}u^{n+1}\right)=0.$
(2.1.2)

The results proved in [11, 19, 17] show that if if $u=u(t,x)$ is a solution of
(1.0.1) such that $u(0,x)=u_{0}(x)$, for some $u_{0}\in H^{s}(\mathbb{R})$,
with $s>3/2$, then the corresponding Cauchy problem has a unique solution
$u\in C^{0}([0,T);H^{s}(\mathbb{R}))\cap C^{1}([0,T);H^{s-1}(\mathbb{R}))$.
Unless otherwise mentioned, throughout this paper by a solution $u$ of (1.0.1)
we mean a function $u\in C^{0}([0,T);H^{s}(\mathbb{R}))\cap
C^{1}([0,T);H^{s-1}(\mathbb{R}))$. This fact will be widely used henceforth.

###  2.2 Main results and outline of the paper

Let $g\in C^{1}(\mathbb{R})$ be an arbitrary function and
$u(t,x):=g(t)e^{\alpha x}$, where $\alpha\in\mathbb{R}$ is a constant.
Recalling that $m=u-u_{xx}$, a simple calculation shows that

$m=(1-\alpha^{2})u\quad\text{and}\quad
u^{n}u_{x}-u^{n-2}u_{x}^{3}=\alpha(1-\alpha^{2})u^{n+1}.$

Then, substituting the expressions above into (1.0.2), we easily see that if
$\alpha=\pm 1$ we then have a solution to (1.0.2). Note that these are strong
solutions, that do not necessarily vanishes at infinity (nor satisfy the
condition of local well-posedness in [11, 19, 17]). Its structure, however,
leads to the natural conjecture if we could "glue" both into a solution with
exponential decaying behaviour at $\pm\infty$. This makes us ask if (1.0.2)
could have peakon solutions, which are very famous solutions for the CH
equation, see [2].

Let $a$ be a constant, $v(z):=e^{-|z|}$, $z=x-ct$, $c>0$ and $u(t,x):=av(z)$.
Then, taking the distributional derivatives of the functions, we have
$\phi^{\prime}(z)=-\text{sgn}\,(z)\phi(z)$,
$\phi^{\prime\prime}(z)=2\delta(z)+\phi(z)$. Then

$\begin{array}[]{l}m=-2a\delta(z),\,\,m_{t}=2ac\delta^{\prime}(z),\,\,m_{x}=-2a\delta^{\prime}(z),\,\,u^{n}u_{x}-u^{n-2}u_{x}^{3}=0.\end{array}$

Again, by substituting the expressions above into (1.0.2) and taking
$a:=c^{1/n}$, we again easily check that we have the (peakon) solution

$u(t,x)=c^{1/n}e^{-|x-ct|},$

which is an orbitally stable solution to (1.0.1), see [12].

In [16] was shown that if an initial data of the CH equation, for large values
of $x$, behaves like the peakon solution, then this property persists for the
corresponding solution. Therefore, it is natural to investigate whether the
solutions of (1.0.2) would inherit the same property using the approach in
[16].

Our next results shed light to the above question. In what follows,
$T_{0}\in(0,T)$ is an arbitrary value for which the solutions is defined and
we denote by ${\cal I}_{0}$ the compact set $[0,T_{0}]$.

######  Theorem 2.1.

Assume that $u\in C^{0}({\cal I}_{0};H^{s}(\mathbb{R}))$, $s>3/2$, is a
solution of (1.0.1) with initial condition $u(0,x)=u_{0}(x)$. Assume that for
some $\theta\in(0,1)$,

$|u_{0}(x)|\sim O(e^{-\theta x})\quad\text{and}\quad|u_{0}^{\prime}(x)|\sim
O(e^{-\theta x})\quad\text{as}\quad x\nnearrow\infty.$ (2.2.1)

Then

$|u(t,x)|\sim O(e^{-\theta x})\quad\text{and}\quad|u_{x}(t,x)|\sim
O(e^{-\theta x})\quad\text{as}\quad x\nnearrow\infty,$ (2.2.2)

uniformly in ${\cal I}_{0}$.

Theorem 2.1 could be inferred from [21, Theorem 1.2]. However, the approach
used in [21] is different of ours. Moreover, Theorem 2.1 is a sine qua non
ingredient for establishing our next result.

######  Theorem 2.2.

Let $u\in C^{0}({\cal I}_{0};H^{s}(\mathbb{R}))$, $s>3/2$, be a solution of
(1.0.1) with initial condition $u(0,x)=u_{0}(x)$. Assume that:

  1. 1.

$n$ is odd,

  2. 2.

For some $\alpha\in(1/(n+1),1)$,

$|u_{0}(x)|\sim o(e^{-x})\quad\text{and}\quad|u_{0}^{\prime}(x)|\sim
O(e^{-\alpha x})\quad\text{as}\quad x\nnearrow\infty,$ (2.2.3)

and

  3. 3.

There exists $t_{1}\in{\cal I}_{0}$, $t_{1}>0$, such that

$|u(t_{1},x)|\sim o(e^{-x})\quad\text{as}\quad x\nnearrow\infty.$ (2.2.4)

Then $u\equiv 0$.

######  Remark 2.1.

In case $n$ is odd we could replace $O(e^{-\theta x})$ by $O(e^{-\theta|x|})$
in (2.2.1)–(2.2.2), as well as $o(e^{-x})$ by $o(e^{-|x|})$ in (2.2.3)–(2.2.4)
and $O(e^{-\alpha|x|})$ in place of $O(e^{-\alpha x})$, and then let
$|x|\rightarrow\infty$. This comes from the following observation: if $n$ is
odd, then $v(t,x):=-u(t,-x)$ would also satisfy (1.0.1) and
$v(0,x):=v_{0}(x)=-u_{0}(-x)$. Therefore, we can extend the result in theorems
2.1 and 2.2 for negative values of $x$ through this transformation by applying
it to $v$.

