# Susceptibility of Adversarial Attack on Medical Image Segmentation Models

Zhongxuan Wang * Both authors contributed equally to this paper. Saint
Andrew’s School
Boca Raton, United States of America
<EMAIL_ADDRESS>Leo Xu  Lynbrook High School
San Jose, United States of America
<EMAIL_ADDRESS>
###### Abstract

The nature of deep neural networks has given rise to a variety of attacks, but
little work has been done to address the effect of adversarial attacks on
segmentation models trained on MRI datasets. In light of the grave
consequences that such attacks could cause, we explore four models from the
U-Net family and examine their responses to the Fast Gradient Sign Method
(FGSM)[1] attack.

We conduct FGSM attacks on each of them and experiment with various schemes to
conduct the attacks. In this paper, we find that medical imaging segmentation
models are indeed vulnerable to adversarial attacks and that there is a
negligible correlation between parameter size and adversarial attack success.
Furthermore, we show that using a different loss function than the one used
for training yields higher adversarial attack success, contrary to what the
FGSM authors suggested. In future efforts, we will conduct the experiments
detailed in this paper with more segmentation models and different attacks. We
will also attempt to find ways to counteract the attacks by using model
ensembles or special data augmentations. Our code is available at
https://github.com/ZhongxuanWang/adv_attk

###### Index Terms:

Adversarial attack, Fast Gradient Sign Method, image segmentation, medical
imaging, U-Net, U-Net++

##  I Introduction

Today, deep convolutional neural networks (CNNs)[2] have become increasingly
popular in medical imaging, playing a role in the classification brain tumors,
detection of organ boundaries, or segmentation of organ tumors. Since CNNs can
exploit the spatial information present in images[2], they have been widely
used in hospitals to provide doctors with valuable insights at an increased
speed. Among all the medical imaging tasks, image segmentation is arguably one
of the more challenging ones since it needs to leverage both global and local
features to create masks for objects. While image classification helps doctors
know the class of the image as a whole, and object detection helps doctors
know the general location of the object, image segmentation allows doctors to
see the boundaries of objects of interest clearly [3]. Their differences are
illustrated in Figure 1.

Figure 1: Difference between image classification, object detection, and image
segmentation methods in the medical imaging field. Predictions are simulated
except for the image segmentation method.

To leverage CNNs in biomedical image segmentation, Ronneberger et al. proposed
U-Net, a revolutionary architecture that consists of a contraction path and a
symmetric expansion path [4]. Recently, many variants of U-Net have been
proposed to achieve state-of-the-art (SOTA) performances in various medical
imaging tasks. Examples of a few include nested U-Nets, U-Nets with dense skip
connections to learn full-scale semantic information [5] [6], and U-Net with
transformer-based encoder or decoder to learn long-range semantic information
[7] [8]. These U-Net variants have all shown superior performances over the
original U-Net.

Although the U-Net family possesses great potential in image segmentation for
MRI data, recent studies have prioritized model performance over security. In
fact, a variety of attacks have recently arisen to intentionally fool models
into making incorrect predictions with high confidence by modifying the
training dataset, testing dataset, model parameters, along others [9]. Given
the confidentiality of medical imaging datasets, it is usually impractical to
poison the training dataset an MRI model was trained on or modify its
parameters. Thus, poisoning inference data is a much larger concern for
doctors. One of the methods for attacking inferencing data is known as a
white-box adversarial attack, which assumes that attackers cannot modify the
training data or the model but know about the model such as its architecture
and weights.

In the context of medical image segmentation, successful adversarial attacks
could incur hefty and irreversible consequences. For instance, if poisoned
tumor segmentations misled doctors, doctors may overlook portions of tumor
tissues that could cause death. If doctors relied on compromised tumor
segmentation images to kill diseased tissues, both benign and vital parts of
the organ may be damaged permanently. Unfortunately, most doctors are not
trained to discern poisoned data from those unaffected, nor are they trained
to take deterrent measures. To make the matter worse, Ma et al. pointed out
that medical imaging models are more vulnerable to adversarial attacks than
other types of models [10]. Thus, studying the effect of adversarial attacks
on medical images has overarching significance.

Most of the current research on adversarial attacks and related defense
techniques only involve image classification [11] [12], which outputs a
confidence score for its classification. In contrast, little research has been
done adversarial attacks on image segmentation tasks. Further, most papers[13]
[14] [15] exploring adversarial attacks on image segmentation datasets have
been done using the ImageNet [16] dataset. However, those works haven’t
explored medical datasets, which are proven to be more vulnerable to
adversarial attacks [10]. Other works using medical imaging segmentation
datasets also do not account for MRI datasets, which could reveal more
detailed features of soft tissues or nerves [17]. In addition, most of such
works have focused on testing lightweight models that are no longer widely
used in modern applications [18] [19] [20], even though more recent models
have proven to be increasingly supceptible to adversarial attacks [21]. An
example of an adversarial attack on an MRI image segmentation model is shown
in Figure 2.

Figure 2: An example of adversarial attack on the medical MRI image
segmentation model.

In light of the grave danger that poisoned MRI data could pose and evident
lack of research in this area, we test the susceptibility of modern MRI image
segmentation models to a popular white-box adversarial attack method called
Fast Gradient Sign Method (FGSM). The main motivation behind this research is
to raise awareness in the academic community on the security of the MRI image
segmentation models. We summarize the main contributions below:

  1. 1.

Through experimenting with different losses to conduct the FGSM attack, we
show that using the BCE loss to conduct the attack leads to greater success
than using the default loss as suggested by the FGSM paper’s author.

  2. 2.

We show that having more parameters does not necessarily make the model more
vulnerable to attacks.

  3. 3.

We show that FGSM can effectively mislead modern image segmentation models
models.

##  II Approaches

In this section, we will describe the models we selected in detail, the
reasons behind those choices, the dataset we used, our adversarial attacking
strategy, and our training hyperparameters.

###  II-A Model Architecture

In this section, we will introduce U-Net and U-Net++. We will also describe
why we chose to use VGG16, ResNeXt-101, and EfficientNet-B7 as our backbones.

####  II-A1 U-Net

Figure 3: U-Net Architecture by Ronneberger et al. Here the architecture
assumes the input is 572 $\times$ 572

U-Net, introduced by Ronneberger et al., is one of the most commonly used
image segmentation architectures for biomedical imaging [4]. U-Net is a
u-shaped network containing a down-sampling path, a bottleneck, and an up-
sampling path (Figure 3). During each level of the down-sampling path, the
dimension of the image is contracted by the max pooling layer, yet the number
of feature channels is expanded by a factor of two, which allows the network
to learn global features better. During each level of the up-sampling path,
the output of the previous layer is concatenated with the output from the same
level’s down-sampling path to simultaneously fuse the global and local
features necessary for segmentation.

Among four of our models, three of them are U-Net based.

  * •

U-Net: The first model is the basic U-Net with no modified backbones. We use
this model as our baseline.

  * •

U-Net w/ ResNeXt-101: The second model is based on the U-Net’s architecture,
but the down-sampling path is replaced by layers from a pre-trained ResNeXt101
32x8d model [22]. ResNeXt101 is a 101-layer variant of ResNeXt, which achieved
second place in the ILSVRC 2016 classification competition [23].

  * •

U-Net w/ EfficientNet-B7: The third model we use is also based on the U-Net’s
architecture, but the encoder layers are replaced by layers from a pre-trained
EfficientNet-B7 model [24]. EfficientNet-B7 is a complex variant of the
EfficientNet family, which achieves SOTA efficiency by outperforming most
models within its domain with much higher efficiency [24]. Today,
EfficientNet-B7 is extensively used in industry and in medical imaging
competitions [25] [26] [27] [28].

Figure 4: UNet++ Architecture by Zhou et al. UNet++ introduced dense skip
connections as highlighted in blue arrows and green arrows shown in the graph.
The model also introduced deep supervision as indicated by $L$, but in our
paper, this is excluded during training.

####  II-A2 UNet++

UNet++ by Zhou et al. introduces an important innovation to the U-Net network
— dense skip connections [5]. As shown in Figure 4, instead of naively
concatenating the feature map from each level of the encoder layer
$X^{1,0},X^{2,0},X^{3,0}$ to the feature maps of the corresponding decoder
layer $X^{2,2},X^{1,3},X^{0,4}$, intermediate dense skip connections are
introduced, as shown in green colored nodes in Figure 4. Dense skip
connections allow the model to learn faster because the image representations
are richer, and the semantic gap is smaller.

The last trained model is U-Net++ based.

  * •

UNet++ w/ EfficientNet-B7: The fourth model we used is based on the U-Net++’s
architecture and has an EfficientNet-B7 backbone [24]. We choose
EfficientNet-B7 because our prior experiments show that a pre-trained
EfficientNet-B7 has a superior performance when used as an encoder.

###  II-B Datasets

All the models we use in our experiements were trained using University of
Wisconsin-Madison’s gastro-intestinal tract (UW-Madison GI Tract) MRI image
segmentation dataset [29], which is publicly available on Kaggle. The dataset
is made up of 272 workable 3D scans and 38208 images that are black-and-white.
Segmentation masks are encoded in the run-length encoding (RLE) format.

There are three classes in this dataset: large bowel (14,017 images), small
bowel (11,129 images), and stomach (8,558 images). Instances chosen for
training and testing datasets were carefully picked to ensure they all have a
similar distribution. To prevent the data leakage problem, slices from
individual scans were grouped together and together either in the training set
or the testing set. The detailed dataset distribution is shown in Table I.

####  II-B1 Pre-processing

During pre-processing, all pixels were normalized to range from $[0,1]$, and
all images are resized to $224\times 224$. ($224=32\times 7$).

TABLE I: Detailed dataset distribution. | Training (Slices) | Testing (Slices) | Total (slices)
---|---|---|---
Large Bowel Tumor | 12,698 | 1,319 | 14,017
Small Bowel Tumor | 9,955 | 1,174 | 11,129
Stomach Tumor | 7,611 | 947 | 8558
Total | 34,432 (90%) | 3,776 (10%) | 38,208

###  II-C Model Training

In this section, we will share the training parameters that we used to conduct
our experiments. All of our models were trained without sufficient fine-tuning
because we prioritized analyzing the impact of adversarial attack over gaining
the best performances on normal input images for all models.

####  II-C1 Hardware and Software

All four of our models were trained on one Nvidia A6000 (48GB) instance with
14vCPUs and 100 GiB RAM.

####  II-C2 Hyperparameters

All models were trained using AdamW with an initial learning rate of $3e-4$
and a weight decay of $1e-3$. We also used a cosine annealing learning rate
scheduler with max iterations of 7,081 and a maximum learning rate of $3e-4$.
We trained all our models for 15 epochs with early stopping. We used a batch
size of 64 and applied data augmentation on training images in the form of
shifting, scaling, and deformation.

For our U-Net models, instead of having the number of kernels starting at 64
and growing by a factor of two until 1024, we modified it to have a kernel
number beginning at 16 and ending at 256 because to make the model more
efficient during training.

####  II-C3 Loss Function

All of the models in this paper were trained using a hybrid loss that combined
the Dice loss and Focal loss, which helped to deal with class imbalance and
improve the performance of our models [30].

Let’s define $y\in\\{0,1\\}$ as the ground truth mask and $\hat{y}\in[0,1]$ as
the predicted mask.

Dice loss (DL) [31] is defined in Formula 1. The Dice loss is basically
$1-DSC$. DSC is expanded in section III-A We add $10^{-6}$ in the numerator
and denominator to avoid division by zero.

$DL=1-2\times\frac{y\hat{y}+10^{-6}}{y+\hat{y}+10^{-6}}$ (1)

Focal loss (FL) [32] is defined in Formula 4. Focal loss improves binary cross
entropy (BCE) loss [33] by dealing with the imbalanced dataset problem. We
derive the focal loss formula firstly by deriving the binary cross entropy
loss formula, as shown in Formula 3.

We define $C$, $W$, and $H$ to indicate the number of channels, the height,
and width of the image. $c\in[0..C)$, $i\in[0..W)$, and $j\in[0..H)$ are
indexes. For example, $y_{c,i,j}$ means the pixel value of the mask $y$ at
channel index $c$, width index $i$, and height index $j$.

So, we define $\hat{y}^{t}$ that for each pixel of $y$,

$\hat{y}^{t}_{c,i,j}=\begin{cases}\hat{y}_{c,i,j},&\text{if }y_{c,i,j}=1\\\
1-\hat{y}_{c,i,j},&\text{if }y_{c,i,j}=0\end{cases}$ (2)

Therefore, binary cross entropy loss for image segmentation can be defined in
the formula below.

$BCE(y,\hat{y})=-\sum_{c=0}^{C}\sum_{i=0}^{W}\sum_{j=0}^{H}\log(\hat{y}^{t}_{c,i,j})$
(3)

Focal loss adds a modulating factor $(1-\hat{y}^{t})^{\gamma}$ to BCE.
$\gamma$ is a tunable hyperparameter. Setting $\gamma>0$ will differentiate
focal loss from binary cross entropy loss. Setting $\gamma>1$ would make the
model less sensitive to class imbalance, and setting $1>\gamma>0$ would make
the model more sensitive to class imbalance.

$FL(y,\hat{y})=-\sum_{c=0}^{C}\sum_{i=0}^{W}\sum_{j=0}^{H}(1-\hat{y}^{t}_{c,i,j})^{\gamma}\cdot\log(\hat{y}^{t}_{c,i,j}))$
(4)

For all our experiments we set $\gamma=2$.

###  II-D Fast Gradient Sign Method

Goodfellow et al. proposed the Fast Gradient Sign Method (FGSM), which would
generate adversarial inputs by nudging the input in the direction of the
gradient with respect to the input space. [1].

Figure 5: Input images and predictions with various $\epsilon$ values for our
U-Net++ Model. Diff value is measured by the clean input metric score minus
the poisoned input metric score divided by the clean input metric score again.
Therefore, the more drastic the difference is, the more successful the attack
would be.

