# Qualitative analysis of solutions
for a degenerate PDE model
of epidemic dynamics

Marina Chugunova1, Roman Taranets2
and Nataliya Vasylyeva3,4,5
${}^{1}\,$Claremont Graduate University,
150 E. 10th Str., Claremont, California 91711, USA
<EMAIL_ADDRESS>
${}^{2}\,$Institute of Applied Mathematics and Mechanics of NASU,
G.Batyuka Str. 19, 84100, Sloviansk, Ukraine
<EMAIL_ADDRESS>
${}^{3}\,$Institute of Applied Mathematics and Mechanics of NASU,
G.Batyuka Str. 19, 84100 Sloviansk, Ukraine
${}^{4}\,$Institute of Hydromechanics of NASU,
Zhelyabova Str. 8/4, 03057 Kyiv, Ukraine
${}^{5}\,$Dipartamento di Matematica, Politecnico di Milano
Via Bonardi 9, 20136 Milano, Italy
<EMAIL_ADDRESS>
###### Abstract

Compartmental models are widely used in mathematical epidemiology to describe
dynamics of infection disease. A new SIS-PDE model, recently derived by Chalub
and Souza, is based on a diffusion-drift approximation of probability density
in a well-known discrete – time Markov chain SIS-DTMC model. This new SIS-PDE
model is conservative due to degeneracy of the diffusion term at the origin.
We analyze a class of degenerate PDE models and obtain sufficient conditions
for existence of classical solutions with certain regularity properties. Also,
we show that under some additional assumptions about coefficients and initial
data classical solutions vanish at the origin at any finite time. Vanishing at
the origin of solutions is consistent with the conservation property of the
model. The main results of this article are: sufficient conditions for
existence of weak solutions, analysis of their asymptotic behavior at the
origin, and the proof of existence of weak solutions convergent to Dirac delta
function. Moreover, we study long-time behavior of solutions and confirm our
analysis by numerical computations.

##  1  Introduction

Regardless of the modern improvement in development of highly efficient
antibiotics and vaccines, infectious diseases still contribute significantly
to deaths worldwide. While the earlier recognized diseases like cholera or the
plague still sometimes create problems in underdeveloped countries erupting
occasionally in epidemics, in the developed countries new diseases are
emerging like AIDS (1981) or hepatitis C and E (1989-1990). New threads
constantly appear like the recent bird flu (SARS) epidemic in Asia, the very
dangerous Ebola virus in Africa and worldwide spread of COVID-19. Overall,
infectious diseases continue to be one of the most important health problems.
Modeling of epidemiological phenomena has a very long history with the first
model for smallpox formulated by Daniel Bernoulli in 1760. From the early
twentieth century, in response to epidemics of various infectious diseases, a
large number of models has been constructed and analyzed, see for example [1,
2, 9, 22, 29, 30] (and references therein).

Compartmental models are a very general modeling technique. They are often
applied to the mathematical modeling of infectious diseases. The population is
assigned to compartments like: susceptible, infectious, and recovered in
widely used SIR model or to: susceptible, infective, and susceptible like in
SIS epidemiological scheme. SIS scheme provides the simplest description of
the dynamics of a disease that is contact-transmitted, and that does not lead
to immunity like for example COVID-19. Discrete-time Markov chain type SIR and
SIS models are considered to be a classical approaches in modern mathematical
modeling in epidemiology. The most recent development in mathematical
epidemiology is based on introduction of a continuous modeling based on
partial differential equations SIR-PDE model like in [12, 14, 15, 16, 17].

In our paper, for $T>0$ and $\Omega=(0,1),$ $\Omega_{T}=\Omega\times(0,T)$, we
consider the following equation (see [14]) in the unknown function
$p:=p(x,t)$: $\bar{\Omega}_{T}\to\mathbb{R}$:

$\frac{\partial p}{\partial t}=\frac{1}{2N}\frac{\partial^{2}}{\partial
x^{2}}(f(x)p)-\frac{\partial}{\partial x}(g(x)p)\quad\text{ in
}\quad\Omega_{T},$ (1.1)

coupled with the boundary condition

$\frac{1}{2N}\bigg{[}(1-R_{0})p(1,t)+\frac{\partial}{\partial
x}p(1,t)\bigg{]}+p(1,t)=0,\quad t\in[0,T],$ (1.2)

and initial data

$p(x,0)=p_{0}(x)\quad\text{in}\quad\bar{\Omega}.$ (1.3)

Addressing to the SIR-PDE model, we conclude that $x\in\bar{\Omega}$ is the
fraction of infected, $N$ is a population of individuals, $p$ is the
probability to find a fraction $x$ at time $t$ in a population of size $N$,
$R_{0}\geqslant 0$ is the basic reproductive factor,

$f(x):=x(R_{0}(1-x)+1),\ \ g(x):=x(R_{0}(1-x)-1).$

This model is degenerate at $x=0$ that is a boundary point of the domain.

It is worth noting that processes defined by similar models were studied by
Feller in the early 1950s and used to great effect by Kimura, et al. in the
1960s and 70s to give quantitative answers to a wide range of questions in
population genetics. Although, rigorous analysis of analytic properties of
these equations is only now getting into the focus of applied mathematicians.
The study of initial or/and initial-boundary value problems for degenerate
equations including Kimura-type operators has a long history. We do not
provide here a complete survey on the published results concerned to these
degenerate equations, but rather present some of them. Indeed, the
investigation of elliptic and parabolic problems to degenerate equations
containing the operators like

$\mathcal{L}:=a(x)\sum_{ij=1}^{n}a_{ij}(x)\frac{\partial^{2}}{\partial
x_{i}\partial x_{j}}+\sum_{i=1}^{n}b_{i}(x)\frac{\partial}{\partial x_{i}}$

with $a(x)\approx|x|^{\alpha},$ $\alpha>0,$ and $a_{ij}$ satisfying
ellipticity conditions, are extensively studied by many authors with various
analytical approaches (see e.g. [7, 31, 32, 33, 34, 38, 39, 44, 45, 46, 47,
49, 50]) including stochastic calculus [4, 5, 13, 19, 23, 24, 27, 36, 43].

Under suitable assumptions on the asymptotic behavior of the operator’s
coefficients at the boundary of the domain, the uniqueness of bounded
solutions was shown in [31, 46] without prescribing any boundary conditions at
the origin. In [45], taking advantage of appropriate super- and subsolutions,
similar uniqueness results have been proved in the case of unbounded solution.
In [39, 47, 48], under some conditions on the coefficients, uniqueness results
for Cauchy problem to parabolic equation like $\frac{\partial u}{\partial
t}-\mathcal{L}u=f$ in the bounded domain in suitable weighted $L^{2}$ spaces
are established.

Epstein et al. [23, 24, 25, 26, 27] (see also references therein) study the
generalized Kimura operators (generalization $\mathcal{L}$ with $\alpha=1$) in
the manifold setting. In particular, they discussed maximum principle and the
Harnack inequality. The integral maximum principles in the whole
$\mathbb{R}^{n}$ for solutions of degenerate parabolic equations are also
obtained in [3]. As for probabilistic approaches, there are abundant theories
on existence and uniqueness of solutions to stochastic differential equations
with degenerate diffusion coefficients (see [19, 23, 24, 28, 36]), besides
well-posedness of the related problems in the case of $\alpha=1$ are discussed
in [4, 5, 13]. It is worth noting that, degenerate diffusion is examined in
the context of measure-valued process (see [11, 20, 41]) as well as via the
semigroup techniques [10, 35, 42]. As for well-posedness of parabolic
degenerate problems, we quote [6, 7, 21, 32, 34, 43, 44, 49, 50], where
existence of weak and classical solutions is established for different value
of $\alpha>0$. In particular, if $\alpha=1$ the solvability of Cauchy problem
in semi-space $\mathbb{R}^{n}_{+,T}$ in the weighted Hölder classes with
boundary and without boundary condition on the part of boundary are discussed
[7, 23, 42], while the case of $\alpha>1$ is analyzed in [18, 21, 50].
Previous researchers like [18, 23] restricted their attention to the solutions
with the best possible regularity properties, which leads to considerable
simplifications. For real applications it is important to consider solutions
with more complicated behavior that is the goal of our paper.

Outline of the paper is as follows: section 2 contains the main notation and
functional setting; in section 3, we show existence of stationary solutions,
confirm numerically their meta-stability and analyze convergence; in section
4, we analyze particular classical and weak solutions; in section 5, we study
a local asymptotic of solutions at the origin and discuss sufficient
conditions which provide the fulfillment of the conservation law.

##  2 Functional setting and notations

Throughout this work, the symbol $C$ will denote a generic positive constant,
depending only on the structural quantities of the problem. We will carry out
our analysis in the framework of the weighted Hölder and Sobolev spaces. To
this end, throughout the paper, let

$\alpha\in(0,1)\quad\text{and}\quad s\in\mathbb{R}$

be arbitrarily fixed.

For any non-negative integer $k$, and any Banach space
$(\mathbf{X},\|\cdot\|_{\mathbf{X}}),$ and any $p\geq 1$, we consider spaces

$\mathcal{C}([0,T],\mathbf{X}),\quad
L^{p}([0,T],\mathbf{X}),\quad\mathcal{C}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{T}).$

The last class so-called the parabolic Hölder space has been used by several
authors, and its definition and properties can be found, for instance, in [37,
(1.10)-(1.12)]. Along the paper, we will also encounter the usual spaces
$\mathcal{C}^{k+\alpha}(\bar{\Omega}),$ $W^{1,p}(\Omega)$, $L^{p}(\Omega)$ and
$L_{\omega}^{2}(\Omega)$. It is worth noting that the last class is a weighted
space $L^{2}$ with a weight $\omega$ and induced norm

$\|v\|_{L_{\omega}^{2}(\Omega)}=\bigg{(}\int\limits_{\Omega}\omega(x)v^{2}(x)\,dx\bigg{)}^{1/2}.$

Moreover, we will use the notations $H^{1}(\Omega)$ and $H_{0}^{1}(\Omega)$
for $W^{1,2}(\Omega)$ and $W_{0}^{1,2}(\Omega)$, respectively.