It is natural to ask if (1.0.1) would have or not compactly supported
solutions, see [4, 13]. In fact, it is well known that the CH equation does
not have such solutions for $t>0$. Despite in [16] the asymptotic behaviour of
the solutions was used to prove a non-existence result of the existence of
compactly supported solutions of the CH equation, unfortunately the same
approach cannot guarantee such fact for (1.0.1) in view of the right hand side
of (1.0.2). Therefore, in order to determine whether we could have solutions
of (1.0.1) with compact support we need to look for a different alternative.

######  Theorem 2.3.

Let $u$ be a solution of (1.0.1) with initial data $u(0,x)=u_{0}(x)$. Assume
that:

  1. 1.

there exists a non-empty open rectangle ${\cal
R}=(t_{0},t_{1})\times(x_{0},x_{1})\subset[0,T)\times\mathbb{R}$ such that
$u\big{|}_{{\cal R}}\equiv 0$.

  2. 2.

$n$ is odd.

Then $u\equiv 0$.

######  Theorem 2.4.

Assume that $u$ is a solution of (1.0.1), with initial data
$u(0,\cdot)=u_{0}(\cdot)\not\equiv 0$, satisfying the condition 2 in Theorem
2.3. Given numbers $a$ and $b$, with $a<b$, then
$u(t,\cdot)\big{|}_{[a,b]}\not\equiv 0$, for any $t\in(0,T)$.

Observe that Theorem 2.4 is stronger than Theorem 2.3 since the former is
implied by the latter. Besides, theorem 2.4 has another strong consequence.

######  Corollary 2.1.

Under the conditions in Theorem 2.4, if $u_{0}(x)$ is not identically zero,
then $u(t,\cdot)$ cannot be compactly supported, for each $t>0$.

Theorems 2.1–2.4 are proved in Section 4 whereas in Section 3 we present
technical results that we need in the demonstration of these theorems. Next,
we discuss our achievements in Section 5 and present our conclusions in
Section 6.

##  3 Preliminaries and technical results

In this section we present a couple of essential steps needed for the proof of
the results of the previous section.

######  Proposition 3.1.

If $u$ is a solution of (1.0.1), then the quantities

${\cal H}_{1}(t):=\int_{\mathbb{R}}u(t,x)dx$ (3.0.1)

and

${\cal
H}(t):=\int_{\mathbb{R}}\frac{u(t,x)^{2}+u_{x}(t,x)^{2}}{2}dx=\frac{1}{2}\|u(t,\cdot)\|_{H^{1}(\mathbb{R})}^{2}$
(3.0.2)

are constant. In particular, if $u$ is either non-negative or non-positive,
then $\|u(t,\cdot)\|_{1}$ is constant.

###### Proof.

Assuming (3.0.1), if $u$ does not change its sign, then $u=\pm|u|$. Therefore,
from (3.0.1) we conclude that ${\cal H}_{1}(t)=\pm\|u(t,\cdot)\|_{1}$.

Let us prove that (3.0.1) and (3.0.2) are constant. A straightforward
calculation shows the following formal identities for any $v=v(t,x)$:

$\begin{array}[]{lcl}\displaystyle{\partial_{t}(v-v_{xx})+\partial_{x}\left[\frac{n+2}{2}v^{n+1}-\frac{n}{2}u^{n-1}v_{x}^{2}-v^{n}v_{xx}\right]}&=&\displaystyle{v_{t}-v_{txx}+\frac{(n+1)(n+2)}{2}v^{n}v_{x}}\\\
\\\
\displaystyle{-\frac{n(n-1)}{2}v^{n-2}v_{x}^{3}+2nv^{n-1}v_{x}v_{xx}+v^{n}v_{xxx}}\end{array}$

and

$\begin{array}[]{lcl}\displaystyle{\partial_{t}\left(\frac{v^{2}+v_{x}^{2}}{2}\right)+\partial_{x}\left(\frac{v^{n+2}}{n+2}+\frac{1-n}{2}v^{n}v_{x}^{2}-v^{n+1}v_{xx}-vv_{tx}\right)=}\\\
\\\
\displaystyle{v\left(v_{t}-v_{txx}+\frac{(n+1)(n+2)}{2}v^{n}v_{x}-\frac{n(n-1)}{2}v^{n-2}v_{x}^{3}+2nv^{n-1}v_{x}v_{xx}+v^{n}v_{xxx}\right)}.\end{array}$

If we choose $v=u$ and integrate the expressions above over $\mathbb{R}$, we
obtain

$\frac{d}{dt}\int_{\mathbb{R}}(u-u_{xx})dx=\frac{d}{dt}\int_{\mathbb{R}}udx=0$

and

$\frac{d}{dt}\int_{\mathbb{R}}\frac{u(t,x)^{2}+u_{x}(t,x)^{2}}{2}dx=0.$

The relations above imply the desired results. ∎

Given a solution $u$ of (1.0.1) , for each $t\in[0,T)$ we define

$f_{t}(x):=\frac{n}{2}u^{n-1}u_{x}^{2}+\frac{n(n+3)}{2(n+1)}u^{n+1},$ (3.0.3)

and

$F_{t}(x):=\partial_{x}\Lambda^{-2}\ast f_{t}(x).$ (3.0.4)

Let us denote the families

${\cal F}_{1}=(f_{t})_{t\in[0,T)},\quad{\cal F}_{2}=(F_{t})_{t\in[0,T)}$
(3.0.5)

and their corresponding sub-families

${\cal F}^{\ast}_{1}=(f_{t})_{t\in(0,T)},\quad{\cal
F}_{2}^{\ast}=(F_{t})_{t\in(0,T)}.$ (3.0.6)

Note that ${\cal F}_{1}\subset C^{0}(\mathbb{R})$ whereas ${\cal F}_{2}\subset
C^{1}(\mathbb{R})$.

######  Proposition 3.2.

Let $u$ be a solution of (1.0.1) and $f_{t}(\cdot)$ be the function given by
(3.0.3). Assume that $n$ is odd. Then we have the equivalences
$f_{t}(\cdot)=0\Leftrightarrow\Lambda^{-2}f_{t}(x)=0\Leftrightarrow{\cal
H}(t)=0$, where ${\cal H}(\cdot)$ is the conserved quantity (3.0.2).