The FGSM attacking formula is provided below, where $adv_{y}$ is the
adversarial image, $\theta$ is the parameters of the model, $x$ is the input
to the model, $y$ is the prediction of the model, $J(\theta,x,y)$ is the loss
function of the model, $sign$ is the sign of the gradient with respect to the
pixels used in the back-propagation stage, and $\epsilon$ is the multiplier of
the noise that can be tuned to achieve a balance between stealthiness and
effectiveness. Figure 5 shows under different $\epsilon$ values, what do the
input image and the predicted mask look like. It shows that the higher the
$\epsilon$ value, the more successful the attack will be, but the input image
will loss the stealthiness as the attack noises would become gradually
visible.

$Adv_{y}=x+\epsilon*sign(\nabla_{x}J(\theta,x,y))$ (5)

FGSM is a simple yet robust adversarial attack. The attack is also illustrated
in Figure 2. In this paper, we compared the performance of all four of our
segmentation models before and after the FGSM attack. In all experiments, we
use $\epsilon=0.009$ because, as shown in Figure 5, $\epsilon=0.009$ achieves
both stealthiness and effectiveness.

In the original FGSM paper, authors Goodfellow, Shlens, and Szegedy, suggested
that $J(\theta,x,y)$ should be the loss function used to train the network
[1]. However, during our experiments, we found that using the original loss
function led to less effective attacks than using an alternative loss. For our
experiments, we found that using binary cross entropy (BCE) loss function led
to significant improvement in attacking success.

##  III Results

The results are shown in Table II. Since there are some false negative ground
truth masks in this dataset, we only test our models’ performance using the
MRI slices that have segmentation masks.

For our experiments, U-Net w/ ResNeXt-101 has the most number of parameters,
followed by U-Net++ w/ EfficientNet-B7 and U-Net w/ EfficientNet-B7 models.

Figure 6: Comparison of the predicted masks of four models. Three colors in
the predictions columns indicate three different objective classes. Model
differences to the ground-truth mask are also illustrated. For the
differences, true positive is highlighted in green, false positive is
highlighted in blue, and false negative is highlighted in red.

For normal inputs, U-Net++ with EfficientNet-B7 and U-Net with EfficientNet-B7
models are the most successful model among all with U-Net with EfficientNet-B7
model performing slightly worse. Their predictions based on a normal image
input and their differences to the original mask are illustrated in Figure 6.
All four models predicted the masks well with few false negatives and false
positives.

Figure 7: Adversarial attacks on four models. The first column contains the
poisoned inputs. The second column contains predictions using the poisoned
inputs. The third column indicates their differences. However, it should be
noted that for the first two models, we used a higher epsilon value,
$\epsilon=0.015$, for illustration purposes because the first two models are
very resilient to the attack. we also talk it briefly in future work section
IV-A. For the differences, true positive is highlighted in green, false
positive is highlighted in blue, and false negative is highlighted in red.

However, all four models were all significantly impacted by FGSM, instantly
making them unreliable for doctors, as shown in Figure 7.

Figure 8: Prediction masks showing the differences between the use of binary
cross entropy (BCE) loss, hybrid loss combining dice and focal loss, and focal
loss for an FGSM attack. Attacks were done on the U-Net++ w/ EfficientNet-B7
model for the same image. Diff value is measured by the clean input metric
score minus the poisoned input metric score divided by the clean input metric
score again.

Since FGSM attacks require a loss function to derive the signs of the
gradients, we also tested out three different loss functions to see how the
attack success varies. It turned out that BCE had the highest success rate,
despite the fact that our original model was trained on a hybrid loss of focal
and dice loss. The comparison between three loss functions is shown in Figure
8 FGSM achieves the highest success in U-Net++ w/ EfficientNet-B7 model in all
three loss functions.

Our results also imply a negligible correlation between the number of
parameters and attack success rate. However, it is worth noting that U-Net w/
EfficientNet-B7 and U-Net++ w/ EfficientNet-B7 models, which had the best
performances for clean inputs, were the most vulnerable to the FGSM attack.
U-Net++ w/ EfficientNet-B7 model has not only the best performance but also
the highest attacking success rate.

In addition, even though the authors of the FGSM paper suggested to use the
cost function used to train the model to conduct the attack, empirical
evidence suggests that it is not true in this case. The original cost function
used to train all four models is the hybrid loss combing focal loss and dice
loss. As shown in Table II, combining focal loss and dice loss to conduct the
attack received the lowest attack success, yet using binary cross entropy
(BCE) loss to conduct the attack yielded significantly higher attacking
success.

###  III-A Evaluation Metric

Dice Similarity Coefficient (DSC): To evaluate the models’ performance on
testing data before and after applying adversarial noise, we used DSC that
would measure the effectiveness of the overlap between the ground truth and
predicted mask. DSC is bounded between $-1$ and $+1$. [34]. Its formula is
defined below. $y$ and $\hat{y}$ are the ground-truth mask and the predicted
mask.

$DSC=2\times\frac{|y\cap\hat{y}|}{|y|+|\hat{y}|}$ (6)

Attacking Success (AS): To evaluate the effectiveness of the adversarial
attack, we created a metric that would measure the percentage change in DSC,
as shown in Formula 7. The resulting value is a percentage between 0% and
100%, and higher the AS the more successful the attack is.

$AS=\frac{\text{DSC Before Attack}-\text{DSC After Attack}}{\text{DSC Before
Attack}}$ (7)

TABLE II: Comparison of the performance before and after a FGSM attack was
done on our models: U-Net with VGG13 (VGG U-Net), U-Net with ResNeXt-101
(ResNeXt U-Net), U-Net with EfficientNet-B7 (EffB7 U-Net), and U-Net++ with
EfficientNet-B7 (EffB7 U-Net++) on the GI Tract dataset. All measured in Dice
Similarity Coefficient (DSC) score. Attacking successes of three losses used
for FGSM attack were compared. The most successful attacks are highlighted in
bold.

|  |  | FGSM Using Loss
---|---|---|---
Model Names | Parameters | Normal | BCE | Focal+Dice | Focal
VGG U-Net | 18.44M | 0.7509 | 0.4063 | 0.5772 | 0.5105
ResNeXt U-Net | 95.76M | 0.7841 | 0.3873 | 0.6197 | 0.5560
EffB7 U-Net | 67.10M | 0.7994 | 0.4576 | 0.5348 | 0.5097
EffB7 U-Net++ | 68.16M | 0.8024 | 0.3750 | 0.4705 | 0.4330

##  IV Discussions and Conclusions

In this paper, we trained four advanced image segmentation models from the
U-Net family and examined the efficacy of FGSM for poisoning MRI data to
understand how vulnerable they are to adversarial attacks. We observe that all
the models in this paper are heavily impacted by FGSM, stressing an urgent
need to enact serious security measures under professional environments. In
addition, we observe that even though the FGSM paper suggests using the loss
function used to train the model, using binary entropy loss as an alternative
to generate attacking noises under this context has consistently demonstrated
better attacking success rates. Lastly, we observe that having more parameters
does not necessarily imply the vulnerability of the model to adversarial
attacks.

###  IV-A Future Work

In future studies, we will test the adversarial robustness of more of U-Net
models using various attacking methods. Doing this would give the academic
community a more complete sense of what models are more susceptible to
adversarial attacks, and what types of adversarial attacks are likely be
successful.

During our experiment, we also noted that some models are resilient to certain
types of images, while others are not. Specifically, we found out image
luminosity seems to be an important factor. We will conduct more experiments
to find out if there is a relationship between image brightness and attack
successes for certain models.

Finally, we will also test out the effectiveness of adversarial attacks by
ensembling all our trained models.

## Acknowledgment

We thank our friends, parents, and teachers for constantly supporting us
throughout this process.

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# An Iterative Energy Estimate for Degenerate Einstein model of Brownian
motion

Isanka Garli Hevage1, Akif Ibraguimov2, Zeev Sobol3

###### Abstract.

We consider the degenerate Einstein’s Brownian motion model for the case when
the time interval ($\tau$) of particle Jumps before collision (free jumps)
reciprocal to the number of particles per unit volume $u(x,t)>0$ at the point
of observation $x$ at time $t$. The parameter $0<\tau\leq C<\infty$, controls
characteristic of the fluid "almost decreases" to $0$ when
$u\rightarrow\infty$. This degeneration leads to the localisation of the
spread of particle propagation in the media. In our report we will present a
structural condition of the time interval of free jumps - $\tau$ and the
frequency of these free jumps $\phi$ as functions of $u$ which guarantees the
finite speed of propagation of $u$.

1 Department of Mathematics and Statistics, Texas Tech University

Lubbock, Texas, USA, e-mail<EMAIL_ADDRESS>
2 Department of Mathematics and Statistics, Texas Tech University,

Lubbock, Texas, USA, e-mail<EMAIL_ADDRESS>
3 Department of Mathematics, University of Swansea,
Fabian Way, Swansea SA1 8EN, UK, e-mail<EMAIL_ADDRESS>
##  1\. Introduction

Consider the sought experiment of the fluid which occupy a bounded domain
$\Omega\subset\mathbb{R}^{N}\ ;\ N=1,2,\dots$ by the particles. Let
$\vec{\Delta}$ be the $N$ dimensional vector of the free jumps which we define
as the particles’ jumps without collisions. Let
$(\Delta_{1},\Delta_{2},\cdots,\Delta_{N})$ are the coordinates of free
jump.In definition of free jump we follow the classical Einstein paradigm.
Note that in many literature instead of free jump term of free pass is used.
From this point of view events of free jump and free pass are synonyms.
In his famous thesis (see [4]), Einstein assumed that process of random motion
of the particles is characterised by events of free jumps of the particles.
Einstein proposed two main parameters which characterise free jumps : $\tau$
-the time interval within which the free jumps occur, and $\varphi(\Delta)$ \-
frequency of the occurrences of free jumps of the length $\Delta$. Einstein
assumed that $\tau$ and the variance
$\displaystyle\sigma^{2}={\int_{-\infty}^{\infty}{\Delta}^{2}\varphi(\Delta)\
d\Delta}$ are to be constants which allowed him to reduce his sought
experiment to the classical heat equation with constant coefficients where
that exhibits the so called effect of infinite speed of the perturbation.
Namely if at some moment $t_{0}$ of time and at any point $x_{0}$
concentration of particles $u(x,t)>0$ then $u(x,t)>0$ for all $x$ and $t\geq
t_{0}$. This property of $u(x,t)$ is very unrealistic. The question which we
asked in this article is as follows: Can Einstein paradigm of free jumps be
generalised in such way that one sees finite speed of propagation of the
particles ? i.e. If $u(x,0)=0$ only at some point $x_{0}$ then $u(x_{0},t)=0$
during time interval $[0,T]$ for some $T>0$.
In this article we will show that if time interval of free jumps $\tau$ is
inverse proportional to the concentration $u$ or/and $u$ is proportional to
$\sigma^{2}$ then, under some assumptions of the proportionality, our solution
of degenerate Einstein equation in IBVP(4.7) will exhibit the finite speed of
propagation property. This property closely relate to so called Barenblatt
solutions for degenerate porous medium equation (see [2]). It is necessary to
state that origin of the porous media equation is differ from sought
experiment of the Einstein paradigm. Namely the first continuity equation
hypothesised in the form:

(1.1)
$L(\rho,\vec{J})\triangleq\rho_{t}+\nabla\cdot\left(\rho\vec{J}\right)=0,$

where $\rho$ is density and $\vec{J}$ is flux in porous media that equals to
the velocity of the flow - $\vec{v}$ (see [1]). Next it is assumed that $\rho$
is function of the pressure $p$ and can be approximated based on thermodynamic
experiment. For example, $\rho$ can be approximated by $p^{\lambda}$ for some
gasses by thermodynamic laws and $\displaystyle\vec{v}=-\frac{k}{\mu}\nabla
p,$ where $k$ \- permeability of porous media, and $\mu$ is viscosity subject
to the experimental Darcy equation (see [9]). Combining these experimental
relations in (1.1) one can get the following divergent equation for scalar
pressure function

(1.2) $\displaystyle L(w)=(w)_{t}-\frac{k}{\mu}\ \frac{1}{\lambda+1}\
\Delta\left(w^{\frac{\lambda+1}{\lambda}}\right)=0.$

where $w=p^{\lambda}$ and $\lambda>0$. In the pioneering work by Barenblat-
Kompaneetz-Zeldovich (see [5]) it was shown that under specific initial and
boundary conditions there exists self-similar solution of the equation (1.2),
which exhibits finite speed of propagation. Evidently, the Einstein operator
$L_{e}$(see (3.7)) is in non-divergent form and based on sought experiment
which allows to interpret the results on observable data in term of the length
of free jumps, and then adjust parameters $\tau$ and $\varphi$ of the model.
In this paper we will follows Einstein approach but will use technique for
divergent equation. Article is organized as follows: In Section 2, we consider
generalization of classical Einstein model of Brownian motion to
$\mathbb{R}^{N}$, when major parameters of the system - time interval $\tau$
of the free jumps with a length $\Delta$ (free passes) and the frequency
$\varphi$ of the free jumps of the length $\Delta$ depend on the concentration
of the number of particle per unite volume. We use the generic mass
conservation law with absorption/reaction and basic stochastic principles to
derive a partial differential inequality (PDI) under the Einstein’s axioms in
Section 3. By introducing local forces and non-locality processes we conclude
the deterministic PDI (3.7) which governs the dynamics of the generalized
Einstein paradigm. Using Hypothesis 1, In Section 4 we define $u$ as the weak
viscous solution to the non-linear initial boundary value problem (4.7) such
that holds (4.9). Introducing the assumption on functions $F,G,P$ in Section
5, we explicitly structures $H$ in (5.7) and $F$ in (5.9), in a way that $u$
conserves the finite speed of propagation, verifying $H$ and $F$ throughout
those assumptions with necessary conditions on $\Lambda$. In section 6 we
establish corroborating Lemmas and introduce Ladyzhenskaya Iterative scheme
which apply in Section 7 to prove the localization property of $u$, based on
De-Georgi’s construction, if the functional $Y_{0}(T)$ preserves the
boundednes in (7.13). Next in Section 8 we show some counter examples of the
functions $F$ and $P$ that attest all constrains on $F,G,H$. We present
auxiliary results of Functional spaces in Section 10 to prove the uniform
boundednesses regarding the classical solution $u_{\epsilon}$ which ensures
its limit as in (4.9). This Degenerate porous media equation is obtained
different from Einstein conservation law which state that rate of growth of
the density is proportional to divergence of the flux.