Let $d(x)$ be the distance from any point $x\in\bar{\Omega}\subset\mathbb{R}$
to the origin and set

$\underline{d}=\min\\{d(x),d(\bar{x})\\},\quad\bar{d}=\max\\{d(x),d(\bar{x})\\},\quad
x,\bar{x}\in\bar{\Omega},$ $d=\begin{cases}\underline{d}\quad\text{for}\quad
s\geq 0,\\\ \bar{d}\quad\text{for}\quad s<0.\end{cases}$

Denoting for any $\gamma\in(0,1)$, and $x\neq\bar{x},$ $t_{1}\neq t_{2}$

$\displaystyle\langle v\rangle_{x,s,\Omega}^{(\gamma)}$
$\displaystyle=\underset{\bar{\Omega}}{\sup}\,d^{-s}\underline{d}^{\gamma}\frac{|v(x)-v(\bar{x})|}{|x-\bar{x}|^{\gamma}},$
$\displaystyle\langle v\rangle_{x,s,\Omega_{T}}^{(\gamma)}$
$\displaystyle=\underset{\bar{\Omega}_{T}}{\sup}\,d^{-s}\underline{d}^{\gamma}\frac{|v(x,t)-v(\bar{x},t)|}{|x-\bar{x}|^{\gamma}},$
$\displaystyle\langle v\rangle_{t,s,\Omega_{T}}^{(\gamma)}$
$\displaystyle=\underset{\bar{\Omega}_{T}}{\sup}\,d^{-s}\frac{|v(x,t_{1})-v(x,t_{2})|}{|t_{1}-t_{2}|^{\gamma}},$

we give the following definition:

######  Definition 2.1.

Functions $v=v(x)$ and $w=w(x,t)$ belong to the spaces
$\mathcal{C}^{k+\alpha}_{\alpha}(\bar{\Omega})$ and
$\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{T})$, for
$k=0,1,2,$ if the norms here below are finite:

$\displaystyle\|v\|_{\mathcal{C}_{\alpha}^{k+\alpha}(\bar{\Omega})}$
$\displaystyle=\sum_{j=0}^{k}\Big{[}\Big{\|}d^{\lfloor
j/2\rfloor}(x)\frac{\partial^{j}v}{\partial
x^{j}}\Big{\|}_{\mathcal{C}(\bar{\Omega})}+\Big{\langle}d^{\lfloor
j/2\rfloor}(x)\frac{\partial^{j}v}{\partial
x^{j}}\Big{\rangle}_{x,\alpha/2,\Omega}^{(\alpha)}\Big{]},$
$\displaystyle\|w\|_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{T})}$
$\displaystyle=\|w\|_{\mathcal{C}([0,T],\mathcal{C}^{k+\alpha}_{\alpha}(\bar{\Omega}))}+\sum_{j=0}^{k}\Big{\langle}\frac{\partial^{j}w}{\partial
x^{j}}\Big{\rangle}_{t,0,\Omega_{T}}^{(\frac{k+\alpha-j}{2})},\quad k=0,1,$
$\displaystyle\|w\|_{\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})}$
$\displaystyle=\|w\|_{\mathcal{C}([0,T],\mathcal{C}^{2+\alpha}_{\alpha}(\bar{\Omega}))}+\|\partial
w/\partial
t\|_{\mathcal{C}^{\alpha,\frac{\alpha}{2}}_{\alpha}(\bar{\Omega}_{T})}$
$\displaystyle+\Big{\langle}\frac{\partial w}{\partial
x}\Big{\rangle}_{t,0,\Omega_{T}}^{(\frac{\alpha}{2})}+\sum_{j=1}^{2}\Big{\langle}d(x)\frac{\partial^{j}w}{\partial
x^{j}}\Big{\rangle}_{t,0,\Omega_{T}}^{(\frac{2+\alpha-j}{2})},$

where $\lfloor j\rfloor$ denotes the floor function of $j$ (i.e. the greatest
integer less than or equal to $j$).

It is apparent that in any domain
$\Omega_{\varepsilon}=(\varepsilon,l)\subset\Omega$, the spaces
$\mathcal{C}_{\alpha}^{k+\alpha}(\bar{\Omega}_{\varepsilon})$ and
$\mathcal{C}^{k+\alpha,\frac{k+\alpha}{2}}_{\alpha}(\bar{\Omega}_{\varepsilon,T})$
coincide with the usual Hölder
$\mathcal{C}^{k+\alpha}(\bar{\Omega}_{\varepsilon})$ and the parabolic Hölder
spaces
$\mathcal{C}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{\varepsilon,T})$.

Moreover, in our analysis, the following weighted Hölder spaces are needed.

######  Definition 2.2.

Functions $V:=V(x)$ and $W:=W(x,t)$ belong to the spaces
$E_{s}^{k+\alpha}(\bar{\Omega})$ and
$E_{s}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{T})$, for $k=0,1,2,$ if the
norms here below are finite:

$\displaystyle\|V\|_{E_{s}^{k+\alpha}(\bar{\Omega})}$
$\displaystyle=\sum_{j=0}^{k}\Big{[}\Big{\|}d^{-s+j}(x)\frac{\partial^{j}V}{\partial
x^{j}}\Big{\|}_{\mathcal{C}(\bar{\Omega})}+\Big{\langle}\frac{\partial^{j}V}{\partial
x^{j}}\Big{\rangle}_{x,s-j,\Omega}^{(\alpha)}\Big{]},$
$\displaystyle\|W\|_{E_{s}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{T})}$
$\displaystyle=\|W\|_{\mathcal{C}([0,T],E_{s}^{k+\alpha}(\bar{\Omega}))}+\sum_{j=0}^{k}\Big{\langle}\frac{\partial^{j}W}{\partial
x^{j}}\Big{\rangle}_{t,s-j,\Omega_{T}}^{(\frac{k+\alpha-j}{2})},$

for $k=0,1$, while for $k=2$

$\displaystyle\|W\|_{E_{s}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})}$
$\displaystyle=\|W\|_{\mathcal{C}([0,T],E_{s}^{2+\alpha}(\bar{\Omega}))}+\|\partial
W/\partial t\|_{E_{s}^{\alpha,\frac{\alpha}{2}}(\bar{\Omega}_{T})}$
$\displaystyle+\sum_{j=1}^{2}\Big{\langle}\frac{\partial^{j}W}{\partial
x^{j}}\Big{\rangle}_{t,s-j,\Omega_{T}}^{(\frac{2+\alpha-j}{2})}.$

######  Definition 2.3.

For $k=0,1,2,$ we define
$\mathcal{C}_{\alpha,0}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{T})$ to be
the space consisting of those functions
$w\in\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{T})$,
satisfying the zero initial conditions

$w|_{t=0}=0\quad\text{and}\quad\frac{\partial w}{\partial t}|_{t=0}=0,\quad
m=0,...,\lfloor k/2\rfloor.$

In a similar manner we introduce the spaces:
$\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\partial\Omega_{T})$ and
$\mathcal{C}_{\alpha,0}^{k+\alpha,\frac{k+\alpha}{2}}(\partial\Omega_{T})$ for
$k=0,1,2$.

##  3 Weak solutions: convergence to steady state and asymptotics as
$T\to+\infty$

In this section, as it is mentioned in Introduction, we discuss the long time
behavior of a weak solution to problem (1.1)-(1.3). To this end, we first
construct the explicit stationary solution
$P_{s}:=P_{s}(x):\bar{\Omega}\to\mathbb{R}$ related to (1.1)-(1.3), and when
we examine the given data which provides the convergence of the weak solution
as $T\to+\infty$. In particular, we consider the case of convergence $p(x,t)$
to $P_{s}(x)$.

###  3.1 Existence of a steady state

Figure 1: These two pictures illustrate the dominant behavior of convection
for short times. (Left) convection moves solution toward the steady state from
the right side to the left one for $R_{0}=2$ and $N=200$, (Right) convection
moves solution toward the steady state from the left side to the right one for
the same parameter values.

Figure 2: These two pictures illustrate short-time dynamics for $R_{0}=0$ and
$N=100$ (on the left) and long time with blow up at the origin (on the right).
The initial data are plotted with a dashed line.

Figure 3: These two pictures illustrate the dominant behavior of convection
for short times. (Left) convection moves solution toward the origin
$R_{0}=0.5$ and $N=100$, where solution blows up. (Right) convection again
moves solution toward the origin $R_{0}=1$ and $N=100$, where it blows up.

Figure 4: These two pictures illustrate the dominant behavior of convection
for short times. (Left) convection moves solution toward the origin $R_{0}=0$
and $N=100$, where solution blows up. (Right) convection again moves solution
toward the origin $R_{0}=0$ and $N=200$, where it blows up.

First, we will start with getting an analytical formula for a stationary
solution for (1.1):

$\frac{1}{2N}\frac{\partial}{\partial x}\bigl{(}\frac{\partial}{\partial
x}(f(x)P_{s})-2Ng(x)P_{s}\bigr{)}=0\quad\text{in}\quad\Omega,$ (3.1)

coupled with the boundary condition:

$\frac{\partial}{\partial x}P_{s}(1)=-(2N-R_{0}+1)P_{s}(1).$ (3.2)

Integrating (3.1) in $x$ and taking into account (3.2), we have

$\frac{\partial}{\partial x}(f(x)P_{s})=2Ng(x)P_{s}.$

It is apparent that this equation has a general solution

$f(x)P_{s}(x)=\mathcal{C}\,F(x),$ (3.3)

where we put

$F(x):=e^{2N\int\limits_{0}^{x}{\tfrac{g(s)}{f(s)}ds}},\ \
\mathcal{C}:=\mathop{\lim}\limits_{x\to 0}f(x)p(x).$

As a result, we obtain the explicit form of the classical stationary solution
to (1.1)-(1.3)

$P_{s}(x)=\tfrac{\mathcal{C}}{\omega(x)}\text{ where
}\omega(x):=\tfrac{f(x)}{F(x)}.$ (3.4)

Hence, the changing sign convection term for $R_{0}=2$ equals to zero at
$x=0.5$ a wave-like solution moves toward this point forming a meta-stable
steady state shape. The illustration that the solution short-time behavior is
driven by the convection, as can be seen in Figures 1–2. It takes a long time
for a meta-stable steady state to move mass toward the origin, this long-time
dynamics is due to a slow diffusion effect and eventually the solution
eventually blows-up at the origin. That, for two different sets of parameter
values, as can be seen in Figures 3–4. All numerical simulations show high
accuracy of mass conservation property even for long time computations that
suggests an existence a delta function type weak solution which acts as a
global attractor in this dynamical system.

###  3.2 Long-time behavior of a weak solution

Assuming that $\omega(x)$ is defined with (3.4) and

$N\geqslant 1,\quad R_{0}\geqslant 0\quad\text{and}\quad 0\leqslant
p_{0}(x)\in L_{\omega}^{2}(\Omega),$

(see Section 2 as to the definition of $L_{\omega}^{2}(\Omega)$), we define a
weak solution of (1.1)-(1.3) in the following sense.

######  Definition 3.1.

A non-negative function $p(x,t)\in L^{\infty}(0,T;L_{\omega}^{2}(\Omega))$ is
a weak solution of problem (1.1)–(1.3) if

$p_{t}\in L^{2}(0,T;H^{-1}(\Omega)),\ \ (\omega(x)p)_{x}\in L^{2}(Q_{T}),$

and satisfies (1.1) in the sense

$\int\limits_{0}^{T}{\Big{\langle}\frac{\partial p}{\partial
t},\psi\Big{\rangle}_{H^{1},H^{-1}}\,dt}+\\\
\iint\limits_{Q_{T}}{\bigl{(}\frac{1}{2N}\frac{\partial fp}{\partial
x}-gp\bigr{)}\frac{\partial\psi}{\partial x}\,dxdt}=0$

for all $\psi\in L^{2}(0,T;H_{0}^{1}(\Omega))$. Here $\langle
u,v\rangle_{H^{1},H^{-1}}$ is a dual pair of elements $u\in H^{1}$ and $v\in
H^{-1}$.

Now we are ready to state our first main result related to asymptotic behavior
of a weak solution to (1.1)-(1.3).