###### Proof.

The proof is straightforward once we note that the function (2.1.1) is
positive and $\Lambda^{-2}f_{t}(x)=(g\ast f_{t})(x)$. ∎

Note that Proposition 3.3 has a very strong and beautiful meaning: if we can
find a value a member of ${\cal F}_{1}^{\ast}$ that vanishes identically, then
${\cal H}(t)=0$ for all $t$, in view of the conservation of the
$H^{1}(\mathbb{R})-$norm. Moreover, this implies that ${\cal F}_{1}=\\{0\\}$,
as well as ${\cal F}_{2}=\\{0\\}$. On the contrary, if it is possible to find
a member of ${\cal F}_{1}^{\ast}$ that does not vanish, then any other member
of this family will not be identically zero.

######  Proposition 3.3.

Let $u$ be a solution of (1.0.1) and assume that condition 2 in Theorem 2.3 is
satisfied. If there exists numbers $t^{\ast}$, $x_{0}$ and $x_{1}$ such that
$\\{t^{\ast}\\}\times[x_{0},x_{1}]\subset(0,T)\times\mathbb{R}$,
$f_{t^{\ast}}\big{|}_{(x_{0},x_{1})}\equiv 0$,
$F_{t^{\ast}}(x_{0})=F_{t^{\ast}}(x_{1})$, then ${\cal F}_{1}={\cal
F}_{2}=\\{0\\}$.

###### Proof.

Since $\partial_{x}^{2}\Lambda^{-2}=\Lambda^{-2}-1$, we have

$F_{t^{\ast}}^{\prime}(x)=\partial_{x}F_{t^{\ast}}(x)=(\partial_{x}^{2}\Lambda^{-2})f_{t^{\ast}}(x)=\Lambda^{-2}f_{t^{\ast}}(x)-f_{t^{\ast}}(x).$

Therefore, as long as $x\in(x_{0},x_{1})$, we conclude that
$F_{t^{\ast}}^{\prime}(x)=\Lambda^{-2}f_{t^{\ast}}(x)$. The Fundamental
Theorem of Calculus jointly the condition
$F_{t^{\ast}}(x_{0})=F_{t^{\ast}}(x_{1})$ yield

$0=F_{t^{\ast}}(x_{1})-F_{t^{\ast}}(x_{0})=\int_{x_{0}}^{x_{1}}\Lambda^{-2}f_{t^{\ast}}(x)dx,$

which forces $\Lambda^{-2}f_{t^{\ast}}(x)=0$. The result follows from
Proposition 3.2. ∎

######  Lemma 3.1.

Let $a$ and $b$ real numbers, with $a<b$, and define
$S_{a,b}:\mathbb{R}\rightarrow\mathbb{R}$ given by

$S_{a,b}=\frac{1}{2}\text{sgn}\,{(a-y)}e^{-|a-y|}-\frac{1}{2}\text{sgn}\,{(b-x)}e^{-|b-x|}.$

Then $S_{a,b}\in L^{1}(\mathbb{R})$, $S_{a,b}(y)>0$ for all
$y\in\mathbb{R}\setminus[a,b]$, and we have the identity

$F_{t}(b)-F_{t}(a)=\int_{\mathbb{R}}S_{a,b}(y)f_{t}(y)dy$ (3.0.7)

for each $t\in[0,T)$.

###### Proof.

See [9, pages 10-11],[18]. ∎

Combining Lemma 3.1, Proposition 3.2 and the fact that if
$f_{t^{\ast}}(\cdot)$ vanishes on $(a,b)$, then $S_{a,b}(y)f_{t^{\ast}}(y)\geq
0$, we have the following result (see also [10]).

######  Lemma 3.2.

Let $u$ be a solution of (1.0.1) and assume that condition 2 in Theorem 2.3 is
satisfied. If there exists numbers $a$ and $b$, with $a<b$, such that
$F_{t^{\ast}}(a)=F_{t^{\ast}}(b)$ and $f_{t^{\ast}}\big{|}_{(a,b)}\equiv 0$,
for some $t^{\ast}\in(0,T)$, then $f_{t^{\ast}}(\cdot)\equiv 0$.

##  4 Proof of the main results

We now combine the results of the previous section to present the
demonstration of the theorems announced in Section 2.

To begin with, we firstly prove theorems 2.3 and 2.4 and then the other
results. Note that the conditions in theorems 2.3 and 2.4 imply that
$f(\cdot)$ is non-negative.

Proof of Theorem 2.3. From (2.1.2) we have

$F_{t}(x)=-(u_{t}+u^{n}u_{x})(t,x),\quad t>0.$ (4.0.1)

Therefore, if $u$ vanishes on ${\cal R}$, then $F_{t}(x)=0$ provided that
$(t,x)\in{\cal R}$. Let us choose $t^{\ast}\in(t_{0},t_{1})$ and real numbers
$a$ and $b$ such that $x_{0}<a<b<x_{1}$. Then

$F_{t^{\ast}}(b)=F_{t^{\ast}}(a)=0.$

Since $f_{t^{\ast}}(x)\geq 0$ and $f_{t^{\ast}}\big{|}_{[a,b]}\equiv 0$, then
Proposition 3.3 combined with Proposition 3.2 imply the result. $\square$

Proof of Theorem 2.4. Let us assume that we could find $t^{\ast}\in(0,T)$ and
an interval $[a,b]$ such that $u(t^{\ast},x)=0$, $x\in[a,b]$. Evaluating
(1.0.2) at $t^{\ast}$ and letting $x\in[a,b]$, we conclude that

$(1-\partial_{x}^{2})u_{t}(t^{\ast},x)=0,$

which implies that $u_{t}(t^{\ast},x)=0$ and by (4.0.1) we have
$F_{t^{\ast}}(a)=u_{t}(t^{\ast},a)=0=u_{t}(t^{\ast},b)=F_{t^{\ast}}(b)$. Lemma
3.2 forces $f_{t^{\ast}}(\cdot)\equiv 0$ and Proposition 3.2 implies that
$u_{0}\equiv 0$. $\square$