##  2\. Generalized Einstein paradigm

Let $x\in\mathbb{R}^{N}$ and $u(x,t)$ be the function which represent the
number of particles per unit volume at point $x$ and at time $t$. Consider a
particle ($P$) of particular type suspended in the medium of interest. Denote
$\mathbb{P}(\tau)$ to be the set of vectors with non-colliding jumps of $P$
corresponding to time interval $\tau$.We call
$\vec{\Delta}=\big{(}\Delta_{1},\cdots,\Delta_{N}\big{)}^{T}$ to be a ” vector
of free jump of particles $P$” if $\vec{\Delta}\in\mathbb{P}(\tau)$. We assume
the following extension of the axioms in classical Einstein Brownian motion:

######  Assumption 1.

  1. (i)

Consider a particle ($P$) of particular type suspended in the medium of
interest. Denote $\mathbb{P}(\tau)$ to be the set of vectors with non-
colliding jumps of $P$ corresponding to time interval $\tau>0$. We call
$\vec{\Delta}=\big{(}\Delta_{1},\cdots,\Delta_{N}\big{)}^{T}$ to be a vector
of free jump of particles $P$ if $\vec{\Delta}\in\mathbb{P}(\tau)$.

  2. (ii)

Time interval of free jumps $\tau$, expected vector $\vec{\Delta}_{e}$ of a
free jump $\vec{\Delta}$ and probability density function of free jump
$\varphi(\vec{\Delta})$ are the only parameters which characterise process of
free jumps. Note that in a view of the definition of the set
$\mathbb{P}(\tau)$, if $\vec{\Delta}\notin\mathbb{P}(\tau)$ then
$\varphi(\vec{\Delta})=0$.

  3. (iii)

The key parameters $\tau$ and $\varphi$ in general can depend on the
concentration of the particles and also space coordinate and time itself.

######  Axiom 1.

Whole universe axiom:

(2.1)
$\displaystyle\int_{\mathbb{P}(\tau)}\varphi(\vec{\Delta})d\vec{\Delta}=1.$

######  Assumption 2.

Symmetry of free jumps

(2.2) $\varphi(\vec{\Delta})=\varphi(-\vec{\Delta}).$

Existence of second moments

(2.3)
$\int_{\mathbb{P(\tau)}}|\vec{\Delta}|^{2}\varphi(\vec{\Delta})d\vec{\Delta}<\infty$

Note that it follows from the preceding assumption that the expectation of
$\vec{\Delta}$ equals zero and that there exists a co-variance matrix
$[\sigma_{ij}^{2}]$ of free jumps:

######  Definition 1.

Standard Co-variance matrix of free jump

(2.4)
$\displaystyle\sigma_{ij}^{2}\triangleq\int_{\mathbb{P}(\tau)}{\Delta_{i}}{\Delta_{j}}\varphi(\vec{\Delta})d\vec{\Delta}\quad\text{
where }i,j=1,2,\dots,N.$

Evidently $\sigma_{ij}(x,t)$ depend on space $x$ and time $t$. We postulate
generalized Einsteins Axiom for the number of particles found at time $t+\tau$
in the control volume $dv$ contained point $x$ by :

######  Axiom of Mass conservation.

(2.5) $u(x,t+\tau)\cdot
dv=\int_{\mathbb{P}(\tau)}u(x+\vec{\Delta},t)\varphi(\vec{\Delta})d\vec{\Delta}\cdot
dv+\tau\cdot\int_{\mathbb{P}(\tau)}M(x+\vec{\Delta},t)\varphi(\vec{\Delta})d\vec{\Delta}\cdot
dv.$

Mass conservation law (2.5) intuitively is easy to interpret, and it says that
at any given point in space $x$ at time $t+\tau$ we will observe density of
all particles with free jumps from the point $x$ at time $t$, "$+$ density" of
particles which "produced" and "$-$ density" of particles which "consumed"
during time interval $[t,t+\tau].$ For comparison see [4] (pages 14) first
formula with integral. We modeled this by adding the term $M(\cdot)$ which
model process of rate of absorption/consumption during time interval
$[t,t+\tau]$ due to particles interaction along of free jumps and therefore it
set to be non-positive.

######  Remark 1.

Einstein definition of density of particles differs from the fundamental
definition of the density of fluid, rather mean the concentration.

##  3\. Derivation of Partial differential Inequality

In this section we will derive partial differential inequality from (2.5)
whose solution will exhibit the feature of finite speed of propagation. Let
$\zeta=(\zeta_{1},\zeta_{2},\cdots,\zeta_{N})$ multi-index and
$x^{\zeta}\triangleq x_{1}^{\zeta_{1}}\cdot x_{2}^{\zeta_{2}}\cdot\dots\cdot
x_{N}^{\zeta_{N}}.$ Assume that $u(x,t)\in C_{x,t}^{2,1}$. By Taylor’s
Expansion (see [7]), and using (2.1) - (2.4) we get

(3.1)
$\displaystyle\int_{\mathbb{P}(\tau)}u(x+\vec{\Delta},t)\varphi(\vec{\Delta})d\vec{\Delta}\text{
= }\text{ }u(x,t)\text{ +
}\frac{1}{2}\sum_{i,j=1}\sigma_{ij}^{2}u_{x_{i}x_{i}}{(x,t)}\text{ +
}R_{\zeta}$

where

(3.2) $\displaystyle
R_{\zeta}\triangleq{\int_{\mathbb{P}(\tau)}\sum_{|\zeta|=2}H_{\zeta}(x,\vec{\Delta},t)(\vec{\Delta})^{\zeta}\varphi(\vec{\Delta})d\vec{\Delta}}$

with $H_{\zeta}$ locally bounded and $\lim_{\vec{\Delta}\rightarrow
0}H_{\zeta}(x,\vec{\Delta},t)=0.$ Using above in $\eqref{Einstein_conserv_eq}$
we get

(3.3)
$u(x,t+\tau)-u(x,t)=\frac{1}{2}\sum_{i,j=1}^{N}\sigma_{ij}{u}_{x_{i}x_{j}}{(x,t)}+R_{\zeta}+\tau\int_{\mathbb{P}(\tau)}M(x+\vec{\Delta},t)\varphi(\vec{\Delta})d\vec{\Delta}.$

Observe that LHS and RHS in the equation (3.5) are defined at different times.
In order to eliminate this ambiguity and derive the equation at the same point
we will assume that $u(x,t)\in C^{2,1}_{x,t}$. By Carathéodory’s criterion
$\exists$ function $\psi^{t}$ such that

(3.4) $\displaystyle u(x,t+\tau)-u(x,t)=\tau\psi^{t}(x,t,\tau)$

where $\lim_{\nu\rightarrow 0}\psi^{t}(x,t,\nu)=u_{t}(x,t)$. Using (3.5) in
(3.5) we get

(3.5) $\tau
u_{t}(x,t)\approx\frac{1}{2}\sum_{i,j=1}^{N}\sigma_{ij}{u}_{x_{i}x_{j}}{(x,t)}+R_{\zeta}+\tau\int_{\mathbb{P}(\tau)}M(x+\vec{\Delta},t)\varphi(\vec{\Delta})d\vec{\Delta}.$

######  Assumption 3.

Mathematically for our forthcoming iterative procedure we will assume that

(3.6) $\displaystyle
R_{\zeta}+\tau\int_{\mathbb{P}(\tau)}M(x+\vec{\Delta},t)\varphi(\vec{\Delta})d\vec{\Delta}\leq
0.$

Using (3.6) in (3.5) and approximating the first order terms, it follows that
the function $u(x,t)$ can be estimated by the following Partial differential
inequality:

(3.7) $\displaystyle L_{e}u=\tau{u_{t}}-\sum_{i,j=1}^{N}$
$\displaystyle\sigma_{ij}u_{x_{i}x_{j}}\leq 0.$

Observe that the partial differential inequality (3.7) is deterministic, while
the stochastic nature is modeled by functions $\tau$ and $\varphi$ (the latter
appears in (3.7) in form of its co-variance matrix [$\sigma_{ij}$]), which are
key characteristics of the process dynamics. In general, they can be functions
of spatial and time variables $x$ and $t$, concentration $u$ and its gradient
$\nabla u$. In this report we present $\tau$ and $\sigma$ satisfy the
following Hypothesis.

######  Hypotheses 1.

In addition to (2.1) and (2.2), let

(3.8)
$c_{1}P(u)|\xi|^{2}\leq\sum\limits_{ij=1}^{N}\frac{1}{\tau}\sigma_{ij}\xi_{i}\xi_{j}\leq
c_{2}P(u)|\xi|^{2},\quad\xi\in\mathbb{R}^{N}$

for some $0<c_{1}<c_{2}$ and $P\in C([0,\infty))$, $P(0)=0$, $0<P(s)<C<\infty$
for $s>0$.

######  Remark 2.

Note that in (3.8) $P(u)$ is a composite parameter of stochastic processes of
free jump, which characterise relative co-variance of free jump with respect
to time interval of latter, which in our scenario degenerate at $0$. Which
mean that dispersion growth slower than time interval of this dispersion, as
the concentration vanishes.

##  4\. Non-linear Degenerate IBVP

Let $\sigma>0$. For simplicity of the argument we will restrict the exposition
to the case $\sigma_{ij}=\sigma\delta_{ij}$ for $i,j=1,\dots,N$ so that
$P=\dfrac{\sigma}{\tau}$. Hence (3.7) takes form

(4.1) $\displaystyle u_{t}\leq P(u)\Delta u.$

######  Definition 2.

  1. (i)

Let $h$ be a positive function such that $h\in C((0,\infty))$ and integrable
at zero w.r.t. $u$. Then we define

(4.2) $H(u)\triangleq\int_{0}^{u}h(s)ds$

  2. (ii)

Let $F\triangleq hP$ and $h$,$P$ be such that $F(0)=0$, and $F$ is
differentiable on $(0,\infty)$ with a locally bounded derivative
$F^{{}^{\prime}}>0$.

  3. (iii)

Let $G$ be s.t. $\sqrt{F^{{}^{\prime}}(u)}=G^{{}^{\prime}}(u)$ and $G(0)=0$.
Then $\displaystyle G(s)\triangleq\int_{0}^{s}\sqrt{F^{{}^{\prime}}(s)}\ ds.$

######  Remark 3.

Note that

(4.3) $0\leq
G(u)=\int_{0}^{u}G^{{}^{\prime}}(s)ds\leq\sqrt{u\int_{0}^{u}F^{{}^{\prime}}(s)ds}=\sqrt{uF(u)}.$

In above, all functions $F,G$ and $H$ are increasing w.r.t. $s$.

Further restriction on $P$ and of $h$ will be presented later. We multiply
(4.1) by $h(u)$ to get.

(4.4) $\displaystyle\dfrac{\partial}{\partial t}H(u)-F(u)\Delta u\quad\mbox{on
}\Omega\times(0,T)\ ,$

for some $T>0$. Here $u$ is positive measurable such that $u(\cdot,t)\to
u(\cdot,0)$ as $t\to 0$ in (locally) measure, and

(4.5) $\displaystyle u\in
L_{loc}^{\infty}\left(\Omega\times[0,T]\right),\quad\nabla u\in
L^{2}_{loc}\left(\Omega\times(0,T]\right),\quad u_{t}\in
L^{1}_{loc}\left(\Omega\times(0,T]\right).$

So $F(u)\Delta u$ is understood in the weak sense: for all $\theta\in
Lip(\Omega)$ with a compact support,

(4.6) $\displaystyle-\int\theta F(u)\Delta udx$
$\displaystyle=\int\left[F(u)(\nabla u)(\nabla\theta)+F^{\prime}(u)|\nabla
u|^{2}\theta\right]dx$ $\displaystyle=\int\left[F(u)(\nabla
u)(\nabla\theta)+\left|\nabla G(u)\right|^{2}\theta\right]dx.$

Then under the Hypotheses 1, the differential inequality (3.7) we defined
$u(x,t)$ as a non-negative solution of the following differential inequality
and initial condition which we for convenience will call as IBVP

(4.7) $\displaystyle\begin{cases}\ \dfrac{\partial}{\partial t}H(u)-F(u)\Delta
u\ \leq 0\ &\text{ in }\Omega\times(0,T)\\\ \hskip 65.44142ptu(x,0)\ =0&\mbox{
in }\Omega^{{}^{\prime}}\Subset\Omega\\\ \hskip 65.44142pt\ u(x,t)=\hskip
2.27626pt\phi(x,t)&\mbox{ on }\ {\partial\Omega\times(0,T)}.\end{cases}$

where $\phi\geq 0$ and $\displaystyle\phi(x,t)=0$ when $t\rightarrow 0$. The
condition on the boundary $\partial\Omega\times(0,T)$ is not essential to
prove that for every ball $B_{R}(x_{0})\Subset\Omega^{{}^{\prime}}$ and $c<1$,
$\exists$ $T^{\prime}=T^{\prime}(x_{0},c)\in(0,T]$ such that $u(x,t)=0$ for
all $(x,t)\in B_{cR}(x_{0})\times[0,T^{\prime}]$ in Theorem 1. We also assume
that $u(x,0)$ is continuous on $\Omega$. Observe that $u$ in above IBVP is
degenerates when $u\rightarrow 0$. Therefore its solution
$u(x,t)\not\in\mathcal{C}^{2,1}_{x,t}(\Omega\times(0,T])$. Our result is
qualitative and does not address the existence of the solutions, but the
obtained property of the solution will be applicable for the weak viscous
solution which one can define as follows. Let us define the regularized IBVPϵ
for some $u_{\epsilon}\in\mathcal{C}^{2,1}_{x,t}(\Omega\times(0,T])$ as
follows:

(4.8) $\displaystyle\begin{cases}\ \dfrac{\partial}{\partial
t}H(u_{\epsilon})-(F(u_{\epsilon})+\epsilon)\Delta u_{\epsilon}\ \leq 0\
&\text{ in }\Omega\times(0,T)\\\ \hskip 105.2751ptu_{\epsilon}(x,0)\
=\epsilon&\mbox{ in }\Omega^{{}^{\prime}}\Subset\Omega\\\ \hskip 105.2751pt\
u_{\epsilon}(x,t)=\hskip 2.27626pt\phi(x,t)+{\epsilon}&\mbox{ on }\
{\partial\Omega\times(0,T)}.\end{cases}$

Assume that $u_{\epsilon}(x,0)$ is continuous in $\Omega$. In this article,
the constants in all estimates for $u_{\epsilon}$ do not depend on $\epsilon$.
This observation allow us to pass to the limit in the final estimates, and
conclude the localization property for the limiting function:

(4.9) $u(x,t)=\lim_{\epsilon\to 0}u_{\epsilon}(x,t),$

which is considered as a weak passage to the limit (see [3]). The obtained
function $u(x,t)$ is called a weak viscous solution of the IBVP (4.7) with
first differential inequality replaced by by the differential equation $Lu=0$
, which will exhibit localisation property. Further details on weak viscous
solution of IBVP with homogeneous boundary function will be presented in
Section 10.