######  Theorem 1.

(i) Let $0\leqslant p_{0}(x)\in L_{\omega}^{2}(\Omega)$ and
$\mathop{\lim}\limits_{x\to 0}\omega(x)p(x,t)=0$ then a weak solution $p(x,t)$
satisfies the relation

$\omega^{\frac{1}{2}}(x)p(x,t)\to 0\text{ strongly in }L^{2}(\Omega)\text{ as
}t\to+\infty.$

Moreover, if $(\omega(x)p_{0}(x))_{x}\in L^{2}(\Omega)$ then there is
convergence

$\omega(x)p(x,t)\to 0\text{ strongly in }W^{1,2}(\Omega)\text{ as
}t\to+\infty.$ (3.5)

(ii) Let $\omega^{\frac{1}{2}}(x)p_{0}(x)\in L_{\omega}^{2}(\Omega)$ and
$\mathop{\lim}\limits_{x\to 0}\omega(x)p(x,t)=\mathcal{C}>0$ then for a weak
solution $p(x,t)$ we have

$\omega(x)p(x,t)\to\mathcal{C}\text{ strongly in }L^{2}(\Omega)\text{ as
}t\to+\infty.$ (3.6)

Moreover, if $\omega(x)(\omega(x)p_{0}(x))_{x}\in L^{2}(\Omega)$ then

$\omega(x)\frac{\partial}{\partial x}(\omega(x)p(x,t))\to 0\text{ strongly in
}L^{2}(\Omega)\text{ as }t\to+\infty.$

Figure 5: These two pictures illustrate convergence of weighted $L^{2}$ norm
of $p(x,t)$ to a constant for $R_{0}=0$ and $N=100$ (on the left) and
$R_{0}=0$ and $N=200$ (on the right).

Figure 6: These two pictures illustrate convergence of weighted $L^{2}$ norm
of $p(x,t)$ for $R_{0}=0.5$ and $N=100$ (on the left) and $R_{0}=2$ and
$N=200$ (on the right).

Numerical simulations in Figures 5–6 illustrate the convergence result (3.6).

######  Remark 3.1.

Note that (3.5) implies

$\int\limits_{\Omega}{p(x,t)\,dx}\to 0\text{ as }t\to+\infty,$

whence we deduce that $p_{\infty}:=p(x,+\infty)=0$ for a. e.
$x\in\bar{\Omega}$. On the other hand, (3.6) does the convergence to the
steady state, i. e. $p_{\infty}=P_{s}$.

###### Proof of Theorem 1.

Introducing a new function $z:=\omega(x)p(x,t),$ we rewrite problem
(1.1)–(1.3) in more suitable form:

$\begin{cases}\omega^{-1}(x)\frac{\partial z}{\partial
t}=\frac{1}{2N}\frac{\partial}{\partial x}\big{(}F(x)\frac{\partial
z}{\partial x}\big{)},\qquad\quad(x,t)\in\Omega_{T},\\\ \frac{\partial
z}{\partial x}|_{x=1}=0\quad\text{ and }\quad z|_{x=0}=0,\qquad t\in[0,T],\\\
z(x,0)=\omega(x)p_{0}(x):=z_{0}(x),\qquad\quad x\in\bar{\Omega}.\end{cases}$
(3.7)

Note that if $z|_{x=0}=\mathcal{C}>0$ then for the new function
$\tilde{z}=z-\mathcal{C}$ we again get problem similar to (3.7).

We begin to verify the point (i) of the theorem. To this end, multiplying the
equation in (3.7) by $z(x,t)$ and integrating over $\Omega$, we obtain

$\frac{1}{2}\frac{d}{dt}\int\limits_{\Omega}{\omega^{-1}(x)z^{2}\,dx}+\tfrac{1}{2N}\int\limits_{\Omega}{F(x)\bigg{(}\frac{\partial
z}{\partial z}\bigg{)}^{2}\,dx}=\frac{1}{2N}F(x)z\frac{\partial z}{\partial
z}\Bigl{|}_{0}^{1}=0.$ (3.8)

Next, we take advantage of the Hardy inequality:

$\int\limits_{\Omega}{\omega^{-1}(x)v^{2}\,dx}\leqslant
C_{H}(R_{0})\int\limits_{\Omega}{F(x)\bigg{(}\frac{\partial v}{\partial
x}\bigg{)}^{2}\,dx}$

with $v(0)=0$. Here the constant $C_{H}(R_{0})$ satisfies the inequalities

$A(R_{0})\leqslant C_{H}(R_{0})\leqslant 4A(R_{0})$

with

$A(R_{0})=\mathop{\sup}\limits_{r\in(0,1)}\Bigl{(}\int\limits_{0}^{r}{\tfrac{dx}{F(x)}}\Bigr{)}\Bigl{(}\int\limits_{r}^{1}{\tfrac{dx}{\omega(x)}}\Bigr{)}.$

Thus, collecting (3.8) with the Hardy inequality, we end up with relations

$\int\limits_{\Omega}{\omega^{-1}(x)z^{2}(x,t)\,dx}\leqslant
e^{-\frac{t}{NC_{H}(R_{0})}}\int\limits_{\Omega}{\omega^{-1}(x)z_{0}^{2}(x)\,dx}\to
0\text{ as }t\to+\infty.$ (3.9)

Multiplying the equation in (3.7) by $-\omega(x)\frac{\partial}{\partial
x}\big{(}F(x)\frac{\partial z}{\partial x}\big{)}$ and integrating over
$\Omega$, we obtain the equality

$\displaystyle\frac{1}{2}\frac{d}{dt}\int\limits_{\Omega}{F(x)\bigg{(}\frac{\partial
z}{\partial
x}\bigg{)}^{2}\,dx}+\frac{1}{2N}\int\limits_{\Omega}{\omega(x)\bigg{(}\frac{\partial}{\partial
x}\big{(}F(x)\frac{\partial z}{\partial x}\big{)}\bigg{)}^{2}\,dx}$
$\displaystyle=F(x)\frac{\partial z}{\partial t}\frac{\partial z}{\partial
x}\Bigl{|}_{0}^{1},$

which in turn provides

$\frac{1}{2}\frac{d}{dt}\int\limits_{\Omega}{F(x)\bigg{(}\frac{\partial
z}{\partial
x}\bigg{)}^{2}\,dx}+\frac{1}{2N}\int\limits_{\Omega}{\omega(x)\bigg{(}\frac{\partial}{\partial
x}\big{(}F(x)\frac{\partial z}{\partial x}\big{)}\bigg{)}^{2}\,dx}=0.$ (3.10)

In order to handle the second term in the left-hand side of this equality, we
will exploit the inequality

$\int\limits_{\Omega}{\tfrac{v^{2}}{F(x)}\,dx}\leqslant
C_{P}(R_{0})\int\limits_{\Omega}{\omega(x)\Big{(}\frac{\partial v}{\partial
x}\Big{)}^{2}\,dx}\text{ with }v(1)=0,$

where

$C_{P}(R_{0})=\int\limits_{\Omega}{\tfrac{1}{F(x)}\Bigl{(}\int\limits_{x}^{1}{\tfrac{dy}{\omega(y)}}\Bigr{)}\,dx}.$

Hence, we end up with relations

$\int\limits_{\Omega}{F(x)\bigg{(}\frac{\partial z}{\partial
x}\bigg{)}^{2}\,dx}\leqslant
e^{-\frac{t}{NC_{P}(R_{0})}}\int\limits_{\Omega}{F(x)\bigg{(}\frac{\partial
z_{0}}{\partial x}\bigg{)}^{2}\,dx}\to 0\text{ as }t\to+\infty.$ (3.11)

As a result, we obtain the convergence

$z(x,t)\to 0\quad\text{ strongly in }\quad W^{1,2}(\Omega)\text{ as
}t\to+\infty.$

if only the inequality holds

$\int\limits_{\Omega}{\bigg{(}\omega^{-1}(x)z_{0}^{2}(x)+F(x)\bigg{(}\frac{\partial
z_{0}}{\partial x}\bigg{)}^{2}\bigg{)}\,dx}<+\infty.$

The simple consequence of this fact and convergence (3.11) is the bound

$z(x,t)\leqslant
x^{\frac{1}{2}}e^{-\frac{t}{2NC_{P}(R_{0})}}\Bigl{(}\int\limits_{\Omega}{F(x)\bigg{(}\frac{\partial
z_{0}}{\partial x}\bigg{)}^{2}\,dx}\Bigr{)}^{\frac{1}{2}},$

which in turn provides the desired relation

$p(x,t)\leqslant\tfrac{x^{\frac{1}{2}}F(x)}{f(x)}e^{-\frac{t}{2NC_{P}(R_{0})}}\Bigl{(}\int\limits_{\Omega}{F(x)\bigg{(}\frac{\partial
z_{0}}{\partial x}\bigg{)}^{2}\,dx}\Bigr{)}^{\frac{1}{2}}.$

At this point we prove statement (ii) of the theorem. Multiplying (3.7) by
$\omega(x)\psi(x)z(x,t)$ (as for the test function $\psi$ see Definition 3.1)
and integrating over $\Omega$, we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\int\limits_{\Omega}{\psi(x)z^{2}\,dx}+\frac{1}{2N}\int\limits_{\Omega}{f(x)\psi(x)\bigg{(}\frac{\partial
z}{\partial x}\bigg{)}^{2}\,dx}$
$\displaystyle=\frac{1}{2N}\bigg{(}f(x)\psi(x)z\frac{\partial z}{\partial
x}-\frac{1}{2}(\omega(x)\psi(x))^{\prime}F(x)z^{2}\bigg{)}\bigg{|}_{0}^{1}$
$\displaystyle+\frac{1}{4N}\int\limits_{\Omega}z^{2}\frac{d}{dx}\bigg{(}F(x)\frac{d}{dx}\bigg{(}\omega(x)\psi(x)\bigg{)}\bigg{)}\,dx.$

Then choosing here

$\psi(x)=\omega^{-1}(x)\int\limits_{0}^{x}{\tfrac{dy}{F(y)}}=\tfrac{F(x)}{f(x)}\int\limits_{0}^{x}{\tfrac{dy}{F(y)}}\to\tfrac{1}{1+R_{0}}\text{
as }x\to 0,$

we arrive at the equality

$\frac{d}{dt}\int\limits_{\Omega}{\psi(x)z^{2}\,dx}+\frac{1}{N}\int\limits_{\Omega}{f(x)\psi(x)\bigg{(}\frac{\partial
z}{\partial x}\bigg{)}^{2}\,dx}+\frac{1}{2N}z^{2}(1,t)=0.$ (3.12)

In order to manage the left hand-side of this equality, we apply the Hardy
inequality

$\int\limits_{\Omega}{\psi(x)v^{2}\,dx}\leqslant
C_{H}(R_{0})\int\limits_{\Omega}{f(x)\psi(x)v^{2}_{x}\,dx}\text{ with
}v(0)=0,$

where $A(R_{0})\leqslant C_{H}(R_{0})\leqslant 4A(R_{0})$ and

$A(R_{0})=\mathop{\sup}\limits_{r\in(0,1)}\Bigl{(}\int\limits_{0}^{r}{\tfrac{dx}{f(x)\psi(x)}}\Bigr{)}\Bigl{(}\int\limits_{r}^{1}{\psi(x)\,dx}\Bigr{)}.$

Thus, we easily conclude that

$\int\limits_{\Omega}{\psi(x)z^{2}(x,t)\,dx}\leqslant
e^{-\frac{t}{NC_{H}(R_{0})}}\int\limits_{\Omega}{\psi(x)z_{0}^{2}(x)\,dx}\to
0\text{ as }t\to+\infty.$ (3.13)

Note that as $\psi(x)$ is bounded from bottom then we have $z\to 0$ in $L^{2}$
as $t\to+\infty$.