Proof of Corollary 2.1. If $u$ were compactly supported, fixed
$t^{\ast}\in(0,T)$, we would then be able find $a,b\in\mathbb{R}$, with $a<b$,
such that $u(t,\cdot)\big{|}_{[a,b]}\equiv 0$, which would be a contradiction
with Theorem 2.4. $\square$

Before proving theorems 2.1 and 2.2, let us define

$M:=\sup_{t\in{\cal I}_{0}}\|u(t,\cdot)\|_{H^{s}(\mathbb{R})}+\sup_{t\in{\cal
I}_{0}}\|u_{x}(t,\cdot)\|_{H^{s}(\mathbb{R})}<\infty.$ (4.0.2)

Also, in what follows $p$ denotes a positive integer, and we would like to
note that $\partial_{x}g\in L^{1}(\mathbb{R})$, where $g$ is given by (2.1.1),
and $u^{n-1}u_{x},\,u^{n+1}\in L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$
in view of the Sobolev Embedding Theorem [20, Proposition 1.3] and the Algebra
Property [20, page 320, Exercise 6]. Therefore, we have

${\cal F}_{1}\subset L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})\cap
C^{0}(\mathbb{R})\quad\text{and}\quad{\cal F}_{2}\subset L^{1}(\mathbb{R})\cap
L^{\infty}(\mathbb{R})\cap C^{1}(\mathbb{R})$

where the families ${\cal F}_{1}$ and ${\cal F}_{2}$ are given by (3.0.5).

In addition, noticing that if $h\in L^{1}(\mathbb{R})\cap
L^{\infty}(\mathbb{R})$, then $h\in L^{r}(\mathbb{R})$, for all $1\leq
r\leq\infty$, from which we have

$\lim_{r\rightarrow\infty}\|h\|_{r}=\|h\|_{\infty}.$

Finally, in the next pages we make use of several claims, whose demonstrations
are left to the Appendix. This is done in order to allow a more fluid reading
and understanding of the demonstration and avoid moderately lengthy technical
demonstrations of parts of the theorems.

Proof of Theorem 2.1 Our strategy for proving Theorem 2.1 is the following: we
firstly show that both $\|u(t,\cdot)\|_{2p}$ and $\|u_{x}(t,\cdot)\|_{2p}$ is
bounded for each positive integer $p$. Therefore, we conclude that both
$\|u(t,\cdot)\|_{\infty}$ and $\|u_{x}(t,\cdot)\|_{\infty}$ are also bounded,
for each $t\in{\cal I}_{0}$. Next, we construct a sequence of piecewise smooth
functions $\phi_{N}(x)$, such that $\phi_{N}(x)\rightarrow e^{\theta x}$
pointwise for $x>0$, such that $\|\phi_{N}(\cdot)u(t,\cdot)\|_{\infty}$ and
$\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{\infty}$ are bounded, for each $N$. More
precisely, we show the existence of $L\geq 0$ such that
$|\phi_{N}(x)u(t,x)|+|\phi_{N}(x)u_{x}(t,x)|$ is bounded from above by $L$ (in
particular, if $u_{0}(x_{0})\neq 0$, for some $x_{0}$, then necessarily
$L>0)$. Then, allowing $N\rightarrow\infty$ and using (2.2.1) we conclude
(2.2.2).

Let us begin the demonstration. Multiplying (2.1.2) by $u^{2p-1}$, using
(3.0.4) and integrating the resulting expression over $\mathbb{R}$ with
respect to $x$, we obtain

$\frac{d}{dt}\int_{\mathbb{R}}u(t,x)^{2p}dx+\int_{\mathbb{R}}u(t,x)^{2p+n-1}u_{x}(t,x)dx+\int_{\mathbb{R}}u^{2p-1}(t,x)F_{t}(x)dx=0.$
(4.0.3)

Let

$I_{1}:=\frac{d}{dt}\int_{\mathbb{R}}u(t,x)^{2p}dx,\,\,I_{2}:=\int_{\mathbb{R}}u(t,x)^{2p+n-1}u_{x}(t,x)dx,\,\,I_{3}:=\int_{\mathbb{R}}u(t,x)^{2p-1}F_{t}(x)dx.$

It is straightforward that

$I_{1}=\|u(t,\cdot)\|_{2p}^{2p-1}\frac{d}{dt}\|u(t,\cdot)\|_{2p}.$

Claim 1: $|I_{2}|\leq M^{n}\|u(t,\cdot)\|_{2p}^{2p}$ and
$|I_{3}|\leq\|u(t,\cdot)\|_{2p}^{2p-1}\|F_{t}(\cdot)\|_{2p}$.

From (4.0.3) we have the inequality $I_{1}\leq|I_{2}|+|I_{3}|$. Therefore,

$\frac{d}{dt}\|u(t,\cdot)\|_{2p}\leq
M^{n}\|u(t,\cdot)\|_{2p}+\|F_{t}(\cdot)\|_{2p},$

that is

$\|u(t,\cdot)\|_{2p}\leq\left(\|u_{0}(\cdot)\|_{2p}+\int_{0}^{t}e^{-M^{n}\tau}\|F_{\tau}(\cdot)\|_{2p}d\tau\right)e^{M^{n}t}.$

Taking the limit $p\rightarrow\infty$, we have

$\|u(t,\cdot)\|_{\infty}\leq\left(\|u_{0}(\cdot)\|_{\infty}+\int_{0}^{t}e^{-M^{n}\tau}\|F_{\tau}(\cdot)\|_{\infty}d\tau\right)e^{M^{n}t}.$
(4.0.4)

Let us now consider (2.1.2). Differentiating it with respect to $x$,
multiplying the result by $u_{x}^{2p-1}$ and rearranging the terms, we have

$u_{x}^{2p-1}u_{tx}+nu^{n-1}u_{x}^{2p+1}+u^{n}u^{2p-1}u_{xx}+u_{x}^{2p-1}\partial_{x}F_{t}(x)=0.$
(4.0.5)

Defining

$I_{4}=u_{x}^{2p-1}u_{tx},\,\,I_{5}=nu^{n-1}u_{x}^{2p+1},\,\,I_{6}=u^{n}u^{2p-1}u_{xx},\,\,I_{7}=u_{x}^{2p-1}\partial_{x}F_{t}(x)$

and integrating (4.0.5) with respect to $x$ over $\mathbb{R}$, we obtain the
inequality

$I_{4}\leq|I_{5}|+|I_{6}|+|I_{7}|.$

Again, it is straightforward the relation

$I_{4}=\|u_{x}(t,\cdot)\|_{2p}^{2p-1}\frac{d}{dt}\|u_{x}(t,\cdot)\|_{2p}$

and similarly as before, we have the following claim:

Claim 2: $|I_{5}|\leq nM^{n}\|u_{x}(t,\cdot)\|_{2p}^{2p}$, $|I_{6}|\leq
nM^{n}\|u_{x}(t,\cdot)\|_{2p}^{2p}$ and
$|I_{7}|\leq\|u_{x}(t,\cdot)\|_{2p}^{2p-1}\|\partial_{x}F_{t}(\cdot)\|_{2p}$.