##  5\. Assumptions on Functions and interpretation

In this section we state main properties and additional assumption, which we
imposed on the functions $H$ and $F$ besides monotonously, we will require for
some finite $M>0$ following assumption holds

######  Assumption 4.

(5.1) $\displaystyle\exists\ C_{1}>0\mbox{ such that }F(s)\leq
C_{1}G^{{}^{\prime}}(s)G(s),~{}s\in[0,M].\ \text{ Here }\
G(s)=\int_{0}^{s}\sqrt{F^{{}^{\prime}}(s)}\ ds.$ (5.2)
$\displaystyle\displaystyle\exists\ \lambda\in(0,2),C_{2}>0\mbox{ such that
}\left(\sqrt{sF(s)}\right)^{\lambda}\leq{C_{2}H(s)},~{}s\in[0,M],$

We will choose $H$ and impose restrictions on $P$ such that Assumption 4 hold.

######  Proposition 1.

Let $F$ and $H$ hold (5.2). Then

(5.3) $\displaystyle{H(s)\geq\left(H^{-\Lambda}(M)+\Lambda
C_{2}^{1+\Lambda}\int_{s}^{M}\frac{1}{\tau
P(\tau)}d\tau\right)^{-\frac{1}{\Lambda}}},$

with $\Lambda+1=\dfrac{2}{\lambda}$ and $\infty>\Lambda>0$.

###### Proof.

We raise (4.3) to power $\lambda$ and use (5.2) to get,

(5.4) $\displaystyle
G^{\lambda}(s)\leq\left(\sqrt{sF(s)}\right)^{\lambda}\leq{C_{2}H(s)}\implies
h(s)P(s)=F(s)\leq
C_{2}^{\frac{2}{\lambda}}\dfrac{H^{\frac{2}{\lambda}}(s)}{s}$

Now we rewrite right hand side of (5.4) as follows:

(5.5) $\displaystyle\frac{h(s)}{H^{\Lambda+1}(s)}$
$\displaystyle\leq\frac{C_{2}^{1+\Lambda}}{sP(s)}$

Integrate (5.5) over $(s,M)$ we get 5.3. ∎

The preceding proposition implies the following choice of function $H$.

######  Definition 3.

Let $P$ in Hypotheses(1) be such that $\displaystyle
0<\int_{M}^{\infty}\frac{ds}{sP(s)}<\infty$, for some finite $M$. Then for
$s\in[0,M]$, let

(5.6) $\displaystyle I(s)$
$\displaystyle\triangleq\int_{s}^{\infty}\frac{d\sigma}{\sigma P(\sigma)}\
;~{}M>s>0,$ (5.7) $\displaystyle\boxed{H(s)\triangleq\left[\Lambda
I(s)\right]^{-\frac{1}{\Lambda}}=\left(\Lambda\int_{s}^{\infty}\frac{1}{\tau
P(\tau)}d\tau\right)^{-\frac{1}{\Lambda}},~{}s>0.}$

Function $P(s)$ and consequently $H(s)$ define only on bounded interval, But
in order notation to be simple we extended $I(s)$ on whole axis in such order
that $I(s)>constant>0$ on the interval $[M,\infty)$.

######  Remark 4.

Note that, $H$ in (5.7) has $H(0)=0$ since $P(0)=0$. Moreover, substitute
(5.7) in (4.2) to get

(5.8) $\displaystyle h(s)=\frac{1}{sP(s)}\left[\Lambda
I(s)\right]^{-\frac{1}{\Lambda}-1}=\frac{1}{sP(s)}H^{(\Lambda+1)}(s)\
,~{}s>0;$

Using (5.8) for (ii) in Definition 2 we get

(5.9)
$\displaystyle\boxed{F(s)=h(s)P(s)=\frac{1}{s}H^{(\Lambda+1)}(s)=\left(\Lambda
s^{\frac{\Lambda}{1+\Lambda}}I(s)\right)^{-\frac{1}{\Lambda}-1}.}$

Consequently,

$\left[sF(s)\right]^{\frac{\lambda}{2}}=H^{(\Lambda+1)\frac{\lambda}{2}}(s)=H(s),$

since $(\Lambda+1)\frac{\lambda}{2}=1$. Thus, (5.2) holds with $C_{2}=1$.

The next proposition imposes restrictions on $P$ such that $F$ as in 5.9, is
non-negatively increasing on $[0,M)$.

######  Proposition 2.

Let $P$ be in Definition 3 and $F$ be in (5.9). Let

(5.10) $\displaystyle A$ $\displaystyle\triangleq\sup_{0<s<M}P(s)I(s)<\infty$
(5.11) $\displaystyle a$ $\displaystyle\triangleq\limsup_{s\rightarrow
0}P(s)I(s)<A.$

If $\frac{\Lambda+1}{\Lambda}>A$ then $F^{\prime}(s)>0$ on $(0,M)$, and if
$\frac{\Lambda+1}{\Lambda}>a$ then $\lim_{s\rightarrow 0}F(s)=0$.

###### Proof.

First we prove that function $F$ is increasing. It suffices to show the map

(5.12) $s\mapsto\displaystyle{s^{\frac{\Lambda}{1+\Lambda}}I(s)}$

decreases on $(0,M)$. Note that

(5.13)
$\displaystyle\displaystyle\frac{d}{ds}\left(s^{\frac{\Lambda}{1+\Lambda}}I(s)\right)=\left(\frac{\Lambda}{\Lambda+1}\frac{1}{P(s)}\right)\left(P(s)I(s)-\frac{\Lambda+1}{\Lambda}\right)s^{-\frac{1}{\Lambda+1}}.$

Thus, provided $\frac{\Lambda+1}{\Lambda}>A$, one has $F^{{}^{\prime}}(s)>0$
for $s\in(0,M)$. Next we show that $\displaystyle
s^{\frac{\Lambda}{1+\Lambda}}I(s)\rightarrow\infty\text{ as }s\rightarrow 0$,
which implies $\lim_{s\rightarrow 0}F(s)=0$. By (5.11), for every $\epsilon>0$
there exists $s_{\epsilon}\in(0,M)$ such that $I(s)P(s)<a+\epsilon$ for
$s\in(0,s_{\epsilon})$, which yields the following chain of inequalities:

$\displaystyle I(s)<(a+\epsilon)\frac{1}{P(s)}$
$\displaystyle=-(a+\epsilon)sI^{{}^{\prime}}(s)$ $\displaystyle\frac{d}{ds}\ln
I(s)$ $\displaystyle<-\frac{1}{a+\epsilon}\cdot\frac{1}{s}$ $\displaystyle
I(s)$
$\displaystyle>I(s_{\epsilon})\left(\frac{s_{\epsilon}}{s}\right)^{\frac{1}{a+\epsilon}}$
$\displaystyle s^{\frac{\Lambda}{\Lambda+1}}I(s)$ $\displaystyle\geq
I(s_{\epsilon})s_{\epsilon}^{\frac{1}{a+\epsilon}}\times
s^{\frac{\Lambda}{\Lambda+1}-\frac{1}{a+\epsilon}}\ ;\ 0<s<s_{\epsilon}.$

Provided $\frac{\Lambda+1}{\Lambda}>a$, one can choose $\epsilon$ such that
$\frac{\Lambda+1}{\Lambda}>a+\epsilon$, hence
$\frac{\Lambda}{\Lambda+1}-\frac{1}{a+\epsilon}<0.$

$\therefore\ \ s^{\frac{\Lambda}{\Lambda+1}}I(s)\to\infty\text{ as }s\to
0.\qed$

######  Remark 5.

If $A$ and $a$ exist then indeed $\displaystyle\frac{\Lambda+1}{\Lambda}>A>a$
in Proposition 2. However, $A<\infty$ only if $a<\infty$ since function
$P(s)I(s)\in C(0,M]$.

######  Definition 4 (Equivalence of functions and almost monotone functions).

  1. (i)

Functions $f$ and $g$ of a set $E$ are equivalent and write $f\asymp g$ on $E$
if there exist a constant $c\geq 1$ such that $\displaystyle\frac{1}{c}\
g(x)\leq f(x)\leq c\ g(x)$ for all $x\in E$.

  2. (ii)

A function $f$ on an interval $I\subset\mathbb{R}$ is called almost decreasing
(almost increasing) if it is equivalent to a non-increasing (non-decreasing)
function of an interval. Equivalently, there exists a constant $c>0$ such that

$f(t)\geq cf(s),\quad t,s\in I,~{}t<s.$

In the following proposition we give assumptions providing for the inequality
(5.1) in Assumption 4.

######  Proposition 3.

Let $I$ be as in Definition 3, $P$ in Proposition 2, and $F$ in (5.9) with
$\Lambda$ as in Remark 5. Then (5.1) holds provided the existence of $\mu>0$
such that $s\mapsto P(s)I^{\mu}(s)$ is an almost decreasing function.

###### Proof.

By the direct computation, it follows from (5.9) that

(5.14) $\displaystyle F(s)\asymp\ $ $\displaystyle\
s^{-1}I^{-\frac{1}{\Lambda}-1}(s)$ (5.15) $\displaystyle
F^{{}^{\prime}}(s)\asymp\ $ $\displaystyle\ \frac{F(s)}{s}[P(s)I(s)]^{-1}.$

Using (5.14),(5.15) and (iii) in Definition (2) as following:

(5.16) $\displaystyle\frac{F(s)}{G(s)G^{{}^{\prime}}(s)}$
$\displaystyle=\frac{\displaystyle
F(s)}{\displaystyle\left(\int_{0}^{s}\sqrt{F^{{}^{\prime}}(t)\
dt}\right)\displaystyle\left(\sqrt{F^{{}^{\prime}}(s)}\right)}$ (5.17)
$\displaystyle\asymp\frac{\displaystyle
F(s)}{\displaystyle\left(\int_{0}^{s}\sqrt{\displaystyle\frac{F(t)}{t}[P(t)I(t)]^{-1}}\
dt\right)\left(\sqrt{\displaystyle\frac{F(s)}{s}[P(s)I(s)]^{-1}}\right)}$
(5.18)
$\displaystyle=\frac{\displaystyle[I(s)]^{-\frac{1}{2\Lambda}-\frac{1}{2}}[P(s)I(s)]^{\frac{1}{2}}}{\displaystyle\int_{0}^{s}t^{-1}[I(t)]^{-\frac{1}{2\Lambda}-\frac{1}{2}}[P(t)I(t)]^{-\frac{1}{2}}\
dt}$ (5.19)
$\displaystyle=\frac{\displaystyle[I(s)]^{-\frac{1}{2\Lambda}-\frac{\mu}{2}}[P(s)I^{\mu}(s)]^{\frac{1}{2}}}{\displaystyle\int_{0}^{s}t^{-1}[I(t)]^{-\frac{1}{2\Lambda}-1}[P(t)]^{-\frac{1}{2}}\
dt}$

By the Cauchy mean value theorem (aka extended mean value theorem), there
exist $t\in(0,s)$ such that

$\frac{\displaystyle[I(s)]^{-\frac{1}{2\Lambda}-\frac{\mu}{2}}}{\displaystyle\int_{0}^{s}t^{-1}[I(t)]^{-\frac{1}{2\Lambda}-1}[P(t)]^{-\frac{1}{2}}\
dt}=\left(\frac{1}{2\Lambda}+\frac{\mu}{2}\right)\frac{\displaystyle[I(t)]^{-\frac{1}{2\Lambda}-1-\frac{\mu}{2}}[tP(t)]^{-1}}{\displaystyle
t^{-1}[I(t)]^{-\frac{1}{2\Lambda}-1}[P(t)]^{-\frac{1}{2}}}\asymp[P(t)I^{\mu}(t)]^{-\frac{1}{2}}.$

Note that $\displaystyle I^{{}^{\prime}}(s)=\ [sP(s)]^{-1}$. Since $t\mapsto
P(t)I^{\mu}(t)$ is almost decreasing one has

(5.20)
$\frac{F(s)}{G(s)G^{{}^{\prime}}(s)}\asymp\left(\frac{P(s)I^{\mu}(s)}{P(t)I^{\mu}(t)}\right)^{\frac{1}{2}}\leq
C\ ;\ 0<t<s.\qed$

The sufficient condition for the existence of $\mu>0$ for the almost
decreasing function $t\mapsto P(t)I^{\mu}(t)$ is as follows

######  Remark 6.

Assume that there exists $\tilde{P}\in C^{1}(0,\infty)$ such that
$P\asymp\tilde{P}$ on $(0,M)$, such that

(5.21) $\limsup_{s\to 0}s\tilde{I}(s)\tilde{P}^{{}^{\prime}}(s)<\infty,$

with

$\tilde{I}(s)=\int_{s}^{\infty}\frac{dt}{t\tilde{P}(t)}.$

Indeed, (5.21) implies that $s\mapsto s\tilde{I}(s)\tilde{P}^{{}^{\prime}}(s)$
is bounded on $(0,M)$. Let

$B=\sup\limits_{0<s<M}s\tilde{I}(s)\tilde{P}^{{}^{\prime}}(s).$

Fix $\mu\geq B$. Then the function $Q(s)=\tilde{P}(s)\tilde{I}^{\mu}(s)$ is
non-increasing since

$Q^{\prime}(s)=\tilde{P}^{{}^{\prime}}(s)\tilde{I}^{\mu}(s)-\mu
s^{-1}\tilde{I}^{\mu-1}(s)=s^{-1}\tilde{I}^{\mu-1}(s)\left[s\tilde{I}(s)\tilde{P}^{{}^{\prime}}(s)-\mu\right]\leq
0.$

Finally, note that $P(s)I^{\mu}(s)\asymp\tilde{P}(s)\tilde{I}^{\mu}(s)$.