Now, multiplying the equation in (3.7) by
$-\omega(x)\phi(x)\frac{\partial}{\partial x}\big{(}F(x)\frac{\partial
z}{\partial x}\big{)}$ and integrating over $\Omega$, we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\int\limits_{\Omega}{\phi(x)F(x)\bigg{(}\frac{\partial
z}{\partial
x}\bigg{)}^{2}\,dx}+\frac{1}{2N}\int\limits_{\Omega}\omega(x)\phi(x)\bigg{(}\frac{\partial}{\partial
x}\bigg{(}F(x)\frac{\partial z}{\partial x}\bigg{)}\bigg{)}^{2}\,dx$
$\displaystyle=\bigg{(}\phi(x)F(x)\frac{\partial z}{\partial t}\frac{\partial
z}{\partial
x}-\frac{1}{4N}\omega(x)\phi^{\prime}(x)F^{2}(x)\bigg{(}\frac{\partial
z}{\partial x}\bigg{)}^{2}\bigg{)}\bigg{|}_{0}^{1}$
$\displaystyle+\frac{1}{4N}\int\limits_{\Omega}(\omega(x)\phi^{\prime}(x))^{\prime}F^{2}(x)\bigg{(}\frac{\partial
z}{\partial x}\bigg{)}^{2}\,dx.$

Taking $\phi(x)$ such that
$(\omega(x)\phi^{\prime}(x))^{\prime}F^{2}(x)=2f(x)\psi(x)$, i. e.

$\phi(x)=2\int\limits_{0}^{x}{\tfrac{1}{\omega(y)}\Bigl{(}\int\limits_{0}^{y}{\tfrac{1}{F(v)}\Bigl{(}\int\limits_{0}^{v}{\tfrac{ds}{F(s)}}\Bigr{)}dv}\Bigr{)}dy}\sim\tfrac{x^{2}}{2(R_{0}+1)}\text{
as }x\to 0,$

we have

$\displaystyle\frac{d}{dt}\int\limits_{\Omega}\phi(x)F(x)\bigg{(}\frac{\partial
z}{\partial
x}\bigg{)}^{2}\,dx+\frac{1}{N}\int\limits_{\Omega}\omega(x)\phi(x)\bigg{(}\frac{\partial}{\partial
x}\bigg{(}F(x)\frac{\partial z}{\partial x}\bigg{)}\bigg{)}^{2}\,dx$
$\displaystyle=\frac{1}{N}\int\limits_{\Omega}f(x)\psi(x)\bigg{(}\frac{\partial
z}{\partial x}\bigg{)}^{2}\,dx.$

Collecting this equality with (3.12), we arrive at

$\displaystyle\frac{d}{dt}\int\limits_{\Omega}\bigg{(}\phi(x)F(x)\bigg{(}\frac{\partial
z}{\partial x}\bigg{)}^{2}+\psi(x)z^{2}\bigg{)}\,dx$
$\displaystyle+\frac{1}{N}\int\limits_{\Omega}\omega(x)\phi(x)\bigg{(}\frac{\partial}{\partial
x}\bigg{(}F(x)\frac{\partial z}{\partial x}\bigg{)}\bigg{)}^{2}\,dx$
$\displaystyle+\frac{1}{2N}z^{2}(1,t)=0.$ (3.14)

Then, applying the estimate

$\int\limits_{\Omega}\frac{\phi(x)}{F(x)}v^{2}\,dx\leqslant
C_{P}(R_{0})\int\limits_{\Omega}\omega(x)\phi(x)\bigg{(}\frac{\partial
v}{\partial x}\bigg{)}^{2}\,dx\text{ with }v(1)=0,$

where

$C_{P}(R_{0})=\int\limits_{\Omega}{\tfrac{\phi(x)}{F(x)}\Bigl{(}\int\limits_{x}^{1}{\tfrac{dy}{\omega(y)\phi(y)}}\Bigr{)}\,dx},$

to (3.2), we conclude

$\displaystyle\int\limits_{\Omega}\phi(x)F(x)\bigg{(}\frac{\partial
z}{\partial
x}\bigg{)}^{2}\,dx\leqslant\Bigl{[}\int\limits_{\Omega}{\phi(x)F(x)\bigg{(}\frac{\partial
z_{0}}{\partial x}\bigg{)}^{2}\,dx}$
$\displaystyle+\tfrac{1}{NC_{P}(R_{0})}\bigg{(}1-\tfrac{NC_{P}(R_{0})C_{H}(R_{0})}{C_{H}(R_{0})-C_{P}(R_{0})}+\tfrac{NC_{P}(R_{0})C_{H}(R_{0})}{C_{H}(R_{0})-C_{P}(R_{0})}e^{\tfrac{C_{H}(R_{0})-C_{P}(R_{0})}{NC_{P}(R_{0})C_{H}(R_{0})}t}\bigg{)}$
$\displaystyle\times\int\limits_{\Omega}\psi(x)z_{0}^{2}(x)\,dx\Bigr{]}e^{-\frac{t}{NC_{P}(R_{0})}}\to
0\quad\text{ as }\quad t\to+\infty.$

As a result,

$z(x,t)\to 0\text{ and }x\frac{\partial z}{\partial x}(x,t)\to 0\text{
strongly in }L^{2}(\Omega)\text{ as }t\to+\infty,$

if only the inequality holds

$\int\limits_{\Omega}{\bigg{(}\psi(x)z_{0}^{2}(x)+\phi(x)F(x)\bigg{(}\frac{\partial
z_{0}}{\partial x}\bigg{)}^{2}\bigg{)}\,dx}<+\infty.$

Finally, the passage to $p$ in these relations completes the proof of
assertion (ii) and, as a consequence, of Theorem 1.

∎

##  4 Particular solutions

In this section, we will illustrate the result of Theorem 1 by particular
solutions. First, we analyze classical solutions to problem (3.7) and then we
discuss particular weak solutions.

###  4.1 Particular classical solutions

Introducing the new variable

$s=\sqrt{2N}\int\limits_{0}^{x}{\tfrac{dy}{f^{\frac{1}{2}}(y)}},$

and denoting

$l(s):=\sqrt{2N}\tfrac{g(x)}{f^{\frac{1}{2}}(x)}=\sqrt{\tfrac{2N}{R_{0}}}\,\tfrac{\sin\bigl{(}\frac{1}{2}\sqrt{\frac{R_{0}}{2N}}s\bigr{)}\bigl{[}R_{0}-1-(R_{0}+1)\sin^{2}\bigl{(}\frac{1}{2}\sqrt{\frac{R_{0}}{2N}}s\bigr{)}\bigr{]}}{|\cos\bigl{(}\frac{1}{2}\sqrt{\frac{R_{0}}{2N}}s\bigr{)}|},$
$s_{1}:=2\sqrt{\tfrac{2N}{R_{0}}}\arcsin\bigl{(}\sqrt{\tfrac{R_{0}}{R_{0}+1}}\bigr{)},$

we rewrite problem (3.7) in the form

$\begin{cases}\frac{\partial z}{\partial t}=\frac{\partial^{2}z}{\partial
s^{2}}+l(s)\frac{\partial z}{\partial s},\qquad s\in(0,s_{1}),\quad
t\in(0,T),\\\ z(0,t)=0,\quad\frac{\partial z}{\partial s}(s_{1},t)=0,\quad
t\in[0,T].\end{cases}$ (4.1)

It is worth noting that to reach (4.1), we incorporate the following simple
verified relations:

$s=\begin{cases}2\sqrt{\tfrac{2N}{R_{0}}}\arcsin\bigl{(}\sqrt{\tfrac{R_{0}}{R_{0}+1}}x^{\frac{1}{2}}\bigr{)},\quad\text{
if }\quad R_{0}>0,\\\ \,\\\ 2\sqrt{2N}x^{\frac{1}{2}},\qquad\quad\text{ if
}\quad R_{0}=0,\end{cases}$

or as consequence

$x=\begin{cases}\tfrac{R_{0}+1}{R_{0}}\sin^{2}\bigl{(}\tfrac{1}{2}\sqrt{\tfrac{R_{0}}{2N}}s\bigr{)},\quad\text{
if }\quad R_{0}>0,\\\ \,\\\ \tfrac{1}{8N}s^{2},\qquad\qquad\text{ if }\quad
R_{0}=0.\end{cases}$

Separating variables in (4.1):

$z(s,t)=T(t)S(s),$

leads to the problems

$\tfrac{T^{\prime}(t)}{T(t)}=\tfrac{S^{\prime\prime}(s)+l(s)S^{\prime}(s)}{S(s)}=-\lambda,$

whence

$T^{\prime}(t)=-\lambda T(t),$ $S^{\prime\prime}(s)+l(s)S^{\prime}(s)=-\lambda
S(s)$ (4.2)

with

$S(0)=0,\ S^{\prime}(s_{1})=0.$

After that, multiplying (4.2) by $p(s):=e^{\int\limits_{0}^{s}{l(y)\,dy}}$, we
immediately obtain the equation

$-(p(s)S^{\prime}(s))^{\prime}=\lambda p(s)S(s).$

Then setting

$U(s)=p^{\frac{1}{2}}(s)S(s)\quad
q(s)=\tfrac{(p^{\frac{1}{2}}(s))^{\prime\prime}}{p^{\frac{1}{2}}(s)}=\tfrac{1}{2}\bigl{(}l^{\prime}(s)+\tfrac{1}{2}l^{2}(s)\bigr{)},$

we arrive at the classical Sturm-Liouville problem with the continuous
potential $q(s)$

$\begin{cases}-U^{\prime\prime}(s)+q(s)U(s)=\lambda U(s),\qquad
s\in(0,s_{1}),\\\ U(0)=0,\quad U^{\prime}(s_{1})=0.\end{cases}$ (4.3)

The standard technical calculations provide the following asymptotic behavior
of eigenvalues and eigenfunctions to problem (4.3)

$\lambda_{k}\sim(\tfrac{\pi}{s_{1}})^{2}\bigl{(}k+\tfrac{1}{2}\bigr{)}^{2},\ \
U_{k}(s)\sim\sin\bigl{(}\tfrac{\pi}{s_{1}}(k+\tfrac{1}{2})s\bigr{)},$

or returning to (4.2)

$\lambda_{k}\sim(\tfrac{\pi}{s_{1}})^{2}\bigl{(}k+\tfrac{1}{2}\bigr{)}^{2},\ \
S_{k}(s)\sim
e^{-\frac{1}{2}\int\limits_{0}^{s}{l(y)\,dy}}\sin\bigl{(}\tfrac{\pi}{s_{1}}(k+\tfrac{1}{2})s\bigr{)}.$

Thus, problem (4.1) has a particular solution

$z(s,t)=\mathop{\sum}\limits_{k=0}^{+\infty}{c_{k}e^{-\lambda_{k}t}S_{k}(s)},$

which in turn means

$z(x,t)=\mathop{\sum}\limits_{k=0}^{+\infty}{c_{k}e^{-\lambda_{k}t}\varphi_{k}(x)},$

where

$\lambda_{k}\sim\tfrac{\pi^{2}}{N}\bigl{(}k+\tfrac{1}{2}\bigr{)}^{2},\ \
\varphi_{k}(x)\sim
e^{-N^{\frac{3}{2}}x}\sin\Bigl{(}\pi(k+\tfrac{1}{2})\tfrac{\arcsin\bigl{(}\sqrt{\frac{R_{0}}{R_{0}+1}}x^{\frac{1}{2}}\bigr{)}}{\arcsin\bigl{(}\sqrt{\frac{R_{0}}{R_{0}+1}}\bigr{)}}\Bigr{)}.$

Finally, keeping in mind the relation $z(x,t)=\omega(x)p(x,t)$, we deduce the
particular classical solution

$p(x,t)=\tfrac{1}{\omega(x)}\mathop{\sum}\limits_{k=0}^{+\infty}{c_{k}e^{-\lambda_{k}t}\varphi_{k}(x)}.$

It is worth noting that the asymptotic behavior of the solution
$\frac{C_{1}}{\sqrt{x}e^{C_{2}t}}$ as $x\to 0^{+}$ has a good agreement with
Theorem 1 (i).