These estimates give the differential inequality

$\frac{d}{dt}\|u_{x}(t,\cdot)\|_{2p}\leq
2nM^{n}\|u_{x}(t,\cdot)\|_{2p}+\|\partial_{x}F_{t}(\cdot)\|_{2p},$

which yields the estimate

$\|u_{x}(t,\cdot)\|_{2p}\leq\left(\|u_{0}^{\prime}(\cdot)\|_{2p}+\int_{0}^{t}e^{-2nM^{n}\tau}\|\partial_{x}F_{\tau}(\cdot)\|_{2p}d\tau\right)e^{2nM^{n}t}.$

By letting $p\rightarrow\infty$, we have

$\|u_{x}(t,\cdot)\|_{\infty}\leq\left(\|u_{0}^{\prime}(\cdot)\|_{\infty}+\int_{0}^{t}e^{-2nM^{n}\tau}\|\partial_{x}F_{\tau}(\cdot)\|_{\infty}d\tau\right)e^{2nM^{n}t}.$

Let $N$ be a positive integer and consider the family
$\phi_{N}:\mathbb{R}\rightarrow\mathbb{R}$, given by

$\phi_{N}(x)=\left\\{\begin{array}[]{lcl}1,&\text{if}&x\leq 0,\\\ \\\
e^{\theta x},&\text{if}&0<x<N,\\\ \\\ e^{\theta N},&\text{if}&x\geq
n,\end{array}\right.$ (4.0.6)

Clearly $(\phi_{N})_{N\in\mathbb{N}}\subset C^{0}(\mathbb{R})$ and, for each
$N$, $\phi_{N}$ is piecewise smooth and $\phi_{N}^{\prime}(x)\leq\phi_{N}(x)$
almost everywhere.

Multiplying (2.1.2) by $\phi_{N}(\phi_{N}u)^{2p-1}$ and again integrating with
respect to $x$ over $\mathbb{R}$, we obtain $I_{8}=-I_{9}-I_{10}$, where

$I_{8}=\|\phi_{N}(\cdot)u(t,\cdot)\|_{2p}^{2p-1}\frac{d}{dt}\|\phi_{N}(\cdot)u(t,\cdot)\|_{2p},$
$I_{9}=\int_{\mathbb{R}}(\phi_{N}(x)u(t,x))^{2p}u(t,x)^{n-1}u_{x}(t,x)dx\quad\text{and}\quad
I_{10}=\int_{\mathbb{R}}(\phi_{N}(x)u(t,x))^{2p-1}\phi_{N}(x)F_{t}(x)dx.$

Claim 3: $|I_{9}|\leq M^{n}\|\phi_{N}(\cdot)u(t,\cdot)\|_{2p}^{2p}$ and
$|I_{10}|\leq\|\phi_{N}(\cdot)u(t,\cdot)\|_{2p}^{2p-1}\|\phi_{N}(\cdot)F_{t}(\cdot)\|_{2p}$.

Since $I_{8}=-I_{9}-I_{10}$, the estimates in the claim gives the differential
inequality

$\frac{d}{dt}\|\phi_{N}(\cdot)u(t,\cdot)\|_{2p}\leq
M^{n}\|\phi_{N}(\cdot)u(t,\cdot)\|_{2p}+\|\phi_{N}(\cdot)F_{t}(\cdot)\|_{2p},$

that jointly with Gronwall’s inequality and letting $p\rightarrow\infty$,
gives

$\|\phi_{N}(\cdot)u(t,\cdot)\|_{\infty}\leq\left(\|\phi_{n}(\cdot)u_{0}(\cdot)\|_{\infty}+\int_{0}^{t}e^{-M^{n}t}\|\phi_{N}(\cdot)F_{\tau}(\cdot)\|_{\infty}d\tau\right)e^{M^{n}t}$
(4.0.7)

Now we multiply (4.0.5) by $\phi_{n}(\phi_{n}u_{x})^{2p-1}$ and proceeding
similarly as the previous cases, we obtain the inequality
$I_{11}\leq|I_{12}|+|I_{13}|+|I_{14}|$, where

$\begin{array}[]{lcl}I_{11}&=&\displaystyle{\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{2p}^{2p-1}\frac{d}{dt}\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{2p},}\\\
\\\
I_{12}&\leq&\displaystyle{\int_{\mathbb{R}}nu(t,x)^{n-1}(\phi_{N}(x)u_{x}(t,x)^{2p})u_{x}(t,x)dx},\\\
\\\
I_{13}&\leq&\displaystyle{\int_{\mathbb{R}}u(t,x)^{n}(\phi_{N}(x)u_{x}(t,x))^{2p-1}(\phi_{N}(x)u_{xx}(t,x))dx},\\\
\\\
I_{14}&\leq&\displaystyle{\int_{\mathbb{R}}\phi_{N}(x)(\phi_{N}(x)u_{x}(t,x)^{2p-1})\partial_{x}F_{t}(x)dx},\end{array}$

and we have our last claim.

Claim 4: $|I_{12}|\leq nM^{n}\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{2p}^{2p}$,
$|I_{13}|\leq(n+1)M^{n}\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{2p}^{2p}$ and
$|I_{14}|\leq\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{2p}^{2p-1}\|\phi_{N}(\cdot)\partial_{x}F_{t}(\cdot)\|_{2p}$.