##  6\. Auxiliary integral estimates

We will start with Auxiliary generic estimates for function $u,$ and cutoff
functions $\theta$ w.r.t. non-linear functions $F,$ and $G$.

######  Lemma 1.

Let $u$ and $\nabla u$ be measurable functions, and let $\theta\in
Lip_{c}(\Omega)$. Let $F,G\in C^{1}(0,M)\cup C[0,M]$ satisfy (5.1). Then

(6.1) $\nabla
u\cdot\nabla\left(\theta^{2}F(u)\right)\geq\frac{1}{2}|\nabla(\theta
G(u))|^{2}-(2C_{1}^{2}+1)G^{2}(u)|\nabla\theta|^{2}.$

###### Proof.

$\displaystyle\nabla u\cdot\nabla\left(\theta^{2}F(u)\right)$
$\displaystyle=\theta^{2}F^{{}^{\prime}}(u)|\nabla u|^{2}+2F(u)\nabla
u\cdot\theta\nabla\theta$ (6.2)
$\displaystyle=\theta^{2}F^{{}^{\prime}}(u)|\nabla
u|^{2}+2\dfrac{F(u)}{G^{{}^{\prime}}(u)}G^{{}^{\prime}}(u)\nabla
u\cdot\theta\nabla\theta\quad;\ G^{{}^{\prime}}(u)>0$

Using (5.1) in right-hand side of above yields

$\displaystyle\nabla u\cdot\nabla\left(\theta^{2}F(u)\right)$
$\displaystyle=\theta^{2}|G^{{}^{\prime}}(u)|^{2}|\nabla
u|^{2}+2\frac{F(u)}{G^{{}^{\prime}}(u)}\left(G^{{}^{\prime}}(u)\nabla
u\cdot\theta+G(u)\nabla\theta\right)\cdot\nabla\theta-2\frac{F(u)}{G^{{}^{\prime}}(u)}G(u)|\nabla\theta|^{2}$
$\displaystyle=|\nabla(\theta
G(u))-G(u)\nabla\theta|^{2}+2\dfrac{F(u)}{G^{{}^{\prime}}(u)}\nabla(\theta
G(u))\cdot\nabla\theta-2\dfrac{F(u)}{G^{{}^{\prime}}(u)}G(u)|\nabla\theta|^{2}$
$\displaystyle=|\nabla(\theta
G(u))|^{2}+2\left(\dfrac{F(u)}{G^{{}^{\prime}}(u)}-G(u)\right)\cdot\nabla\theta\cdot\nabla(\theta
G(u))-2\left(\dfrac{F(u)}{G^{{}^{\prime}}(u)}-G(u)\right)|\nabla\theta|^{2}G(u)$
(6.3) $\nabla u\cdot\nabla\left(\theta^{2}F(u)\right)\\\ \geq|\nabla(\theta
G(u))|^{2}-\left|2\left(\dfrac{F(u)}{G^{{}^{\prime}}(u)}-G(u)\right)\cdot\nabla\theta\cdot\nabla(\theta
G(u))\right|-2\left(\dfrac{F(u)}{G^{{}^{\prime}}(u)}-G(u)\right)|\nabla\theta|^{2}G(u).$

Note that by Cauchy’s Inequality,

$\displaystyle
2\left(\dfrac{F(u)}{G^{{}^{\prime}}(u)}-G(u)\right)\nabla\theta\cdot\nabla(\theta
G(u))\leq
2\left(\dfrac{F(u)}{G^{{}^{\prime}}(u)}-G(u)\right)^{2}|\nabla\theta|^{2}+\dfrac{1}{2}|\nabla(\theta
G(u))|^{2}.$

Then inequality (6.3) becomes

$\displaystyle\nabla u\cdot\nabla\left(\theta^{2}F(u)\right)$
$\displaystyle\geq\dfrac{1}{2}|\nabla(\theta
G(u))|^{2}-2\left(\dfrac{F(u)}{G^{{}^{\prime}}(u)}-G(u)\right)^{2}\cdot|\nabla\theta|^{2}-\left(2\dfrac{F(u)}{G^{{}^{\prime}}(u)}-G(u)\right)|\nabla\theta|^{2}G(u)$
(6.4) $\displaystyle=\dfrac{1}{2}|\nabla(\theta
G(u))|^{2}-\left[2\left(\dfrac{F(u)}{G^{{}^{\prime}}(u)}\right)^{2}-2\dfrac{F(u)}{G^{{}^{\prime}}(u)}G(u)+G^{2}(u)\right]|\nabla\theta|^{2}$
(6.5) $\displaystyle\geq\frac{1}{2}|\nabla(\theta
G(u))|^{2}-(2C_{1}^{2}+1)G^{2}(u)|\nabla\theta|^{2}.\qed$

######  Lemma 2.

Assume (5.2). Let $u$ be a measurable function of $\Omega$, $0<u<M$, $\nabla
u\in L^{2}_{loc}(\Omega\times(0,T))$ and $\nabla G(u)\in
L^{2}_{loc}(\Omega\times[0,T])$. Let $\displaystyle j=\frac{2}{N-2}>0$ for
Gagliardo–Nirenberg–Sobolev inequality

(6.6) $\displaystyle||\psi||_{L^{2+2j}}^{2}\leq S||\nabla\psi||_{L^{2}}^{2}.$

Let $K\subset\Omega$ be compact and $\theta_{n}\in Lip_{c}(\Omega)$ such that
$\theta_{n}=1$ on $K$. Then,

(6.7) $\displaystyle\int_{0}^{t}\int_{K}G^{2}(u)dxdt\leq
c_{1}t^{1-(1+j)k}\left[\sup_{0\leq\tau\leq
T}\int_{\Omega}\theta_{n}^{2}H(u(\tau))dx+\int_{0}^{t}\int_{\Omega}|\nabla(\theta_{n}u)|^{2}dxd\tau\right]^{1+jk}$

where $\displaystyle k=\frac{2-\lambda}{2+2j-\lambda}$,
$c_{1}=C_{2}^{1-k}S^{k(1+j)}$ with $C_{2}$ as in (5.2).

###### Proof.

Note that $\displaystyle\lambda=[2-(2+2j)k]/[1-k]$. By (5.2) recall that
$\left(\sqrt{sF(s)}\right)^{\lambda}\leq{C_{2}H(s)}$. Then one has

(6.8) $\displaystyle G^{2}(u)\leq C_{2}^{1-k}G^{2(1+j)k}(u)H^{1-k}(u).$

Integrate both side of (6.8) over $K\times(0,t)$.

$\displaystyle\int_{0}^{t}\int_{K}G^{2}(u)dxdt\leq$ $\displaystyle\
C_{2}^{1-k}\int_{0}^{t}\int_{K}G^{2(1+j)k}(u)H^{1-k}(u)dxd\tau$
$\displaystyle\leq$ $\displaystyle\
C_{2}^{1-k}\int_{0}^{t}\int_{K}\left(|\theta_{n}G(u)|^{2(1+j)}\right)^{k}\left(\theta_{n}^{2}H(u)\right)^{1-k}dxd\tau$
$\displaystyle\leq$ $\displaystyle\
C_{2}^{1-k}\int_{0}^{t}\left[\int_{\Omega}|\theta_{n}G(u)|^{2(1+j)}dx\right]^{k}\left[\int_{\Omega}\theta_{n}^{2}H(u)dx\right]^{1-k}d\tau$
$\displaystyle\leq$ $\displaystyle\
C_{2}^{1-k}S^{k(1+j)}\int_{0}^{t}\left[\int_{\Omega}|\nabla(\theta_{n}G(u))|^{2}dx\right]^{(1+j)k}d\tau\left[\sup_{0\leq\tau\leq
T}\int_{\Omega}\theta_{n}^{2}H(u)dx\right]^{1-k}$

by (6.6). We apply the Holder inequality for time integral and then using the
estimate $x^{v}y^{w}\leq(x+y)^{v+w}\ ;\ x,y,v,w>0$ to get the following:

$\displaystyle\int_{0}^{t}\int_{K}G^{2}(u)dxdt$ $\displaystyle\leq
C_{2}^{1-k}S^{k(1+j)}\left[\sup_{0\leq\tau\leq
T}\int_{\Omega}\theta_{n}^{2}H(u)dx\right]^{1-k}t^{1-k(1+j)}\left[\int_{0}^{t}\int_{\Omega}|\nabla(\theta_{n}G(u))|^{2}dxd\tau\right]^{(1+j)k}$
$\displaystyle\leq C_{2}^{1-k}S^{k(1+j)}t^{1-k(1+j)}\left[\sup_{0\leq\tau\leq
T}\int_{\Omega}\theta_{n}^{2}H(u)dx+\int_{0}^{t}\int_{\Omega}|\nabla(\theta_{n}G(u))|^{2}dxd\tau\right]^{1+jk}$

∎

##  7\. Localization of the solution of degenerate Einstein Equation

######  Definition 5.

Let $R>0$ and $b>2$. Let $R_{n}$ be a sequence such that

(7.1) $\displaystyle R_{0}=$ $\displaystyle R,$ (7.2) $\displaystyle R_{n}=$
$\displaystyle
R_{n-1}-Rb^{-n}=R\left(\frac{b-2+b^{-n}}{b-1}\right);n=1,2,\dots,$ (7.3)
$\displaystyle R_{\infty}=$
$\displaystyle\lim\limits_{n\to\infty}R_{n}=R\left(\frac{b-2}{b-1}\right).$

Let $\theta_{n}(x)\in Lip(\Omega)$ such that

(7.4)
$\displaystyle\theta_{n}(x)\triangleq\min\left\\{\dfrac{\left(R_{n}-||x-x_{0}||\right)_{+}}{R_{n}-R_{n+1}},1\right\\}=\begin{cases}0\
;\ x\notin B_{R_{n}}(x_{0})\\\ 1\ ;\ x\in B_{R_{n+1}}(x_{0})\end{cases}$

for $x_{0}\in{\Omega}^{{}^{\prime}}\Subset\Omega$ and $B_{n}\triangleq
B_{R_{n}}(x_{0})$.

Then one can show that

(7.5) $\displaystyle||\nabla\theta_{n}(x)||_{\infty}\leq\dfrac{b^{n+1}}{R}.$

######  Theorem 1.

Let $u$ be a positive solution of IBVP (4.7). Let
$\Omega^{\prime}\subset\Omega$ be such that $u(x,0)=0$ for
$x\in\Omega^{\prime}$. Then for every ball
$B_{R}(x_{0})\Subset{\Omega}^{{}^{\prime}}$ and every $R^{\prime}\in(0,R)$ ,
there exist $T^{\prime}>0$ such that $u(x,t)=0$ for $(x,t)\in
B_{R^{\prime}}(x_{0})\times[0,T^{\prime}]$.

###### Proof.

Let us choose $b>2$ such that

(7.6) $\displaystyle\displaystyle\dfrac{b-2}{b-1}=\dfrac{R^{\prime}}{R}$

We develop the iteration scheme $R_{n},B_{n},\theta_{n}$ as given in
Definition 5. Note that $R_{\infty}=R^{\prime}$. We start by multiplying
inequality in IBVP (4.7) by $\theta_{n}^{2}$ and integrate over
$\Omega\times(0,t)$,

(7.7) $\displaystyle\int_{\Omega}\theta_{n}^{2}H(u)\text{
}dx+\int_{0}^{t}\int_{\Omega}\nabla u\nabla(\theta_{n}^{2}F(u))\text{
}dxd\tau\leq 0$

Using (6.1) in Lemma 1 in above we get

(7.8) $\displaystyle\int_{\Omega}\theta_{n}^{2}H(u)\text{
}dx+\frac{1}{2}\int_{0}^{t}\int_{\Omega}|\nabla(\theta_{n}G(u))|^{2}\text{
}dxd\tau\leq(2C_{1}^{2}+1)\int_{0}^{t}\int_{B_{n}}G^{2}(u)|\nabla\theta_{n}|^{2}\text{
}dxd\tau\vspace{- 0.8 cm}$

In particular, $\nabla G(u)\in L^{2}_{loc}(\Omega\times[0,T])$. Now using
(6.7) in Lemma 2,

(7.9) $\displaystyle\int_{\Omega}$ $\displaystyle\theta_{n}^{2}H(u)\text{
}dx+\frac{1}{2}\int_{0}^{t}\int_{\Omega}|\nabla(\theta_{n}G(u))|^{2}\text{
}dxd\tau$ $\displaystyle\leq$ $\displaystyle
D\cdot(b^{2})^{n-1}t^{1-k(1+j)}\left[\sup_{0\leq\tau\leq
t}\int_{B_{n}}\theta_{n}^{2}H(u)dx+\int_{0}^{t}\int_{B_{n}}|\nabla(\theta_{n}G(u))|^{2}dxd\tau\right]^{1+jk}$

where $D\triangleq\dfrac{b^{4}}{R^{2}}(2C_{1}^{2}+1)C_{2}^{1-k}S^{k(1+j)}$. As
$0<t\leq T^{\prime}$ we take supremum over $t$

(7.10) $\sup_{0\leq\tau\leq T^{\prime}}\int_{B_{n}}\theta_{n}^{2}H(u)\text{
}dx+\int_{0}^{T^{\prime}}\int_{B_{n}}|\nabla(\theta_{n}G(u))|^{2}\text{
}dxd\tau\\\ \leq
D({b^{2}})^{n-1}(T^{\prime})^{1-k(1+j)}\left[\sup_{0\leq\tau\leq
T^{\prime}}\int_{B_{n}}\theta_{n}^{2}H(u)dx+\int_{0}^{T^{\prime}}\int_{B_{n}}|\nabla(\theta_{n}G(u))|^{2}dxd\tau\right]^{1+jk}$

Note that $\gamma\triangleq\dfrac{1-(1+j)k}{kj}$ then
$\gamma+1-(1+j)k=\gamma(1+kj)$.