###  4.2 Particular weak solutions

Figure 7: These two pictures illustrate existence of delta function type
solution for a symmetrised problems with $R_{0}=0$ and $N=10$.

First of all we remark that with zero on the boundary the integral
$\int_{0}^{a}f(z)\delta(z)dz$ is a priori not well defined. Then, we denote
positive and non-negative cut of functions by $s1_{x>0}$ and $s1_{x\geq
0}$,respectively. It corresponds to integrating $\delta$ function against the
function $f1_{x>0}$ (or possibly $f1_{x\geq 0}$ ) which is not continuous at
the origin $x=0$, where the support of delta function lies. Some formal not
analytically justified expressions have been used in a technical literature
like
$\int_{0}^{a}f(z)\delta(z)dz=\int_{-a}^{0}f(z)\delta(z)dz=\frac{1}{2}f(0)$.
This is the reason why to justify a delta function type solution for our
problem we will use a symmetrization method by considering a problem of
extended domain $[-1,1]$. Now we look for a solution to a symmetrically
extended problem (1.1)–(1.3) on the interval $(-1,1)$ in the form
$p(x,t)=\eta(t)\delta_{0}(x)$.

Multiplying symmetrized equation (1.1) by $\phi(x)\in C^{2}[-1,1]$ with
compact support and $\phi(0)\neq 0$, after integrating by parts in
$Q_{T}:=(-1,1)\times(0,T)$, we have

$\iint\limits_{Q_{T}}{\frac{\partial p}{\partial
t}\phi(x)\,dxdt}=\frac{1}{2N}\iint\limits_{Q_{T}}{\bigl{(}\tilde{f}(x)p\phi^{\prime\prime}(x)+2N\tilde{g}(x)p\phi^{\prime}(x)\bigr{)}\,dxdt},$

where $\tilde{f}$ and $\tilde{g}$ are even continuation of $f$ and $g$,
accordingly. Taking $p(x,t)=\eta(t)\delta_{0}(x)$ in the last equality, we
deduce that

$(\eta(T)-\eta(0))\phi(0)=(\tfrac{1}{2N}f(0)\phi^{\prime\prime}(0)+g(0)\phi^{\prime}(0))\int\limits_{0}^{T}{\eta(t)\,dt}=0.$

Thanks to the inequality $\phi(0)\neq 0$, we have

$\eta(T)=\eta(0)=M>0.$

As a result, symmetrized equation (1.1) has the following solution:

$p(x,t)=M\delta_{0}(x)\text{ for all }(x,t)\in(-1,1)\times(0,+\infty).$

Convergence of solution toward delta function is illustrated in Figure 7. It
is interesting to mention that a non-smooth change of variables
$y=2\,\sqrt{x}$ (for the case $R_{0}=0$) will remove the degeneracy from the
equation but the whole long-time dynamics will not be recovered as a in terms
of $y$ as a global attractor type solution $Ce^{t}$ that satisfied a no-flux
boundary conditions in terms of variable $y$ will not be satisfying no-flux
boundary conditions in terms of variable $x$. Although the $Ce^{t}$ solves the
original problem with Neumann boundary conditions (which make the original
problem ill-posed) it is unstable and a slight perturbation will drive the
dynamics toward delta function.

##  5 Local classical solvability and asymptotic behavior of classical
solutions near $x=0$

Here, we will discuss existence and behavior of a classical solution near the
degenerate point $x=0$. Besides, we will analyze sufficient conditions on the
given data in the model which provides the fulfillment of the conservation
law. To this end, we consider the initial-boundary value problem to more
general linear degenerate equation and the boundary condition than in the
original problem (1.1)–(1.3).

Let $\Omega=(0,l)\subseteq\mathbb{R}_{+}=\\{x:x>0\\}$ be a segment with a
boundary $\partial\Omega=\\{x=0\\}\cup\\{x=l\\}$. In our further
consideration, we also use the notation $\Gamma_{1}$ for the right point of
the boundary, i.e. $\Gamma_{1}=\\{x=l\\}$. For an arbitrary fixed time $T>0$,
we denote

$\partial\Omega_{T}=\partial\Omega\times[0,T]\quad\text{and}\quad\Gamma_{1,T}=\Gamma_{1}\times[0,T].$

We analyze the linear degenerate equation in the unknown function
$u=u(x,t):\Omega_{T}\to\mathbb{R},$

$\frac{\partial u}{\partial t}-\mathcal{L}u=f(x,t)$ (5.1)

supplemented with the initial condition

$u(x,0)=u_{0}(x)\quad\text{in}\quad\bar{\Omega},$ (5.2)

subject to the condition of the third kind (3BC):

$\mathcal{M}u=\psi(x,t)\quad\text{on}\quad\Gamma_{1,T},$ (5.3)

where the functions $f,u_{0}$ and $\psi$ are prescribed.

Coming to the operator involved, $\mathcal{L}$ is the linear degenerate
elliptic operator of the second order with time-dependent coefficients,
namely,

$\mathcal{L}=xa_{0}(x,t)\frac{\partial^{2}}{\partial
x^{2}}+a_{1}(x,t)\frac{\partial}{\partial x}+a_{2}(x,t),$

while the operator $\mathcal{M}$ reads as

$\mathcal{M}=b_{1}(x,t)\frac{\partial}{\partial x}+b_{2}(x,t).$

###  5.1 Solvability of (5.1)-(5.3) in smooth classes and behavior of the
solution at $x=0$

We start by introducing some general assumptions:

H1 (Conditions on the coefficients):

There exist positive $\delta_{0}$ and $\delta_{1}$ such that

$a_{0}(x,t)\geq\delta_{0}\,\text{ for any
}(x,t)\in\bar{\Omega}_{T},\,a_{1}(0,t)\geq\delta_{1}\text{ for any
}t\in[0,T].$ (5.4)

Besides,

$a_{0},a_{1},a_{2}\in\mathcal{C}_{\alpha}^{\alpha,\alpha/2}(\bar{\Omega}_{T}),\,b_{1},b_{2}\in\mathcal{C}^{1+\alpha,\frac{1+\alpha}{2}}(\Gamma_{1,T}),$
$b_{1}(l,t)\neq 0\quad\text{for all }t\in[0,T].$

H2 (Conditions on the given functions):

$u_{0}(x)\in\mathcal{C}_{\alpha}^{2+\alpha}(\bar{\Omega}),\quad
f\in\mathcal{C}_{\alpha}^{\alpha,\alpha/2}(\bar{\Omega}_{T}),\quad\psi\in\mathcal{C}^{1+\alpha,\frac{1+\alpha}{2}}(\Gamma_{1,T}).$

H3 (Compatibility conditions):

When $x\in\Gamma_{1}$, the compatibility condition at $t=0$ holds

$\mathcal{M}u_{0}(x)|_{t=0}=\psi(x,0).$

######  Remark 5.1.

It is worth mentioning that assumption (5.4) on the coefficient $a_{1}$ and
Fiker condition (see. i.e. [40, (1.13), p.17]) guarantee well-posedness in
smooth classes of problem (5.1)-(5.3) (i.e. the initial-boundary value problem
is stated without a boundary condition at the point $x=0$).

We are now in the position to state our main result concerning the solvability
of (5.1)-(5.3).

######  Theorem 2.

Let $T>0$ be arbitrarily fixed and let assumptions H1-H3 hold. Then problem
(5.1)-(5.3) admits a unique classical solution $u=u(x,t)$ on the space-time
rectangle $\Omega_{T}$, satisfying regularity
$u\in\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})$.
Besides, the following estimate holds

$\displaystyle\|u\|_{\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})}$
$\displaystyle\leq
C_{1}[\|u_{0}\|_{\mathcal{C}_{\alpha}^{2+\alpha}(\bar{\Omega})}+\|f\|_{\mathcal{C}_{\alpha}^{\alpha,\alpha/2}(\bar{\Omega}_{T})}+\|\psi\|_{\mathcal{C}^{1+\alpha,\frac{1+\alpha}{2}}}]$
$\displaystyle:=C_{1}\mathfrak{F}(u_{0},f,\psi)$ (5.5)

with some $C_{1}>0$ being independent of the right-hand sides of (5.1)-(5.3).

If, in addition, the following assumption holds.

H4: There is a real $s_{0}$ such that

$\begin{cases}s_{0}>\alpha,\\\ \quad 2s_{0}a_{0}(0,t)+a_{1}(0,t)>0,\\\
\underset{x\to
0}{\lim}\bigg{|}\frac{a_{1}(x,t)+(s_{0}-1)a_{0}(x,t)}{x}\bigg{|}<C,\end{cases}$
(5.6)

for all $t\in[0,T]$ and

$u_{0}\in E_{s_{0}}^{2+\alpha}(\bar{\Omega}),\quad f\in
E_{s_{0}}^{\alpha,\alpha/2}(\bar{\Omega}_{T}).$

Then, the classical solution $u$ satisfies the regularity

$(x^{-s_{0}}u)\in\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T}),$

and

$\|x^{-s_{0}}u\|_{\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})}\leq
C_{2}[\|u_{0}\|_{E_{s_{0}}^{2+\alpha}(\bar{\Omega})}+\|f\|_{E_{s_{0}}^{\alpha,\alpha/2}(\bar{\Omega}_{T})}+\mathfrak{F}(u_{0},f,\psi)].$
(5.7)

######  Remark 5.2.

It is apparent that estimate (5.7) where $s_{0}>\alpha$ provides the equality

$u(0,t)=0\quad\text{for all}\quad t\in[0,T].$ (5.8)

######  Example 5.1.

The following is an example of coefficients $a_{0},a_{1}$ and the weight
$s_{0}$ satisfying condition (5.6) in the case of $\alpha\in(0,1/2)$:

$s_{0}=\frac{1}{2}\quad\text{and}\quad
a_{1}(x,t)=\frac{a_{0}(x,t)}{2},\quad(x,t)\in\bar{\Omega}_{T},$

where $a_{0}$ meets requirement H1.