Therefore, we have the differential inequality

$\frac{d}{dt}\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{2p}\leq
2(n+1)M^{n}\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{2p}+\|\phi_{N}(\cdot)\partial_{x}F_{t}(\cdot)\|_{2p},$

which yields, after using the Gronwall inequality, and taking
$p\rightarrow\infty$:

$\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{\infty}\leq\left(\|\phi_{N}(\cdot)u_{0}^{\prime}(\cdot)\|_{\infty}+\int_{0}^{t}e^{-2(n+1)M\tau}\|\phi_{N}(\cdot)\partial_{x}F_{\tau}(\cdot)\|_{\infty}d\tau\right)e^{2(n+1)Mt}.$
(4.0.8)

We not recall that there exists a positive constant $c$ such that

$\begin{array}[]{lcl}\displaystyle{\|\phi_{N}(\cdot)F_{\tau}(\cdot)\|_{\infty}}&\leq&\displaystyle{c\left(\|\phi_{N}(\cdot)u(t,\cdot)\|_{\infty}+\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{\infty}\right)}\\\
\\\
\displaystyle{\|\phi_{N}(\cdot)\partial_{x}F_{\tau}(\cdot)\|_{\infty}}&\leq&\displaystyle{c\left(\|\phi_{N}(\cdot)u(t,\cdot)\|_{\infty}+\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{\infty}\right).}\end{array}$
(4.0.9)

Also, we recall that $e^{-M^{n}t},e^{-2(n+1)t}\in(0,1)$. Adding (4.0.8) to
(4.0.7), and taking into account that $t\in{\cal I}_{0}=[0,T_{0}]$, we have

$\begin{array}[]{l}\displaystyle{\|\phi_{N}(\cdot)u(t,\cdot)\|_{\infty}+\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{\infty}}\leq\displaystyle{e^{2(n+1)MT_{0}}\Big{(}\|\phi_{N}(\cdot)u_{0}(\cdot)\|_{\infty}+\|\phi_{N}(\cdot)u^{\prime}_{0}(\cdot)\|_{\infty}\Big{)}}\\\
\\\
\displaystyle{+ce^{2(n+1)MT_{0}}\int_{0}^{t}\left(\|\phi_{N}(\cdot)u(t,\cdot)\|_{\infty}+\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{\infty}\right)d\tau}.\end{array}$

Let $K=\max\\{e^{2(n+1)MT_{0}},ce^{2(n+1)MT_{0}}\\}$. The latter inequality
can then be rewritten as

$\begin{array}[]{lcl}\displaystyle{\|\phi_{N}(\cdot)u(t,\cdot)\|_{\infty}+\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{\infty}}&\leq&\displaystyle{K\Big{[}\left(\|\phi_{N}(\cdot)u_{0}(\cdot)\|_{\infty}+\|\phi_{N}(\cdot)u^{\prime}_{0}(\cdot)\|_{\infty}\right)}\\\
\\\
&&\displaystyle{+\int_{0}^{t}\left(\|\phi_{N}(\cdot)u(t,\cdot)\|_{\infty}+\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{\infty}\right)d\tau\Big{]}},\end{array}$

which, after application of Gronwall inequality, yields

$\|\phi_{N}(\cdot)u(t,\cdot)\|_{\infty}+\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{\infty}\leq\kappa\left(\|\phi_{N}(\cdot)u_{0}(\cdot)\|_{\infty}+\|\phi_{N}(\cdot)u^{\prime}_{0}(\cdot)\|_{\infty}\right),$
(4.0.10)

where $\kappa:=KT_{0}$.

On the other hand, we have

$|\phi_{N}(x)u_{0}(x)|\leq|\max\\{1,e^{\theta
x}\\}u_{0}(x)|\leq\|\max\\{1,e^{\theta\cdot}\\}u_{0}(\cdot)\|_{\infty}=:L_{1}/2$

and

$|\phi^{\prime}_{N}(x)u_{0}(x)|\leq|\max\\{1,e^{\theta
x}\\}u^{\prime}_{0}(x)|\leq\|\max\\{1,e^{\theta\cdot}\\}u^{\prime}_{0}(\cdot)\|_{\infty}=:L_{2}/2$

Let $L:=\max\\{\kappa L_{1},\kappa L_{2}\\}$. The inequalities above, jointly
with (4.0.10), implies that

$|\phi_{N}(x)u_{0}(x)|+|\phi^{\prime}_{N}(x)u_{0}(x)|\leq\|\phi_{N}(\cdot)u(t,\cdot)\|_{\infty}+\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{\infty}\leq
L.$ (4.0.11)

Note that $L$ does not depend on $N$ and if $x>0$, taking the limit
$N\rightarrow\infty$, we conclude that

$\phi_{N}u(t,x)\rightarrow e^{\theta
x}u(t,x)\quad\text{and}\quad\phi_{N}u_{x}(t,x)\rightarrow e^{\theta
x}u_{x}(t,x).$

Therefore, if we take the limit $N\rightarrow\infty$ and if $x>0$, (4.0.11)
reads

$|e^{\theta x}u(t,x)|+|e^{\theta x}u_{x}(t,x)|\leq L,$

which implies the result. In particular, note that if $u_{0}(x_{0})\neq 0$ for
some $x_{0}$, then (4.0.11) implies that $L$ is positive. $\square$

Proof of Theorem 2.2: Integrating (4.0.1) with respect to $t$ we have

$-\int_{0}^{t_{1}}F_{\tau}(x)d\tau=u(t_{1},x)-u_{0}(x)+\int_{0}^{t_{1}}(uu^{n})(\tau,x)d\tau=:J.$
(4.0.12)

By Theorem 2.1, we conclude that $(uu^{n})(\tau,x)\sim O(e^{-\alpha(n+1)x})$.
Moreover, since $1/(n+2<\alpha<1)$, then $1-\alpha(n+1)<0$. Therefore,

$\lim_{x\rightarrow\infty}\frac{|(uu^{n})(t,x)|}{e^{-x}}=\lim_{x\rightarrow\infty}\Big{(}\underbrace{\frac{|(uu)^{n}(t,x)|}{e^{-\alpha(n+1)x}}}_{bounded}\,\,\underbrace{\frac{e^{-\alpha(n+1)x}}{e^{-x}}}_{\rightarrow
0}\Big{)}=0,$

meaning that $(uu^{n})(\tau,x)\sim o(e^{-x})$. Therefore, it is immediate that
$J\sim o(e^{-x})$.