Multiply above inequality by $(T^{\prime})^{\gamma}$. So (7.10) yields,

$\displaystyle(T^{\prime})^{\gamma}\sup_{0\leq\tau\leq
T^{\prime}}\int_{\Omega}\theta_{n}^{2}(x)H(u)\
dx+(T^{\prime})^{\gamma}\int_{0}^{T}\int_{\Omega}|\nabla(\theta_{n}(x)G(u))|^{2}\text{
}dxd\tau$ $\displaystyle\leq
D\cdot({b^{2}})^{n-1}\left[(T^{\prime})^{\gamma}\sup_{0\leq\tau\leq
T^{\prime}}\int_{B_{n}}\theta_{n}^{2}(x)H(u)dx+(T^{\prime})^{\gamma}\int_{0}^{T^{\prime}}\int_{B_{n}}|\nabla(\theta_{n}(x)G(u))|^{2}dxd\tau\right]^{1+jk}$
$\displaystyle\leq
D\cdot({b^{2}})^{n-1}\left[(T^{\prime})^{\gamma}\sup_{0\leq\tau\leq
T^{\prime}}\int_{B_{n-1}}\theta_{n-1}^{2}(x)H(u)dx+(T^{\prime})^{\gamma}\int_{0}^{T^{\prime}}\int_{B_{n-1}}|\nabla(\theta_{n-1}(x)G(u))|^{2}dxd\tau\right]^{1+jk}$

Let us introduce

(7.11) $\displaystyle
Y_{n}(T^{\prime})\triangleq(T^{\prime})^{\gamma}\sup_{0\leq\tau\leq
T^{\prime}}\int_{B_{n}}\theta_{n}^{2}(x)H(u)dx+(T^{\prime})^{\gamma}\int_{0}^{T^{\prime}}\int_{B_{n}}|\nabla(\theta_{n}(x)G(u))|^{2}dxd\tau$

then the above becomes

(7.12) $\displaystyle Y_{n}(T^{\prime})\leq
D\cdot({b^{2}})^{n-1}Y_{n-1}^{1+kj}(T^{\prime})$

By Ladyzhenskaya-Uraltceva iterative Lemma 1, if

(7.13) $Y_{0}(T^{\prime})\leq D^{-\dfrac{1}{kj}}b^{-\dfrac{2}{k^{2}j^{2}}}$

then

(7.14) $\displaystyle Y_{n}(T^{\prime})\rightarrow 0\text{ whenever
}n\rightarrow\infty.$

Since (7.8) and $\gamma>0$, we can choose $T^{\prime}$ small enough to satisfy
(7.13). So the assertion holds. ∎

######  Remark 7.

In fact, it was not necessary to assume for $u$ certain bounds and positively
on $B_{R}(x_{0})\times[0,T]$. It is enough to assume that non-negative
solution belong to the class of bounded functions in $R^{N}\times[0,\infty],$
and then using maximum principle, prove that it $u$ bounded by initial data at
$R^{N}\times\\{0\\}$.

##  8\. Example of functions $P$ and $F$ for which constrains holds

Without loss of generality in this section we will assume that $0\leq u\leq
1$, and we will show some generic examples of the function $P$ for which all
constrains on the functions $F$, $G$, $H$ holds. Note that general case $0\leq
u<M$ can be reduced with appropriate scaling in equation. Evidently this will
lead to smaller time interval $[0,T]$ over which solution in $B(x_{0},R)$ is
be equal $0$.

######  Remark 8.

Let $P,I\in C^{1}(0,\infty)$ be as in Definition 3. Assume that

(8.1) $\displaystyle\limsup\limits_{s\to 0}P(s)I(s)$ $\displaystyle<\infty,$
(8.2) $\displaystyle\limsup\limits_{s\to 0}sP^{{}^{\prime}}(s)I(s)$
$\displaystyle<\infty.$

Then Assumption 4 hold, by Propositions (2), (3) and Remark 6. So Theorem 1
about finite speed of propagation holds.

Below we will construct examples of the function $P$ as in Remark 8.

######  Example 1.

$P(s)=s^{\gamma}\ ;\ s\in[0,\infty),\gamma>0.$.

(8.3) $\displaystyle I(s)$ $\displaystyle=\int_{s}^{\infty}t^{-\gamma-1}dt=\
\tfrac{1}{\gamma}s^{-\gamma}$ (8.4) $\displaystyle P(s)I(s)$
$\displaystyle=\tfrac{1}{\gamma}$ (8.5) $\displaystyle sP^{\prime}(s)I(s)$
$\displaystyle=1.$

######  Example 2.

$P(s)=\exp\left(-\dfrac{1}{s^{\gamma}}\right),s\in[0,1),\gamma>0.\ P(0)=0\
;P\in C([0,\infty))$.

(8.6) $\displaystyle
I(s)=\int_{s}^{1}t^{-1}\exp\left(\dfrac{1}{t^{\gamma}}\right)dt.$ (8.7)
$\displaystyle\displaystyle P(s)I(s)=$
$\displaystyle{\exp\left(-\dfrac{1}{s^{\gamma}}\right)}\left[\int_{s}^{1}t^{-1}\exp\left(\dfrac{1}{t^{\gamma}}\right)dt\right]$

By L’Hôpital’s rule

(8.8) $\displaystyle\displaystyle\lim_{s\to 0}P(s)I(s)\equiv\ \lim_{s\to
0}s^{\gamma}\equiv\ \ 0,$

verifies that exist $\ \Lambda$ s.t.
$P(s)I(s)<\displaystyle\frac{\Lambda+1}{\Lambda}\ \forall\ s\in[0,1)$. Next
using (5.21)

(8.9) $\displaystyle sI(s)P^{/}(s)=\ $ $\displaystyle\
s\left[\int_{s}^{1}t^{-1}\exp\left(\dfrac{1}{t^{\gamma}}\right)dt\right]\exp\left(-\frac{1}{s^{\gamma}}\right)s^{\gamma-1}$
(8.10) $\displaystyle\lim_{s\to 0}sI(s)P^{/}(s)=$ $\displaystyle\lim_{s\to
0}\frac{\displaystyle\left[\int_{s}^{1}t^{-1}\exp\left(\frac{1}{t^{\gamma}}\right)dt\right]s^{-\gamma}}{\gamma\displaystyle\exp\left(\frac{1}{s^{\gamma}}\right)}$
(8.11) $\displaystyle\equiv$ $\displaystyle\ 1+\lim_{s\to
0}\frac{\left[\displaystyle\int_{s}^{1}t^{-1}\exp\left(\frac{1}{t^{\gamma}}\right)dt\right]}{\displaystyle\exp\left(\frac{1}{s^{\gamma}}\right)}\quad\text{(by
L'H\^{o}pital's rule)}$ (8.12) $\displaystyle\equiv$ $\displaystyle\
1+\lim_{s\to 0}s^{\gamma}\quad\quad\quad\hskip 85.35826pt\text{(by
L'H\^{o}pital's rule)}$ (8.13) $\displaystyle\equiv$ $\displaystyle\ 1$

By (5.21) $\lim_{s\to 0}{F(s)}[G(s)G^{/}(s)]^{-1}\equiv\ 1$ which verifies
existence of $\ C_{1}>0$ in Assumption (5.1).

######  Example 3.

$\displaystyle
P(s)=\exp\left(-\int_{s}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right),s\in(0,1]\ ;\
P(0)=0\ ;P\in C([0,\infty))$. Provided that $\ 0<k_{1}<\zeta(s)<k_{2}$.

(8.14) $\displaystyle\displaystyle
I(s)=\int_{s}^{1}t^{-1}\exp\left(\int_{t}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right)\
dt.$

Then

(8.15) $\displaystyle\displaystyle P(s)I(s)=$ $\displaystyle\
\frac{\displaystyle\left[\int_{s}^{1}t^{-1}\exp\left(\int_{t}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right)\
dt\right]}{\displaystyle\left[\exp\left(\int_{s}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right)\right]}$

By L’Hôpital’s rule

(8.16) $\displaystyle\lim_{s\to 0}P(s)I(s)\equiv$ $\displaystyle\ \lim_{s\to
0}\frac{1}{\zeta(s)}\equiv 1$

which verifies that $\exists\ \Lambda$ s.t.
$P(s)I(s)<\displaystyle\frac{\Lambda+1}{\Lambda}\ \forall\ s\in[0,1)$. Note
that $\zeta(s)\equiv 1$. Next using (5.21)

(8.17) $\displaystyle sI(s)P^{/}(s)=$ $\displaystyle\
{\displaystyle\left[\int_{s}^{1}t^{-1}\exp\left(\int_{t}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right)\
dt\right]}{\displaystyle\exp\left(-\int_{t}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right)\zeta(s)}$
(8.18) $\displaystyle\lim_{s\to 0}sI(s)P^{/}(s)\equiv\ $
$\displaystyle\lim_{s\to
0}\frac{\displaystyle\left[\int_{s}^{1}t^{-1}\exp\left(\int_{s}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right)\
dt\right]}{\displaystyle\exp\left(\int_{t}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right)}$

We apply L’Hôpital’s rule to get

(8.19) $\displaystyle\lim_{s\to 0}sI(s)P^{/}(s)\equiv\
\frac{1}{\zeta(s)}\equiv\ 1$

By (5.21) $\lim_{s\to 0}{F(s)}[G(s)G^{/}(s)]^{-1}\equiv\ 1$ which verifies
$\exists\ C_{1}>0$ in Assumption (5.1).

######  Example 4.

$\displaystyle
P(s)=\exp\left(-\int_{s}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right),s\in(0,1]\ ;\
P(0)=0\ ;P\in C([0,\infty))$with $\displaystyle\lim_{\tau\to
0}\zeta(\tau)=\infty$. We will assume $\zeta$ be such that $\displaystyle 0<\
k_{3}\leq\frac{\zeta}{\zeta_{0}}\leq k_{4}\ ;\text{for
function}s.t.\zeta_{0}^{/}\leq 0\ ;\
\sup_{0<s<1}\frac{s|\zeta_{0}^{/}|}{\zeta_{0}}=C_{0}<\infty$. Once again
recollect

(8.20) $\displaystyle\displaystyle
I(s)=\int_{s}^{1}t^{-1}\exp\left(\int_{t}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right)\
dt.$

Similarly in(8.16) we get

(8.21) $\displaystyle\displaystyle\lim_{s\to
0}P(s)I(s)\equiv\frac{1}{\zeta(s)}=0.$

verifies the existence of $\ \Lambda$ s.t.
$P(s)I(s)<\displaystyle\frac{\Lambda+1}{\Lambda}\ \forall\ s\in[0,1)$. Note
that $\zeta(s)\equiv\zeta_{0}(s)$.

Next using (5.21) we will get

(8.22) $\displaystyle sI(s)P^{/}(s)=$ $\displaystyle
s\left[\int_{s}^{1}t^{-1}\exp\left(\int_{t}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right)\
dt\right]\exp\left(-\int_{t}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right)\zeta(s)$

Moreover,

$\displaystyle\lim_{s\to 0}sI(s)P^{/}(s)=$ $\displaystyle\lim_{s\to
0}\frac{\displaystyle\left[\int_{s}^{1}t^{-1}\exp\left(\int_{t}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right)\
dt\right]\zeta_{0}(s)}{\displaystyle\exp\left(\int_{s}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right)}$
$\displaystyle\equiv$ $\displaystyle\ 1+\lim_{s\to
0}\frac{\left[\displaystyle\int_{s}^{1}t^{-1}\exp\left(\int_{t}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right)\
dt\right]\frac{\displaystyle
s|\zeta_{0}^{/}(s)|}{\displaystyle\zeta(s)}}{\displaystyle\exp\left(\int_{s}^{1}\frac{\zeta(\tau)}{\tau}d\tau\right)\
dt}\quad\text{(by L'H\^{o}pital's rule)}$ $\displaystyle\equiv$
$\displaystyle\ 1+\lim_{s\to 0}\frac{C_{0}}{\zeta(s)}\hskip
170.71652pt\text{(by L'H\^{o}pital's rule)}$ (8.23) $\displaystyle\equiv$
$\displaystyle\ 1$

By (5.21) $\lim_{s\to 0}{F(s)}[G(s)G^{/}(s)]^{-1}\equiv\ 1$ which verifies
existence of $\ C_{1}>0$ in Assumption (5.1).

##  9\. Auxiliary properties of Functional spaces, and a priory estimates for
the solution

######  Theorem 2.

Let $X\subset Y$ be Banach spaces. Let $X^{*}$ and $Y^{*}$ be the dual spaces
of $X$ and $Y$ respectively. Then $Y^{*}\subset X^{*}$.

###### Proof.

Let $f\in Y^{*}$. Then $f(x)$ is well defined for all $x\in X\subset Y$.
Moreover

(9.1) $\displaystyle\sup_{||x||_{X}=1}f(x)$
$\displaystyle\leq\sup_{||x||_{X}=1}||f||_{Y^{*}}||x||_{Y}\leq||f||_{Y^{*}}\implies||f||_{X^{*}}\leq||f||_{Y^{*}}.\qed$

######  Theorem 3.

Let $X,Y$ be Banach spaces and $Z$ be a topological vector space such that
$X,Y\subset Z$. Let $\displaystyle X+Y\triangleq\\{z=x+y:x\in X,y\in Y\\}$
with $\displaystyle||z||_{X+Y}=\inf\\{||x||_{X}+||y||_{Y}:x\in X,y\in
Y,x+y=z\\}.$ Then

  1. i.

$X+Y$ is a Banach space.

  2. ii.

$X,Y\subset X+Y$.

  3. iii.