It is easy to verify that condition (5.6) is not satisfied in the case of
original problem (1.1)–(1.3). That means, estimate (5.7) does not provid
equality (5.8). Nevertheless, our next result states this equality under
weaker assumptions on the coefficients than (5.6). Namely, we will assume
additionally to H1-H3 the following conditions.

H5

Let $u_{0}\in E_{s}^{2+\alpha}(\bar{\Omega}),$ $f\in
E_{s-1}^{\alpha,\alpha/2}(\bar{\Omega}_{T})$ with $s>1+\alpha$.

H6

Let $a_{0},a_{1}\in\mathcal{C}([0,T],\mathcal{C}^{1}(\bar{\Omega}))$. Besides,
there is $\delta_{2}>0$ and the function
$a(x,t)\in\mathcal{C}([0,T],\mathcal{C}^{1}(\bar{\Omega}))\cap\mathcal{C}_{\alpha/2}^{\alpha,\alpha/2}(\bar{\Omega}_{T})$
(including the case $a\equiv 0$) satisfying the inequality

$\underset{\bar{\Omega}_{T}}{\sup}\frac{|a(x,t)|}{x}\leq C,$

such that there holds

$a_{1}(x,t)-a_{0}(x,t)=\delta_{2}+a(x,t).$

######  Theorem 3.

Let $\psi\in\mathcal{C}^{1+\alpha,\frac{1+\alpha}{2}}(\Gamma_{1,T})$. Under
assumptions H1, H3, H5 and H6, classical solution (5.1)-(5.3) vanishes at
$x=0$ for all $t\in[0,T]$, i.e. equality (5.8) holds.

The rest part of this section is devoted to the proof of Theorems 2 and 3.

###  5.2 Some technical results

In this section we provide some technical results which are key points in the
proof of Theorem 2 and as a consequence of Theorem 3.

####  5.2.1 Some auxiliary estimates

First we describe some properties of the solution to the initial-value problem
for the degenerate linear equation, which will be key point in the proof of
Theorem 2 in Section 5.3.

Let the function $v_{1}=v_{1}(x,t)$ solves the Cauchy problem

$\begin{cases}\frac{\partial v_{1}}{\partial
t}-x\frac{\partial^{2}v_{1}}{\partial x^{2}}-A_{1}\frac{\partial
v_{1}}{\partial
x}=F(x,t)\quad\text{in}\quad\mathbb{R}_{+,T}=\mathbb{R}_{+}\times(0,T),\\\
v_{1}(x,0)=v_{1,0}(x),\quad\text{in}\quad\mathbb{R}_{+},\end{cases}$ (5.9)

where $F$ and $v_{1,0}$ are some given functions, and $A_{1}$ is a given
positive number.

The classical solvability of problem (5.9) was studied in [7, §2-3]. In
particularly, the following result, stated as a lemma, subsumes Theorem 3.1
and estimate (2) in [7].

######  Lemma 5.1.

Let $A_{1}$ and
$v_{1,0}\in\mathcal{C}_{\alpha}^{2+\alpha}(\bar{\mathbb{R}}_{+})$,
$F\in\mathcal{C}_{\alpha}^{\alpha,\alpha/2}(\bar{\mathbb{R}}_{+,T})$. Besides,
we assume that there exists a positive number $r_{0}$ such that

$v_{1,0}(x)=F(x,t)=0\quad\text{if}\quad x>r_{0}.$

Then, for any fixed $T>0$, there is a unique classical solution $v_{1}(x,t)$
to problem (5.9). In addition, the following estimate holds:

$\|v_{1}\|_{\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\mathbb{R}}_{+,T})}\leq
C[\|F\|_{\mathcal{C}_{\alpha}^{\alpha,\alpha/2}(\bar{\mathbb{R}}_{+,T})}+\|v_{1,0}\|_{\mathcal{C}_{\alpha}^{2+\alpha}(\bar{R}_{+})}].$

Here, the generic constant $C$ is independent of the right-hand side in (5.9)
and depends only on $A_{1}$ and $T$.

The next lemma allows us to reduce (5.1)-(5.3) to problem with homogenous
initial data.

######  Lemma 5.2.

There exists a universal constant $C^{*}>0$ with the following property: for
any functions
$\mathcal{V}_{0}=\mathcal{V}_{0}(x)\in\mathcal{C}_{\alpha}^{2+\alpha}(\bar{\Omega})$
and
$\mathcal{V}_{1}=\mathcal{V}_{1}(x)\in\mathcal{C}_{\alpha}^{\alpha}(\bar{\Omega})$,
there exists a function
$\mathcal{V}=\mathcal{V}(x,t)\in\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})$
such that

$\mathcal{V}(x,0)=\mathcal{V}_{0}(x),\quad\frac{\partial\mathcal{V}}{\partial
t}(x,0)=\mathcal{V}_{1}(x),$

and

$\|\mathcal{V}\|_{\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})}\leq
C(\|\mathcal{V}_{0}\|_{\mathcal{C}_{\alpha}^{2+\alpha}(\bar{\Omega})}+\|\mathcal{V}_{1}\|_{\mathcal{C}_{\alpha}^{\alpha}(\bar{\Omega})}).$

###### Proof.

Denoting

$\mathcal{F}(x):=\mathcal{V}_{1}(x)-x\frac{\partial^{2}\mathcal{V}_{0}}{\partial
x^{2}}(x)-A_{1}\frac{\partial\mathcal{V}_{0}}{\partial x},$

we build (see [37, Theorem 4.1]) extensions $\bar{\mathcal{V}}_{0}$ and
$\bar{\mathcal{F}}$ of the functions $\mathcal{V}_{0}$ and $\mathcal{F}$ on
$\mathbb{R}_{+}$ such that

$\displaystyle\|\bar{\mathcal{V}}_{0}\|_{\mathcal{C}_{\alpha}^{2+\alpha}(\bar{\mathbb{R}}_{+})}$
$\displaystyle\leq
C\|\mathcal{V}_{0}\|_{\mathcal{C}_{\alpha}^{2+\alpha}(\bar{\Omega})},$ (5.10)
$\displaystyle\|\bar{\mathcal{F}}\|_{\mathcal{C}_{\alpha}^{\alpha}(\bar{\mathbb{R}}_{+})}$
$\displaystyle\leq
C\|\mathcal{F}\|_{\mathcal{C}_{\alpha}^{\alpha}(\bar{\Omega})}\leq
C(\|\mathcal{V}_{0}\|_{\mathcal{C}_{\alpha}^{2+\alpha}(\bar{\Omega})}+\|\mathcal{V}_{1}\|_{\mathcal{C}_{\alpha}^{\alpha}(\bar{\Omega})}),$

and, besides, the functions $\bar{\mathcal{V}}_{0}$ and $\bar{\mathcal{F}}$
have compact supports. After that, we define the function $\mathcal{V}(x,t)$
to be the solution to the Cauchy problem

$\begin{cases}\frac{\partial\mathcal{V}}{\partial
t}-x\frac{\partial^{2}\mathcal{V}}{\partial
x^{2}}-A_{1}\frac{\partial\mathcal{V}}{\partial
x}=\bar{\mathcal{F}}(x)\quad\text{in }\mathbb{R}_{+,T},\\\
\mathcal{V}(x,0)=\bar{\mathcal{V}}_{0}(x).\end{cases}$

Applying Lemma 5.1 to this problem, and taking into account (5.10), the claim
is proven. ∎

####  5.2.2 Domains. Some auxiliary propositions.

In order to prove the first part of Theorem 2, we will construct a regularizer
(see [37, Section 4]). To this end, we need a special covering of the domain
$\Omega$. We take two collections of open sets $\\{\varpi^{m}\\}$ and
$\\{\Omega^{m}\\}$, which consist of a finite number $\varpi^{m}$ and
$\Omega^{m}$ possessing the following properties for each small number
$\lambda>0$ and any point $x^{m}\in\bar{\Omega}$:

  1. 1.

Denoting by $\mathfrak{B}_{r}(x^{m})$ the ball about $x^{m}$ of radius $r$, we
have that

$\varpi^{m}=\mathfrak{B}_{\lambda/2}(x^{m})\cap\bar{\Omega},\qquad\Omega^{m}=\mathfrak{B}_{\lambda}(x^{m})\cap\bar{\Omega},$

and

$\overline{\varpi^{m}}\subset\Omega^{m}\subset\bar{\Omega},\qquad\bigcup_{m}\varpi^{m}=\bigcup_{m}\Omega^{m}=\bar{\Omega}.$

  2. 2.

There exists a number $\mathcal{N}_{0}$ independent of $\lambda$ such that the
intersection of any $\mathcal{N}_{0}+1$ distinct $\Omega^{m}$ (and
consequently any $\mathcal{N}_{0}+1$ distinct $\varpi^{m}$) is empty.

The index $m$ belongs to one of three sets: $\mathfrak{M}$, $\mathfrak{N}_{0}$
or $\mathfrak{N}_{1}$, where

$\displaystyle m\in\mathfrak{M}\quad$ $\displaystyle\text{if
}\overline{\Omega^{m}}\cap\partial\Omega=\emptyset,$ $\displaystyle
m\in\mathfrak{N}_{0}\quad$ $\displaystyle\text{if
}\overline{\varpi^{m}}\cap\\{x=0\\}\neq\emptyset,$ $\displaystyle
m\in\mathfrak{N}_{1}\quad$ $\displaystyle\text{if
}\overline{\varpi^{m}}\cap\Gamma_{1}\neq\emptyset.$

Finally, denoting
$\partial\Omega^{m}=\partial\Omega\cap\mathfrak{B}_{\lambda}(x^{m})$ and
$\Gamma_{1}^{m}=\Gamma_{1}\cap\mathfrak{B}_{\lambda}(x^{m})$, we conclude that
the covering $\\{\varpi^{m}\\}$ and $\\{\Omega^{m}\\}$ define a partition of
unity for the domain $\Omega$. Let $\xi^{m}(x):$ $\Omega\to[0,1]$ be a smooth
function such that

$\displaystyle\xi^{m}(x)=1$ $\displaystyle\quad\text{if
}x\in\overline{\varpi^{m}},$ $\displaystyle\xi^{m}(x)=0$
$\displaystyle\quad\text{if }x\in\bar{\Omega}\backslash\overline{\Omega^{m}},$
$\displaystyle\xi^{m}(x)\in(0,1)$ $\displaystyle\quad\text{if
}x\in\Omega^{m}\backslash\varpi^{m}.$

Moreover,

$|D_{x}^{j}\xi^{m}|\leq C\lambda^{-|j|},\qquad 1\leq|j|,\quad
1\leq\sum_{m}(\xi^{m})^{2}\leq\mathcal{N}_{0}.$

Then, taking advantage of $\xi^{m}$, we define the function

$\zeta^{m}:=\frac{\xi^{m}}{\sum_{j}(\xi^{j})^{2}}.$ (5.11)

The properties of the functions $\xi^{m}$ tell us that the functions
$\zeta^{m}$ vanish for $x\in\bar{\Omega}\backslash\overline{\Omega^{m}}$; and
in addition, $|D_{x}^{j}\zeta^{m}|\leq C\lambda^{-|j|}$. Hence, the product
$\zeta^{m}\xi^{m}$ defines the partition of unity via the formula

$\sum_{m}\zeta^{m}\xi^{m}=1.$

After that, in the spaces
$\mathcal{C}_{\alpha,0}^{k+\alpha,\frac{k+\alpha}{2}\nu}(\bar{\Omega}_{T}),$
$k=0,1,2,$ we introduce the norms associated with the covering
$\\{\Omega^{m}\\}$:

$\langle\langle
w\rangle\rangle_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{T})}:=\underset{m}{\sup}\|w\|_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\overline{\Omega^{m}}_{T})}.$

Repeating the arguments of Chapter 4 in [37], we can assert the following.