We will prove the result by contradiction: we assume that $u$ is not
identically zero. Then we show that if this is true, we can find a positive
constant $K$ such that

$-\int_{0}^{t_{1}}F_{\tau}(x)d\tau\geq Ke^{-x}$

which contradicts $J\sim o(e^{-x})$.

We begin by recalling that $F_{t}(x)=(\partial_{x}g)\ast f_{t}(x)$, where
$f_{t}(x)$ is given by (see (3.0.3))

$f_{t}(x)=\frac{n}{2}u(t,x)^{n-1}\left(u_{x}(t,x)^{2}+\frac{n+3}{n+1}u(t,x)^{2}\right)\sim
o(e^{-(n+1)\alpha x}).$

Let us define

$\rho(x):=\int_{0}^{t_{1}}f_{\tau}(x)dx.$

Then we have the following identity:

$-\int_{0}^{t_{1}}F_{\tau}(x)d\tau=(\partial_{x}g)\ast\rho(x).$ (4.0.13)

We also observe that

$(\partial_{x}g)\ast\rho(x)=-\frac{1}{2}e^{-x}\int_{-\infty}^{x}e^{y}\rho(y)dy+\frac{1}{2}e^{x}\int_{x}^{\infty}e^{-y}\rho(y)dy.$
(4.0.14)

We note that

$e^{x}\int_{x}^{\infty}\rho(y)dy=o(1)e^{x}\int_{x}^{\infty}e^{-(n+2)\alpha
y}dy\sim o(1)e^{-(n+1)x}=o(e^{-(n+1)x})\sim o(e^{-x}).$

On the other hand, if $n$ is odd, then $\rho(y)>0$ (recall that $u\not\equiv
0$) and

$x\mapsto\int_{-\infty}^{x}e^{y}\rho(y)dy$

is increasing, so that for $x$ sufficiently large, we can find a constant
$k>0$ such that

$\int_{-\infty}^{x}e^{y}\rho(y)dy\geq k>0.$

The latter result, jointly with (4.0.14) and (4.0.13) imply that

$-\int_{0}^{t_{1}}F_{\tau}(x)d\tau\geq\frac{k}{2}e^{-x},$

which shows the desired contradiction. $\square$

##  5 Discussion

As far as the author knows, the first (and unique) work considering asymptotic
properties of (1.0.1) with $n\neq 1$ is [21], in which the ideas introduced by
Brandolese [1] for the studying the CH equation for (1.0.1) were extended. In
the present work we study these properties in a different way, using the ideas
from [16] for proving theorems 2.1 and 2.2.

We note that our results are mostly concerned when $n$ is odd. Such condition
comes from the necessity that the function (3.0.3) be non-negative.
Essentially, this also explains why the assumption that $n$ is odd appears in
[21, Theorem 1.3]. We could replace this hypothesis by assuming that $u$ is
either non-negative or non-positive, however, whether (1.0.1) has non-negative
or non-positive solutions for $n>1$ is an open question. For the CH equation
this is proved by using a diffeomorphism constructed from the its solutions
[3, Theorem 3.1]. Although we could find a similar diffeomorphism for (1.0.1),
we do not have a similar result as [3, Lemma 3.2]. Moreover, the impossibility
of the extension of such result to (1.0.1) make us also unable to proceed
similarly as in [16, Theorem 1.4] to prove that (1.0.1) does not have
compactly supported solutions corresponding to an initial data compactly
supported.

Our way to face this issue was then use the ideas recently introduced in [18]
for dealing with unique continuation of solutions of the CH equation to
address the question under consideration and then proceed similarly as [9] for
the study of solutions compactly supported.

As pointed out in [8, 9, 10], these ideas are essentially geometric, see these
references for a better discussion.

##  6 Conclusion

In this paper we studied persistence and asymptotic properties of an extension
of the CH equation, given by (1.0.1). Our main results are theorems 2.1–2.4.
In particular, we complement the results in [21] by showing that (1.0.1) does
not have compactly supported solutions, as well as we extend to (1.0.1) (for
odd $n$) the results proved in [18], see also [9, 10], regarding unique
continuation of solutions.

## Acknowledgements

CNPq is also thanked for financial support, through grant nº 404912/2016-8.

## References

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##  Appendix A Proof of Claim 1

Note that

$\begin{array}[]{lcl}|I_{2}|&\leq&\displaystyle{\|u_{x}(t,\cdot)\|_{\infty}\int_{\mathbb{R}}|u(t,x)|^{2p+n-1}dx\leq\|u(t,\cdot)\|_{\infty}\|u_{x}(t,\cdot)\|_{\infty}\int_{\mathbb{R}}|u(t,x)|^{2p}dx}\\\
\\\
&=&\displaystyle{\|u(t,\cdot)\|_{\infty}\|u_{x}(t,\cdot)\|_{\infty}\|u(t,\cdot)\|_{2p}^{2p}}\end{array}$

The estimate follows from the inequality above jointly with (4.0.2).