$(X+Y)^{*}=X^{*}\cap Y^{*}$

  1. (i)

###### Proof.

Let $z_{n}\in X+Y$ be such that $\displaystyle\sum_{n}||z_{n}||_{X+Y}<\infty$.
For every $n$, let $x_{n}\in X$, $y_{n}\in Y$ be such that $x_{n}+y_{n}=z_{n}$
and

(9.2) $\displaystyle||x_{n}||_{X}+||y_{n}||_{Y}<||z_{n}||_{X+Y}+2^{-n}.$

Then
$\sum_{n}||x_{n}||_{X}+\sum_{n}||y_{n}||_{Y}<\sum_{n}||z_{n}||_{X+Y}+2^{-n}$.
Since $X$ and $Y$ are Banach spaces there are $x\in X$ and $y\in Y$ such that
$\displaystyle x=\sum_{n}x_{n}\ \ y=\sum_{n}y_{n}$. Thus,
$\displaystyle\sum_{n}z_{n}=x+y.$ ∎

  2. (ii)

###### Proof.

For $x\in X$ then $x\in X+Y$, and by definition $||x||_{X+Y}\leq||x||_{X}$.
Similarly,

$||y||_{X+Y}\leq||y||_{Y}.\qed$

  3. (iii)

###### Proof.

Since $X\subset X+Y$ and $y\subset X+Y$, it follows from Theorem (2)

$(X+Y)^{*}\subset X^{*}\ \text{ and }\ (X+Y)^{*}\subset Y^{*}$

So that $(X+Y)^{*}\subset(X^{*}\cap Y^{*})$. Now show the converse. Let $f\in
X^{*}\cap Y^{*}$, $z\in X+Y$. For any $n\in\mathbb{N}$, let $x_{n}\in X$ and
$y_{n}\in Y$ be such that $z=x_{n}+y_{n}$ and

(9.3)
$\displaystyle\sum_{n}||x_{n}||_{X}+\sum_{n}||y_{n}||_{Y}<\sum_{n}||z_{n}||_{X+Y}+\frac{1}{n}.$

Then $f(z)=f(x)+f(y)$ is well-defined and

(9.4) $\displaystyle|f(z)|\leq$
$\displaystyle||f||_{X^{*}}||x_{n}||_{X}+||f||_{Y^{*}}||y_{n}||_{Y}$ (9.5)
$\displaystyle\leq$ $\displaystyle\max\\{||f||_{X^{*}},||f||_{Y^{*}}\\}\
(||x_{n}||_{X}+||y_{n}||_{Y})$ (9.6) $\displaystyle\leq$
$\displaystyle||f||_{X^{*}\cap Y^{*}}\left(||z||_{X+Y}+\frac{1}{n}\right)$

As $n\rightarrow\infty$, $|f(z)|\leq||f||_{X^{*}\cap Y^{*}}\ ||z||_{X+Y}$. So
$f\in(X+Y)^{*}$ and  $||f||_{(X+Y)^{*}}\leq||f||_{X^{*}\cap Y^{*}}.\qed$

######  Theorem 4.

Let $B_{0}\Subset B_{1}\subset B_{2}$ be Banach Spaces. Let
$I\subset\mathbb{R}$ and $\mathbb{F}\subset L^{P}(I,B_{0})$ such that
$\displaystyle\sup_{f\in\mathbb{F}}||f||_{L^{P}(I,B_{0})}<\infty\ ,\
\sup_{f\in\mathbb{F}}||f^{{}^{\prime}}||_{L^{1}(I,B_{2})}<\infty$. Then
$\mathbb{F}$ is compact in $L^{P}(I,B_{1})$.

See corollary 4 of Theorem 5 in [6]

##  10\. Weak viscous solution of degenerate Einstein equation.

######  Theorem 5.

Let $u_{\epsilon}(x,t)\in C^{2,1}_{x}(\bar{\Omega}\times(0,T])\ ;\
0<\epsilon\leq u_{\epsilon}\leq K<\infty$ be classical solution of the
$IBVP_{\epsilon}$ (4.8) with initial data on $W^{1,2}(\Omega)$. Let
$\displaystyle\tilde{H}(u_{\epsilon})=\int_{\epsilon}^{u_{\epsilon}}\sqrt{{h_{\epsilon}(s)}/{F_{\epsilon}(s)}}\
ds.$ Then for any $\tau>0$,

(10.1)
$\displaystyle\int_{0}^{\tau}\int_{\Omega}\left(\frac{\partial\tilde{H}(u_{\epsilon}(x,t))}{\partial
t}\right)^{2}dxdt+\int_{\Omega}|\nabla
u_{\epsilon}(x,\tau)|^{2}dx=\int_{\Omega}|\nabla g(x)|^{2}dx$

and

(10.2)
$\int_{0}^{T}\int_{\Omega}\left(\frac{\partial\tilde{H}(u_{\epsilon}(x,t))}{\partial
t}\right)^{2}dxdt+\int_{0}^{T}\int_{\Omega}|\nabla u_{\epsilon}(x,t)|^{2}\
dxdt.\\\ +\int_{0}^{T}\int_{\Omega}|u_{\epsilon}(x,t)|^{2}\ dxdt\leq
C(\Omega,T)\int_{\Omega}|\nabla g(x)|^{2}\ dx.$

###### Proof.

Recall that by (4.2) we write $\displaystyle
H(u_{\epsilon})\triangleq\int_{\epsilon}^{u_{\epsilon}}h(s)\ ds$. First
observe that if $u_{\epsilon}(x,t)$ is solution of (4.8) then one can have the
following:

(10.3)
$\displaystyle\frac{h(u_{\epsilon})}{F(u_{\epsilon})}(u_{\epsilon})_{t}-\Delta
u_{\epsilon}=$ $\displaystyle 0\hskip
39.83368pt\text{on}\quad\Omega\times(0,T),$ (10.4) $\displaystyle
u_{\epsilon}(x,0)=$ $\displaystyle\epsilon+g(x)\hskip
11.38092pt\text{on}\quad\Omega,$ (10.5) $\displaystyle u_{\epsilon}(x,t)=$
$\displaystyle\epsilon\cdot\psi(x)\quad\text{on}\quad\partial\Omega\times(0,T).$

with some $g(x)\geq 0,\ g(x)\in W^{1,2}(\Omega)$ and $\psi(x)\geq 0$. Multiply
equation (10.3) by ${(u_{\epsilon})}_{t}$ and integrate over
$\Omega\times(0,\zeta_{1})$ for $0<\zeta_{1}\leq T$.

(10.6)
$\displaystyle\int_{0}^{\zeta_{1}}\int_{\Omega}\left(\frac{\partial\tilde{H}(u_{\epsilon}(x,t))}{\partial
t}\right)^{2}dxdt-\int_{0}^{\zeta_{1}}\int_{\Omega}(u_{\epsilon})_{t}\Delta
u_{\epsilon}$ $\displaystyle=0$

By using (10.5) we get

(10.7) $\displaystyle\int_{\Omega}(u_{\epsilon})_{t}\Delta
u_{\epsilon}=\int_{\partial\Omega}(u_{\epsilon})_{t}\nabla
u_{\epsilon}\cdot\mu^{i}ds-\int_{\Omega}(\nabla u_{\epsilon})_{t}\nabla
u_{\epsilon}\ dx=-\frac{1}{2}\int_{\Omega}\left({|\nabla
u_{\epsilon}|^{2}}\right)_{t}\ dx$

Integrate over time

(10.8) $\displaystyle\int_{0}^{\zeta_{1}}\int_{\Omega}(u_{\epsilon})_{t}\Delta
u_{\epsilon}\ dxdt=-\frac{1}{2}\int_{\Omega}|\nabla
u_{\epsilon}(x,\zeta_{1})|^{2}\ dx+\frac{1}{2}\int_{\Omega}|\nabla g(x)|^{2}\
dx$

Using (10.8) in (10.6) we get

(10.9)
$\int_{0}^{\zeta_{1}}\int_{\Omega}\left(\frac{\partial\tilde{H}(u_{\epsilon}(x,t))}{\partial
t}\right)^{2}dxdt+\frac{1}{2}\int_{\Omega}|\nabla
u_{\epsilon}(x,\zeta_{1})|^{2}dx=\frac{1}{2}\int_{\Omega}|\nabla g(x)|^{2}dx\
;\ 0<\zeta_{1}\leq T$

As $u\in W^{1,2}(\Omega)$ by integrating Friedrich’s inequality in time over
$(0,\zeta_{1})$

(10.10)
$\displaystyle\int_{0}^{\zeta_{1}}\int_{\Omega}(u_{\epsilon}(x,t))^{2}\ dxdt-
C_{F}(\Omega)\epsilon^{2}\zeta_{1}\int_{\partial\Omega}|\psi(x)|^{2}\ ds\leq
C_{F}(\Omega)\int_{0}^{\zeta_{1}}\int_{\Omega}|\nabla u_{\epsilon}(x,t)|^{2}\
dxdt$

By (10.9)

(10.11)
$\displaystyle\int_{0}^{\zeta_{1}}\int_{\Omega}\left(\frac{\partial\tilde{H}(u_{\epsilon}(x,t))}{\partial
t}\right)^{2}dxdt\leq\int_{\Omega}|\nabla g(x)|^{2}dx$

and

(10.12) $\displaystyle\int_{0}^{\zeta_{1}}\int_{\Omega}|\nabla
u_{\epsilon}(x,t)|^{2}dxdt\leq\zeta_{1}\int_{\Omega}|\nabla g(x)|^{2}dx\ ;\
0<\zeta_{1}\leq T$

By adding (10.11) and (10.12) we get

(10.13)
$\int_{0}^{\zeta_{1}}\int_{\Omega}\left(\frac{\partial\tilde{H}(u_{\epsilon}(x,t))}{\partial
t}\right)^{2}dxdt+\int_{0}^{\zeta_{1}}\int_{\Omega}|\nabla
u_{\epsilon}(x,t)|^{2}\ dxdt\leq(1+{\zeta_{1}})\int_{\Omega}|\nabla g(x)|^{2}\
dx$

Let $0<\epsilon_{0}<1$ be fixed. Using (10.10) in above

(10.14)
$\int_{0}^{\zeta_{1}}\int_{\Omega}\left(\frac{\partial\tilde{H}(u_{\epsilon}(x,t))}{\partial
t}\right)^{2}dxdt+\epsilon_{0}\int_{0}^{\zeta_{1}}\int_{\Omega}|\nabla
u_{\epsilon}(x,t)|^{2}\
dxdt+\frac{(1-\epsilon_{0})}{{C_{F}}(\Omega)}\int_{0}^{\zeta_{1}}\int_{\Omega}|u_{\epsilon}(x,t)|^{2}\
dxdt\\\ \leq(1+{\zeta_{1}})\int_{\Omega}|\nabla g(x)|^{2}\
dx+(1-\epsilon_{0}){\zeta_{1}}\epsilon^{2}\int_{\partial\Omega}|\psi(x)|^{2}\
ds\ ;\ \zeta_{1}\in(0,T]$

Setting $\zeta_{1}=T$,

(10.15)
$\int_{0}^{T}\int_{\Omega}\left(\frac{\partial\tilde{H}(u_{\epsilon}(x,t))}{\partial
t}\right)^{2}dxdt+\int_{0}^{T}\int_{\Omega}|\nabla u_{\epsilon}(x,t)|^{2}\
dxdt+\int_{0}^{T}\int_{\Omega}|u_{\epsilon}(x,t)|^{2}\ dxdt\\\ \leq
C(\Omega,T)\left(\int_{\Omega}|\nabla g(x)|^{2}\
dx+\epsilon^{2}\int_{\partial\Omega}|\psi(x)|^{2}\ ds\right)$

where $\displaystyle
C(\Omega,T)\triangleq{[\max(1+T,(1-\epsilon_{0})T)}]/{[\min(\epsilon_{0},{(1-\epsilon_{0})}/{C_{F}(\Omega)})]}\
;\ 0<\epsilon_{0}<1$. As $\epsilon\rightarrow 0$ it follows the result

(10.16)
$\int_{0}^{T}\int_{\Omega}\left(\frac{\partial\tilde{H}(u_{\epsilon}(x,t))}{\partial
t}\right)^{2}dxdt+\int_{0}^{T}\int_{\Omega}|\nabla u_{\epsilon}(x,t)|^{2}\
dxdt\\\ +\int_{0}^{T}\int_{\Omega}|u_{\epsilon}(x,t)|^{2}\ dxdt\leq
C(\Omega,T)\int_{\Omega}|\nabla g(x)|^{2}\ dx\qed$

######  Remark 9.

Note that $\displaystyle{\partial\tilde{H}(u)}/{\partial t}$ is uniformly
bounded in $L^{2}(\Omega\times(0,T))$ and $u$ is uniformly bounded in
$L({I,W^{1,2}(\Omega)})$

######  Theorem 6.

Let $u_{\epsilon}$ be a family of strong solutions to $IBVP_{\epsilon}$ (4.8)
such that ${\partial H(u_{\epsilon})}/{\partial t},\Delta u_{\epsilon}\in
L^{2}_{loc}(\Omega\times(0,T))$, satisfying the estimate $\displaystyle
0<\epsilon\leq u_{\epsilon}\leq K<\infty$. Then for every $\theta\in
C_{c}^{1}(\Omega)$, $u_{\epsilon}$ satisfy the following uniform estimate

(10.17) $\int_{\Omega\times(0,T)}|\nabla(\theta G(u_{\epsilon}))|^{2}\
dxdt\leq c\int_{\Omega}\theta^{2}|G(u_{\epsilon}(x,0))|^{2}\
dx+c\int_{\Omega\times(0,T)}|\nabla\theta|^{2}(G(u_{\epsilon}))^{2}\ dxdt.$

Thus $G(u_{\epsilon})$ is uniformly bounded in
$L^{2}(I,W^{1,2}_{loc}(\Omega))$ for each $\epsilon$.