######  Proposition 5.1.

Let
$w\in\mathcal{C}_{\alpha,0}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})$.
Then for $k=0,1,2,$

$\displaystyle\|w\|_{\mathcal{C}([0,T],\mathcal{C}_{\alpha}^{k}(\bar{\Omega}))}$
$\displaystyle\leq CT^{\frac{2+\alpha-k}{2}}\bigg{[}\langle\partial w/\partial
t\rangle^{(\alpha/2)}_{t,0,\Omega_{T}}$
$\displaystyle+\bigg{\langle}x\frac{\partial w}{\partial
x}\bigg{\rangle}_{t,0,\Omega_{T}}^{(\frac{\alpha}{2})}+\sum_{j=1}^{2}\bigg{\langle}x\frac{\partial^{j}w}{\partial
x^{j}}\bigg{\rangle}_{t,0,\Omega_{T}}^{(\frac{2+\alpha-j}{2})}\bigg{]},$

and, besides, for $k=0,1,$ there holds

$\|w\|_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{T})}\leq
CT^{\frac{2+\alpha-k}{2}}\bigg{[}\langle\partial w/\partial
t\rangle^{(\alpha/2)}_{t,0,\Omega_{T}}+\sum_{j=1}^{2}\bigg{\langle}x\frac{\partial^{j}w}{\partial
x^{j}}\bigg{\rangle}_{t,0,\Omega_{T}}^{(\frac{2+\alpha-j}{2})}\bigg{]},$
$\|\partial w/\partial t\|_{\mathcal{C}(\bar{\Omega}_{T})}\leq
CT^{\frac{\alpha}{2}}\langle\partial w/\partial
t\rangle^{(\alpha/2)}_{t,0,\Omega_{T}},$

where the positive constant $C$ does not depend on $T$.

Next, for $\tau\in[0,T]$ and an arbitrarily given $0<\mathfrak{d}<1$, we set

$\tau=\lambda^{2}\mathfrak{d}.$ (5.12)

######  Proposition 5.2.

Let (5.12) hold. Then for $k=0,1,2,$ and any function
$w\in\mathcal{C}_{\alpha,0}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{\tau})$,
there is the following norm equivalence:

$\langle\langle
w\rangle\rangle_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{\tau})}\leq\|w\|_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{\tau})}\leq
C\langle\langle
w\rangle\rangle_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{\tau})},$

where the positive constant $C$ is independent of $\lambda$ and $\tau$.

######  Proposition 5.3.

Let a function $\Phi_{m}(x)$ defined in $\Omega^{m}$ have the property

$\bigg{|}\frac{\partial^{j}\Phi_{m}}{\partial x^{j}}(x)\bigg{|}\leq
C\lambda^{-j},\quad 0\leq j\leq 2,$

and the numbers $\tau$ and $\lambda$ are related via (5.12). Then for any
function
$w\in\mathcal{C}_{\alpha,0}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{\tau})$,
$k=0,1,2$,

$\|\Phi_{m}w\|_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}\leq
C\|w\|_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}^{m}_{\tau})},$

where the positive constant $C$ does not depend on $\lambda$ and $\tau$.

Moreover, for $m\in\mathfrak{N}_{1}$ there holds

$C\|\Phi_{m}w\|_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}\leq\|w\|_{\mathcal{C}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}\leq
C\|\Phi_{m}w\|_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}^{m}_{\tau})},$

and

$C\|w\|_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}\leq\|w\|_{\mathcal{C}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}\leq
C\|w\|_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}.$

######  Proposition 5.4.

Let (5.12) hold and

$w(x,t)=\sum_{m\in\mathfrak{M}\,\cup\mathfrak{N}\,_{0}\cup\mathfrak{N}\,_{1}}w^{m}(x,t),$

where
$w^{m}\in\mathcal{C}_{\alpha,0}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}^{m}_{\tau})$
vanishes outside $\Omega^{m}$, $k=0,1,2$. Then

$\langle\langle
w\rangle\rangle_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}_{\tau})}\leq
C\underset{m\in\mathfrak{M}\,\cup\mathfrak{N}\,_{0}\cup\mathfrak{N}\,_{1}}{\sup}\|w^{m}\|_{\mathcal{C}_{\alpha}^{k+\alpha,\frac{k+\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}.$

###  5.3 Proof of Theorem 2

It is worth noting that the second part Theorem 2 which concerns to estimate
(5.7) is a simple consequence of (2) and (5.6). Indeed, in order to verify
(5.7), it is enough to substitute the new unknown function
$\mathcal{U}(x,t)=x^{-s_{0}}u(x,t)$ to (5.1)-(5.3) and then, taking into
account assumptions H4, apply the first part of Theorem 2 and estimate (2).
Thus, we are left to prove that (5.1)-(5.3) has a unique solution
$u\in\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})$
satisfying (2). To this end, following arguments of §7-8 in Chapter 4 [37], we
will construct an operator $\mathfrak{R}$ so-called a regularizer to
(5.1)-(5.3) which allow us to construct the classical solution to this
problem.

First of all, using Lemma 5.2 with $\mathcal{V}_{0}=u_{0}$ and
$\mathcal{V}_{1}=\mathcal{L}u_{0}(x)|_{t=0}+f(x,0),$ $x\in\Omega$, we reduce
(5.1)-(5.3) to the problem with homogenous initial data. Thus, we look for a
solution to (5.1)-(5.3) in the form

$u(x,t)=\mathcal{V}(x,t)+v(x,t)$ (5.13)

where $\mathcal{V}(x,t)$ is built in Lemma 5.2. Coming to the new unknown
function $v=v(x,t)$, it solves the following initial-boundary value problem

$\begin{cases}\frac{\partial v}{\partial
t}-\mathcal{L}v=\bar{f}\quad\text{in}\quad\Omega_{T},\\\
v(x,0)=0\quad\text{in}\quad\bar{\Omega},\\\
\mathcal{M}v=\bar{\psi}\quad\text{on}\quad\Gamma_{1,T},\end{cases}$ (5.14)

where we put

$\bar{f}=f(x,t)-\frac{\partial\mathcal{V}}{\partial
t}+\mathcal{L}\mathcal{V},\quad\bar{\psi}=\psi(x,t)-\mathcal{V}|_{\Gamma_{1,T}}.$

It is worth noting that assumptions H1-H3 and Lemma 5.2 provide the following
smoothness

$\bar{f}\in\mathcal{C}_{\alpha,0}^{\alpha,\alpha/2}(\bar{\Omega}_{T})\quad\text{and}\quad\bar{\psi}\in\mathcal{C}_{0}^{1+\alpha,\frac{1+\alpha}{2}}(\Gamma_{1,T}).$
(5.15)

After that, for the sake of convenience, we rewrite problem (5.14) in more
compact form

$\mathcal{A}v=h\quad\text{with}\quad h=(\bar{f},\bar{\psi}),$ (5.16)

where $\mathcal{A}$ is the linear operator defined by the left-hand side of
(5.14). In other words,
$\mathcal{A}v=\\{\mathcal{A}_{0}v,\mathcal{A}_{1}v|_{\Gamma_{1,T}}\\}$, where
$\mathcal{A}_{0}$ is given by the left-hand side of the equation in (5.14)
while $\mathcal{A}_{1}$ is defined by the left-hand side of 3BC in (5.14).

Setting now for $m\in\mathfrak{M}\cup\mathfrak{N}_{0}\cup\mathfrak{N}_{1}$,

$\displaystyle a_{i}^{m}:=a_{i}(x^{m},0),\qquad\,\,b_{j}^{m}:=b_{j}(x^{m},0),$
$\displaystyle
f_{m}:=\xi^{m}(x)\bar{f}(x,t),\quad\psi_{m}:=\xi^{m}(x)\bar{\psi}(x,t),$

(where $\xi^{m},$ $\mathfrak{M}$, $\mathfrak{N}_{0}$, $\mathfrak{N}_{1}$ are
defined in Subsection 5.2.2), we introduce the functions $V^{m}:=V^{m}(x,t)$,
$m\in\mathfrak{M}\cup\mathfrak{N}_{0}\cup\mathfrak{N}_{1},$ as solutions to
the following problems for $\tau\in(0,T]$ satisfying (5.12).

If $m\in\mathfrak{M}$, then

$\begin{cases}\frac{\partial V^{m}}{\partial
t}-x^{m}a_{0}^{m}\frac{\partial^{2}V^{m}}{\partial
x^{2}}-a_{1}^{m}\frac{\partial V^{m}}{\partial x}=f_{m}\quad\text{in}\
\mathbb{R}_{\tau},\\\ V^{m}(x,0)=0,\quad\text{in }\mathbb{R}.\end{cases}$
(5.17)

Instead, if $m\in\mathfrak{N}_{1}$, then $V^{m}$ solves the initial-boundary
value problem:

$\begin{cases}\frac{\partial V^{m}}{\partial
t}-x^{m}a_{0}^{m}\frac{\partial^{2}V^{m}}{\partial
x^{2}}-a_{1}^{m}\frac{\partial V^{m}}{\partial x}=f_{m}\quad\text{in}\
\mathbb{R}_{+,\tau},\\\ V^{m}(x,0)=0,\quad\text{in }\mathbb{R}_{+},\\\
b_{1}^{m}\frac{\partial V^{m}}{\partial
x}+b_{2}^{m}V^{m}=\psi^{m}\quad\text{in}\quad\partial\mathbb{R}_{+,\tau}.\end{cases}$
(5.18)

Finally, in the case of $m\in\mathfrak{N}_{0}$, we define $V^{m}$ as a
solution of the Cauchy problem:

$\begin{cases}\frac{\partial V^{m}}{\partial
t}-xa_{0}^{m}\frac{\partial^{2}V^{m}}{\partial x^{2}}-a_{1}^{m}\frac{\partial
V^{m}}{\partial x}=f_{m}\quad\text{in}\ \mathbb{R}_{+,\tau},\\\
V^{m}(x,0)=0\quad\text{in }\mathbb{R}_{+}.\end{cases}$ (5.19)

It is easy to see that problems (5.17) and (5.18) are stated for usual (non-
degenerate) parabolic linear equation, while problem (5.19) contains the
degenerate parabolic equation similar to (5.1) with the coefficients
satisfying to the Fiker condition.