With respect to $|I_{3}|$, let $q$ the conjugate of $2p$, that is

$q=\frac{2p}{2p-1}.$

By the Hölder inequality, we have

$|I_{3}|\leq\|u(t,\cdot)^{2p-1}\|_{q}\|F_{t}(\cdot)\|_{2p}.$

On the other hand, we have

$\|u(t,\cdot)^{2p-1}\|_{q}=\Big{(}\int_{\mathbb{R}}|u(t,x)|^{(2p-1)q}\Big{)}^{1/q}=\Big{(}\int_{\mathbb{R}}u(t,x)^{(2p)}\Big{)}^{1/q}=\|u(t,\cdot)\|_{2p}^{2p/q}.$

##  Appendix B Proof of Claim 2

Considering $I_{5}$, we have

$|I_{5}|\leq
n\int_{\mathbb{R}}|u(t,x)|^{n-1}|u_{x}(t,x)||u_{x}(t,x)|^{2p}dx\leq
n\|u(t,\cdot)\|_{\infty}^{n-1}\|u_{x}(t,\cdot)\|_{\infty}\|u_{x}(t,\cdot)\|^{2p}_{2p}.$

Integrating $I_{6}$ by parts, we have

$I_{6}=-\frac{n}{2p}\int_{\mathbb{R}}u^{n-1}(t,x)u_{x}(t,x)u_{x}(t,x)^{2p}dx.$

Therefore, we have

$|I_{6}|\leq
n\|u(t,\cdot)\|_{\infty}^{n-1}\|u_{x}(t,\cdot)\|_{\infty}\|u(t,\cdot)\|_{2p}^{2p}.$

Finally, we have

$|I_{7}|\leq\int_{\mathbb{R}}|u_{x}(t,x)|^{2p-1}|\partial_{x}F_{t}(x)|dx\leq\|u_{x}(t,\cdot)\|_{2p}^{2p-1}\|\partial_{x}F_{t}(x)\|_{2p},$

where we have used again the Hölder’s inequality. The result is a consequence
of the inequalities above and (4.0.2).

##  Appendix C Proof of Claim 3

Note that

$|I_{9}|\leq\|u(t,\cdot)\|_{\infty}^{n-1}\|u_{x}(t,\cdot)\|_{\infty}\int_{\mathbb{R}}(\phi_{N}(x)u(t,x))^{2p}dx\leq\|u(t,\cdot)\|_{\infty}^{n-1}\|u_{x}(t,\cdot)\|_{\infty}\|\phi_{N}(\cdot)u(t,\cdot)\|_{2p}^{2p}$

and, from Hölder’s inequality, we have

$|I_{10}|\leq\int_{\mathbb{R}}|\phi_{N}(x)u(t,x)|^{2p-1}|\phi_{N}(x)F_{t}(x)|dx\leq\|\phi_{N}(\cdot)u(t,\cdot)\|_{2p}^{2p-1}\|\phi_{N}(\cdot)F_{t}(\cdot)\|_{2p}.$

The result is again a consequence of the inequalities above and (4.0.2).

##  Appendix D Proof of Claim 4

It is straightforward that

$|I_{12}|\leq
n\|u(t,\cdot)\|_{\infty}^{n-1}\|u_{x}(t,\cdot)\|_{\infty}\int_{\mathbb{R}}(\phi_{N}(x)u_{x}(t,x))^{2p}dx$

and

$\int_{\mathbb{R}}(\phi_{N}(x)u_{x}(t,x))^{2p}dx\leq\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{2p}^{2p}.$

In regard to $I_{13}$, integration by parts gives

$I_{13}=-\frac{n}{2p}\int_{\mathbb{R}}u(t,x)^{n-1}u_{x}(t,x)(\phi_{N}(x)u_{x}(t,x))^{2p}dx-\int_{\mathbb{R}}u(t,x)^{n}(\phi_{N}(x)u_{x}(t,x))^{2p-1}\phi^{\prime}_{n}(x)u_{x}(t,x)dx,$

which implies after noticing that $\phi_{N}(x)\leq\phi_{N}(x)$ a.e.,

$|I_{13}|\leq(n+1)\Big{(}\|u(t,\cdot)\|_{\infty}^{n-1}\|u_{x}(t,\cdot)\|_{\infty}+\|u(t,\cdot)\|_{\infty}^{n}\Big{)}\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{2p}^{2p}.$

With respect to $I_{14}$, using the Hölder inequality we have

$|I_{14}|\leq\int_{\mathbb{R}}|\phi_{N}(x)u_{x}(t,x)^{2p-1}||\phi_{N}(x)\partial_{x}F_{t}(x)|dx\leq\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{2p}^{2p-1}\|\phi_{N}(\cdot)\partial_{x}F_{t}(\cdot)\|_{2p}.$

The inequalities above jointly with (4.0.2) complete the demonstration of
Claim 4.

##  Appendix E Proof of estimates (4.0.9)

We begin by noticing that
$|\phi_{N}(x)F_{\tau}(x)|=|\phi_{N}(x)((\partial_{x}g)\ast f_{\tau})(x)|$. By
(3.0.3) and (4.0.2), and [15, Lemma 2.2], which shows that

$\phi_{N}(x)\int_{\mathbb{R}}\frac{e^{-|x-y|}}{\phi_{N}(y)}dy=\frac{2-\theta}{1-\theta}=:c_{0},$

we have

$\begin{array}[]{lcl}\displaystyle{|\phi_{N}(x)F_{\tau}(x)|}&\leq&\displaystyle{k\|u\|_{\infty}^{n-1}\phi_{N}(x)\int_{\mathbb{R}}e^{-|x-y|}\left(u^{2}+u_{x}^{2}\right)(\tau,y)dy}\\\
\\\
&\leq&\displaystyle{kM^{n-1}\Bigg{[}\left(\phi_{N}(x)\int_{\mathbb{R}}\frac{e^{-|x-y|}}{\phi_{N}(y)}dy\right)\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{\infty}\|u_{x}(t,\cdot)\|_{\infty}}\\\
\\\
&&\displaystyle{\left(\phi_{N}(x)\int_{\mathbb{R}}\frac{e^{-|x-y|}}{\phi_{N}(y)}dy\right)\|\phi_{N}(\cdot)u(t,\cdot)\|_{\infty}\|u(t,\cdot)\|_{\infty}\Bigg{]}}\\\
\\\
&\leq&\displaystyle{kM^{n}c_{0}(\|\phi_{N}(\cdot)u(t,\cdot)\|_{\infty}+\|\phi_{N}(\cdot)u_{x}(t,\cdot)\|_{\infty}).}\end{array}$

Since $\partial_{x}^{2}g=g-\delta$, where $\delta$ is the Dirac $\delta$
distribution, the same procedure shows the second estimate in (4.0.9) and for
this reason its proof is omitted.