###### Proof.

Let $\Omega_{T}\triangleq\Omega\times(0,T)$. Multiply both sides of the
inequality (4.8) by $\theta^{2}$ and integrate over $\Omega_{T}$ we get,

$\displaystyle\int_{\Omega}[H(u_{\epsilon}(x,T))-H(u_{\epsilon}(x,0))]\theta^{2}\
dx$ $\displaystyle=-\int_{\Omega_{T}}(\nabla
u_{\epsilon})^{2}F^{{}^{\prime}}(u_{\epsilon})\theta^{2}\
dxdt-\int_{\Omega_{T}}F(u_{\epsilon})\nabla
u_{\epsilon}\cdot\nabla(\theta^{2})\ dxdt$

Rearrange above and using (i) in Proposition (1),

(10.18) $\displaystyle\int_{\Omega}H(u_{\epsilon 0})\theta^{2}\
dx+\int_{\Omega_{T}}|F(u_{\epsilon})\nabla
u_{\epsilon}\cdot\nabla(\theta^{2})|\ dxdt$
$\displaystyle\geq\int_{\Omega_{T}}|\nabla
u_{\epsilon}G^{{}^{\prime}}(u_{\epsilon})|^{2}\theta^{2}\ dxdt$

Let $0<\epsilon<\frac{1}{2}$ be fixed. By Cauchy Inequality

(10.19) $\displaystyle|\nabla G(u_{\epsilon})|^{2}\theta^{2}\geq$
$\displaystyle(1-2\epsilon)|\nabla(\theta
G(u_{\epsilon}))|^{2}-\left(\frac{1}{2\epsilon}-1\right)(G(u_{\epsilon})\nabla\theta)^{2}$

using (10.19) in (10.18) we get

(10.20) $\int_{\Omega}H(u_{\epsilon 0})\theta^{2}\
dx+\left(\frac{1}{2\epsilon}-1\right)\int_{\Omega_{T}}|G(u_{\epsilon})\nabla\theta)|^{2}\
dxdt\\\
+2||\nabla\theta||_{L^{\infty}}\int_{\Omega_{T}}|F(u_{\epsilon})||\nabla
u_{\epsilon}|\ dxdt\geq(1-2\epsilon)\int_{\Omega_{T}}|\nabla(\theta
G(u_{\epsilon}))|^{2}\ dxdt\qed$

######  Remark 10.

Note that $F(u_{\epsilon})\in L^{2}_{\Omega_{T}}$. By Remark 9,
$\nabla(u_{\epsilon})\in L^{2}_{\Omega_{T}}$. Then (10.20) yields

(10.21) $||H(u_{\epsilon
0})||_{L^{1}_{\Omega}}+||G(u_{\epsilon})\nabla\theta||_{L^{2}_{\Omega_{T}}}^{2}+||F(u_{\epsilon})||_{L_{\Omega_{T}}^{2}}\cdot||\nabla
u_{\epsilon}||_{L_{\Omega_{T}}^{2}}\geq K_{3}||\nabla(\theta
G(u_{\epsilon}))||^{2}_{L^{2}_{\Omega_{T}}}$

where $\displaystyle
K_{3}\triangleq{(1-2\epsilon)}/[{\max\\{({1}/{2\epsilon})-1\ ,\
{2b^{n+1}}/{R}\\}}]$ ; $0<\epsilon<\frac{1}{2}$. Here
$\mathbf{1}_{\operatorname{supp}\theta}H(u_{\epsilon})\in L^{1}(\Omega)$. Thus
$G(u_{\epsilon})$ are uniformly bounded in $L^{2}(I,W^{1,2}_{loc}(\Omega))$.

######  Proposition 4.

Assume all conditions in Proposition (1) and Proposition (2).
If

$\displaystyle\liminf_{s\rightarrow 0}{P(s)\left(\int_{s}^{M}\frac{1}{\sigma
P(\sigma)}\ d\sigma\right)^{\frac{1}{\Lambda}}}>0$

then

$\sup\limits_{\epsilon>0}\int_{I}\int_{\Omega}\theta|\nabla
H(u_{\epsilon})|^{2}\leq C_{\theta}<\infty$

for every $\theta\in C^{1}_{c}(\Omega)$.

###### Proof.

Note that $\displaystyle|\nabla
H(u_{\epsilon})|\leq\left|\frac{H^{\prime}(u_{\epsilon})}{G^{\prime}(u_{\epsilon})}\right||\nabla
G(u_{\epsilon})|$. Hence, by Theorem 6, it suffices to show that
$\displaystyle\sup_{0<s<K}\left|\frac{H^{\prime}(s)}{G^{\prime}(s)}\right|<\infty.$
For this end it suffices to verify $\displaystyle\limsup_{s\rightarrow
0}\left|\frac{H^{\prime}(s)}{G^{\prime}(s)}\right|<\infty.$

Recall (i) in Proposition 1. By (LABEL:F-I) we have

(10.22) $\displaystyle F(s)=$
$\displaystyle\Lambda^{-\frac{1}{\Lambda}-1}s^{-1}(I(s))^{-\frac{1}{\Lambda}-1}$
(10.23) $\displaystyle F^{{}^{\prime}}(s)=$ $\displaystyle
Bs^{-2}{(I(s))^{-\frac{1}{\Lambda}-2}}({P(s)})^{-1}\left(\frac{1+\Lambda}{\Lambda}-P(s)I(s)\right)\
;\ B\triangleq\Lambda^{-\frac{1}{\Lambda}-1}$ (10.24)
$\displaystyle{G^{{}^{\prime}}(s)}=$
$\displaystyle\sqrt{B}s^{-1}{(I(s))^{-\frac{1}{2\Lambda}-1}}({P(s)})^{-\frac{1}{2}}\sqrt{\frac{1+\Lambda}{\Lambda}-P(s)I(s)}$

Next we write $H(s)$ in term of $I(s)$. By (5.6) in (LABEL:H-Result)

(10.25) $\displaystyle H(s)=$ $\displaystyle(\Lambda
I(s))^{-\frac{1}{\Lambda}}$ (10.26) $\displaystyle H^{{}^{\prime}}(s)=$
$\displaystyle B{(I(s))^{-\frac{1}{\Lambda}-1}}s^{-1}(P(s))^{-1}$

Using (10.26) and (10.24)

(10.27)
$\displaystyle\left|\frac{H^{\prime}(s)}{G^{\prime}(s)}\right|=\frac{1}{\sqrt{B}}\frac{1}{\sqrt{(I(s))^{\frac{1}{\Lambda}}P(s)}}\frac{1}{\sqrt{\frac{1+\Lambda}{\Lambda}-I(s)P(s)}}$

By Proposition 2 we have $\displaystyle\frac{1+\Lambda}{\Lambda}>I(s)P(s)>0\
;\ s\in[0,M).$

Thus, by (10.27) $\displaystyle\lim_{s\rightarrow
0}\left|\frac{H^{\prime}(s)}{G^{\prime}(s)}\right|<\infty$ whenever
$\lim_{s\rightarrow 0}\inf{P(s)(I(s))^{\frac{1}{\Lambda}}}>0$. ∎

######  Theorem 7.

Let $u_{\epsilon}$ be a family of strong solutions to (4.8) that satisfies all
conditions in Theorem (6). Let
$\Omega^{{}^{\prime}}_{I}\triangleq\Omega^{{}^{\prime}}\times I$ ;
$\Omega^{\prime}\Subset\Omega$. Then $u_{\epsilon}$ holds the following
uniform estimates

(10.28) $\displaystyle\left|\int_{I}\int_{\Omega}\Psi\frac{\partial
H(u_{\epsilon})}{\partial t}\ dxdt\right|\leq
C(\Omega^{\prime},K)\left(||\Psi||_{L^{\infty}(\Omega\times
I)}+||\nabla\Psi||_{L^{2}(\Omega\times I)}\right),$

and

(10.29) $\displaystyle\int_{I}\int_{\Omega^{\prime}}|\nabla
H(u_{\epsilon})|^{2}\ dxdt\leq C(\Omega^{\prime},K).$

for each $\Psi\in C^{1}_{c}(\Omega^{{}^{\prime}}_{I})$ and for each
$\Omega^{\prime}\Subset\Omega$.

###### Proof.

Multiply both sides of the inequality in (4.8) by $\Psi$ and integrate over
$\Omega^{{}^{\prime}}_{I}$ we get,

$\displaystyle\left|\int_{\Omega^{{}^{\prime}}_{I}}\Psi\frac{\partial
H(u_{\epsilon})}{\partial t}\ dxdt\right|$
$\displaystyle\leq\int_{\Omega^{{}^{\prime}}_{I}}\left|\nabla
u_{\epsilon}\cdot\nabla(F(u_{\epsilon})\Psi)\right|\ dxdt$
$\displaystyle\leq\int_{\Omega^{{}^{\prime}}_{I}}\left|F(u_{\epsilon})\nabla
u_{\epsilon}\cdot\nabla\Psi\right|+\left|\Psi
F^{{}^{\prime}}(u_{\epsilon})(\nabla u_{\epsilon})^{2}\right|\ dxdt$
$\displaystyle\leq
C_{2}\int_{\Omega^{{}^{\prime}}_{I}}\left|G(u_{\epsilon})G^{{}^{\prime}}(u_{\epsilon})\nabla
u_{\epsilon}\cdot\nabla\Psi\right|\
dxdt+\int_{\Omega^{{}^{\prime}}_{I}}\left|\Psi(G^{{}^{\prime}}(u_{\epsilon}))^{2}(\nabla
u_{\epsilon})^{2}\right|\ dxdt$

Compute the right hand side of above

$\displaystyle\leq
C_{2}||G(u_{\epsilon})||_{L^{\infty}(\Omega^{{}^{\prime}}_{I})}\int_{\Omega^{{}^{\prime}}_{I}}|\nabla
G(u_{\epsilon})||\nabla\Psi|\
dxdt+||\Psi||_{L^{\infty}(\Omega^{{}^{\prime}}_{I})}\int_{\Omega^{{}^{\prime}}_{I}}|\nabla
G(u_{\epsilon})|^{2}\ dxdt$ $\displaystyle\leq
C_{2}||G(u_{\epsilon})||_{L^{\infty}(\Omega^{{}^{\prime}}_{I})}\cdot||\mathbf{1}_{\operatorname{supp}\Psi}\nabla
G(u_{\epsilon})||_{L^{2}(\Omega^{{}^{\prime}}_{I})}\cdot||\nabla\Psi||_{L^{2}(\Omega^{{}^{\prime}}_{I})}+||\Psi||_{L^{\infty}(\Omega^{{}^{\prime}}_{I})}\cdot||\mathbf{1}_{\operatorname{supp}\Psi}\nabla
G(u_{\epsilon})||_{L^{2}(\Omega^{{}^{\prime}}_{I})}^{2}$ $\displaystyle\leq
C_{2}||G(u_{\epsilon})||_{L^{\infty}(\Omega^{{}^{\prime}}_{I})}\cdot||\mathbf{1}_{\operatorname{supp}\Psi}\nabla
G(u_{\epsilon})||_{L^{2}(\Omega^{{}^{\prime}}_{I})}\cdot(||\nabla\Psi||_{L^{2}(\Omega^{{}^{\prime}}_{I})}+||\Psi||_{L^{\infty}(\Omega^{{}^{\prime}}_{I})})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad+(||\nabla\Psi||_{L^{2}(\Omega^{{}^{\prime}}_{I})}+||\Psi||_{L^{\infty}(\Omega^{{}^{\prime}}_{I})})\cdot||\mathbf{1}_{\operatorname{supp}\Psi}\nabla
G(u_{\epsilon})||_{L^{2}(\Omega^{{}^{\prime}}_{I})}^{2}$ $\displaystyle\leq
C_{2}\left(||\nabla\Psi||_{L^{2}(\Omega^{{}^{\prime}}_{I})}+||\Psi||_{L^{\infty}(\Omega^{{}^{\prime}}_{I})}\right)\cdot\left(||G(u_{\epsilon})||_{L^{\infty}(\Omega^{{}^{\prime}}_{I})}\cdot||\nabla
G(u_{\epsilon})||_{L^{2}(\Omega^{{}^{\prime}}_{I})}+||\nabla
G(u_{\epsilon})||_{L^{2}(\Omega^{{}^{\prime}}_{I})}^{2}\right)$

Recall that $0<\epsilon\leq u_{\epsilon}\leq K<\infty$ and $G(u_{\epsilon})$
are uniformly bounded.

(10.30) $\displaystyle\therefore
C_{2}||G(u_{\epsilon})||_{L^{\infty}(\Omega^{{}^{\prime}}_{I})}\cdot||\nabla
G(u_{\epsilon})||_{L^{2}(\Omega^{{}^{\prime}}_{I})}+||\nabla
G(u_{\epsilon})||_{L^{2}(\Omega^{{}^{\prime}}_{I})}^{2}\leq
C(\Omega^{{}^{\prime}},K)$

which follows the result $\eqref{L1-bound}$ ∎

From above theorem follows important statement

######  Corollary 1.

If we obtain an estimate analogous to (10.17), replacing $\nabla
G(u_{\epsilon})$ with $\nabla H(u_{\epsilon})$, then we can apply Theorem (4)
to conclude compactness of $\\{H(u_{\epsilon})\\}$ in
$L^{2}(I,L^{q}_{loc}(\Omega))$ with $q<\frac{2N}{N-2}$. Since
$H:\mathbb{R}\to\mathbb{R}$ is a homeomorphism, and $u_{\epsilon}$ is
uniformly bounded, it follows that $\\{u_{\epsilon}\\}$ is compact in
$L^{q}_{loc}(\Omega\times I)$, $q\geq 1$.

  * •

Estimate (10.17) implies $G(u_{\epsilon})$ are uniformly bounded in
$L^{2}(I,W^{1,2}_{loc}(\Omega))$.

  * •

Estimate (LABEL:H-t-1) together with (10.17) implies that ${\partial
H(u_{\epsilon})}/{\partial t}$ is uniformly bounded in
$L^{1}(I,L_{loc}^{1}(\Omega))+L^{2}(I,W_{loc}^{-1,2}(\Omega))$ by Theorem
LABEL:interpolation*(3).

  * •

To obtain for sufficient conditions under which last Corollary 1 holds w.r.t.
input functions, it is suffice to have Proposition 4.

##  11\. Appendix

In the prove of the main theorem 1 on localisation we used basic iterative
inequality and conclusion from it, which explicitly formulated by Ladyženskaja
- Solonnikov- Ural’ceva in [8].

######  Ladyzhenskaya-Uraltceva iterative Lemma 1.

Let sequence $y_{n}\ ;\ n=0,1,2,...$ be non-negative, satisfying the recursion
inequality $\displaystyle y_{n+1}\leq c\text{ }b^{n}\text{
}y_{n}^{1+\epsilon}$ with some constants $c,\epsilon>0\text{ and }b\geq 1$.
Then

$\displaystyle y_{n}\leq c^{\frac{(1+\epsilon)^{n}-1}{\epsilon}}\text{
}b^{\frac{(1+\epsilon)^{n}-1}{\epsilon^{2}}-\frac{n}{\epsilon}}\text{
}y_{0}^{(1+\epsilon)^{n}}.$

In particular $\displaystyle\text{if}\quad
y_{0}\leq\theta_{L}=c^{\frac{-1}{\epsilon}}\text{
}b^{\frac{-1}{\epsilon^{2}}}\text{ and }{b>1}\text{ then
}y_{n}\leq\theta\text{ }b^{\frac{-n}{\epsilon}}$ and consequently

$\displaystyle y_{n}\rightarrow 0\text{ when }n\rightarrow\infty.$

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