######  Definition 5.1.

Let $\tau\in(0,T]$. An operator $\mathfrak{R}$ is called a regularizer, on the
time-interval $[0,\tau]$, if

$\mathfrak{R}:\mathcal{C}_{\alpha,0}^{\alpha,\frac{\alpha}{2}}(\bar{\Omega}_{\tau})\times\mathcal{C}_{0}^{1+\alpha,\frac{1+\alpha}{2}}(\Gamma_{1,\tau})\to\mathcal{C}_{\alpha,0}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{\tau}),$

and

$\mathfrak{R}h=\sum_{m\in\mathfrak{M}\cup\mathfrak{N}\,_{1}\cup\mathfrak{N}\,_{0}}\zeta^{m}(x)V^{m}(x,t),$

where the functions $\zeta^{m}(x)$ and $V^{m}(x,t)$ are given in (5.11) and
(5.17)-(5.19), respectively.

The operator $\mathfrak{R}$ enables us to build an inverse operator to
$\mathcal{A}$. At this point, we state the key lemma.

######  Lemma 5.3.

Let $\tau\in(0,T]$ satisfy (5.12) and assume the hypotheses of Theorem 2.
Then, setting

$\mathcal{H}=\mathcal{C}_{\alpha,0}^{\alpha,\frac{\alpha}{2}}(\bar{\Omega}_{\tau})\times\mathcal{C}_{0}^{1+\alpha,\frac{1+\alpha}{2}}(\Gamma_{1,\tau}),$

for any

$h\in\mathcal{H}\qquad\text{and}\qquad
v\in\mathcal{C}_{\alpha,0}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{\tau})$

the following hold:

  * (i)

$\mathfrak{R}$ is a bounded operator:

$\|\mathfrak{R}h\|_{\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{\tau})}\leq
C\|h\|_{\mathcal{H}},$

where the positive constant $C$ is independent of $\lambda$ and $\tau$.

  * (ii)

There exist operators $\mathfrak{T}_{1}$ and $\mathfrak{T}_{2}$ such that

$\mathcal{A}\mathfrak{R}h=h+\mathfrak{T}_{1}h,\qquad\mathfrak{R}\mathcal{A}v=v+\mathfrak{T}_{2}v,$
(5.20)

where

$\|\mathfrak{T}_{1}h\|_{\mathcal{H}}\leq\frac{1}{2}\|h\|_{\mathcal{H}}\quad\text{and}\quad\|\mathfrak{T}_{2}v\|_{\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{\tau})}\leq\frac{1}{2}\|v\|_{\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{\tau})}.$

###### Proof.

First, we verify statement (i) of this claim. Simple linear changes of
variables allow us to conclude that the results of Lemma 5.1 above and Theorem
6.1 in [37] hold in the case of problems (5.17)-(5.19). Thus, collecting this
results with Propositions 5.1-5.4, we arrive at the estimate

$\displaystyle\|\mathfrak{R}h\|_{\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{\tau})}\leq
C\underset{m\in\mathfrak{M}\,\bigcup\mathfrak{N\,}\,_{1}\bigcup\mathfrak{N\,}\,_{0}}{\sup}\|V^{m}\|_{\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}$
$\displaystyle\leq
C\Big{[}\underset{m\in\mathfrak{M}\,\bigcup\mathfrak{N\,}\,_{1}\bigcup\mathfrak{N\,}\,_{0}}{\sup}\|\xi^{m}\bar{f}\|_{\mathcal{C}_{\alpha}^{\alpha,\frac{\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}+\underset{m\in\mathfrak{N\,}\,_{1}}{\sup}\|\xi^{m}\bar{\psi}\|_{\mathcal{C}^{1+\alpha,\frac{1+\alpha}{2}\nu}(\Gamma^{m}_{1,\tau})}\Big{]}$
$\displaystyle\leq C\|h\|_{\mathcal{H}},$

with the constant $C$ satisfying the assumptions of the present lemma. Thus,
we complete the proof of (i).

Let us verify (ii) of this claim. Here we restrict ourself verification of the
first equality in (5.20). The second one in (5.20) is checked in the same
manner with using analogous arguments from [37, Chapter 4, $\S$ 7]. Collecting
definition of the operator $\mathcal{A}$ (see (5.16)) with the properties of
$\zeta^{m}$ and $\xi^{m}$, we conclude that

$\mathcal{A}\mathfrak{R}h=\\{\mathcal{A}_{0}\mathfrak{R}h,\mathcal{A}_{1}\mathfrak{R}h\big{|}_{\Gamma_{1,\tau}}\\},$

where

$\mathcal{A}_{0}\mathfrak{R}h=\bar{f}+\mathfrak{T}_{1}^{1}h,\quad\mathcal{A}_{1}\mathfrak{R}h=\bar{\psi}+\mathfrak{T}_{1}^{2}h,$

and

$\displaystyle\mathfrak{T}_{1}^{1}h$
$\displaystyle=a_{2}(x,t)\mathfrak{R}h-\sum_{m}\bigg{[}2xa_{0}(x,t)\frac{\partial\zeta^{m}}{\partial
x}\frac{\partial V^{m}}{\partial
x}+xa_{0}(x,t)V^{m}\frac{\partial^{2}\zeta^{m}}{\partial x^{2}}\bigg{]}$
$\displaystyle-\sum_{m\in\mathfrak{M}\,\bigcup\mathfrak{N\,}\,_{1}}\zeta^{m}[xa_{0}(x,t)-x^{m}a_{0}(x^{m},0)]\frac{\partial^{2}V^{m}}{\partial
x^{2}}$
$\displaystyle-\sum_{m\in\mathfrak{N\,}_{0}}\zeta^{m}x[a_{0}(x,t)-a_{0}(x^{m},0)]\frac{\partial^{2}V^{m}}{\partial
x^{2}};$ $\displaystyle\mathfrak{T}_{1}^{2}h$
$\displaystyle=b_{2}(x,t)\mathfrak{R}h|_{\Gamma_{1,\tau}}+\sum_{m\in\mathfrak{N\,}\,_{1}}b_{1}(x,t)\frac{\partial\zeta^{m}}{\partial
x}V^{m}|_{\Gamma_{1,\tau}}$
$\displaystyle+\sum_{m\in\mathfrak{N\,}\,_{1}}\zeta^{m}[b_{1}(x,t)-b_{1}(x^{m},0)]\frac{\partial
V^{m}}{\partial x}|_{\Gamma_{1,\tau}}.$

After that, appealing to Propositions 5.1-5.4, Lemma 5.1 above and Theorem
6.1. from [37], and taking into account assumptions H2 with the following
easily verified inequalities:

$\bigg{\|}x\zeta^{m}\frac{\partial^{2}V^{m}}{\partial
x^{2}}\bigg{\|}_{\mathcal{C}_{\alpha}^{\alpha,\frac{\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}\leq
C[1+\mathfrak{d}^{\alpha/2}]\bigg{\|}x\frac{\partial^{2}V^{m}}{\partial
x^{2}}\bigg{\|}_{\mathcal{C}_{\alpha}^{\alpha,\frac{\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}\,m\in\mathfrak{N}_{0},$
$\bigg{\|}\zeta^{m}\frac{\partial^{2}V^{m}}{\partial
x^{2}}\bigg{\|}_{\mathcal{C}_{\alpha}^{\alpha,\frac{\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}\leq
C[1+\mathfrak{d}^{\alpha/2}]\bigg{\|}\frac{\partial^{2}V^{m}}{\partial
x^{2}}\bigg{\|}_{\mathcal{C}_{\alpha}^{\alpha,\frac{\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}\,m\in\mathfrak{M}\cup\mathfrak{N}_{1},$
$\displaystyle\bigg{\|}x\frac{\partial\zeta^{m}}{\partial x}\frac{\partial
V^{m}}{\partial
x}\bigg{\|}_{\mathcal{C}_{\alpha}^{\alpha,\frac{\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}$
$\displaystyle\leq
C\bigg{[}(\mathfrak{d}^{\frac{1+\alpha}{2}}+\mathfrak{d}^{\frac{1}{2}})\bigg{\langle}x\frac{\partial
V^{m}}{\partial
x}\bigg{\rangle}_{t,0,\Omega^{m}_{\tau}}^{(\frac{1+\alpha}{2})}$
$\displaystyle+\mathfrak{d}^{\frac{\alpha}{2}}\bigg{\langle}x\frac{\partial^{2}V^{m}}{\partial
x^{2}}\bigg{\rangle}_{t,0,\Omega^{m}_{\tau}}^{(\frac{\alpha}{2})}\bigg{]},$
$\displaystyle\bigg{\|}xV^{m}\frac{\partial^{2}\zeta^{m}}{\partial
x^{2}}\bigg{\|}_{\mathcal{C}_{\alpha}^{\alpha,\frac{\alpha}{2}}(\bar{\Omega}^{m}_{\tau})}$
$\displaystyle\leq
C\bigg{[}(1+\lambda^{\alpha})\mathfrak{d}^{\frac{2+\alpha}{2}}\bigg{\langle}\frac{\partial
V^{m}}{\partial t}\bigg{\rangle}_{t,0,\Omega^{m}_{\tau}}^{(\frac{\alpha}{2})}$
$\displaystyle+\mathfrak{d}^{\frac{1+\alpha}{2}}\bigg{\langle}x\frac{\partial
V^{m}}{\partial
x}\bigg{\rangle}_{t,0,\Omega^{m}_{\tau}}^{(\frac{1+\alpha}{2})}\bigg{]},$

where the positive constant $C$ is independent of $\lambda$ and $\tau$, we end
up with the estimates:

$\|\mathfrak{T}_{1}^{1}h\|_{\mathcal{C}_{\alpha}^{\alpha,\alpha/2}(\bar{\Omega}_{\tau})}\leq\frac{1}{4}\|h\|_{\mathcal{H}},\quad\|\mathfrak{T}_{1}^{2}h\|_{\mathcal{C}^{1+\alpha,\frac{1+\alpha}{2}}(\Gamma_{1,\tau})}\leq\frac{1}{4}\|h\|_{\mathcal{H}}.$

In virtue of
$\mathfrak{T}_{1}h=\\{\mathfrak{T}_{1}^{1}h,\mathfrak{T}_{1}^{2}h\\}$, the
last inequalities provide the first equality (5.20). This finishes the proof
of point (ii) and, consequence, the proof of Lemma 5.3. ∎

Lemma 5.3 and, namely, relation (5.20) means that the operators
$(I+\mathfrak{T}_{1})$ and $(I+\mathfrak{T}_{2})$ (here $I$ is the identity)
are invertible for a suitable small time $\tau$, and
$(I+\mathfrak{T}_{j})^{-1},$ $j=1,2,$ are bounded. Hence, this arrives at the
equalities

$\mathcal{A}\mathfrak{R}(I+\mathfrak{T}_{1})^{-1}h=h,\quad(I+\mathfrak{T}_{1})^{-1}\mathfrak{R}v=v,$

which imply that $\mathcal{A}$ has bounded right and left inverse operators
such that

$\mathfrak{R}(I+\mathfrak{T}_{1})^{-1}=(I+\mathfrak{T}_{2})^{-1}\mathfrak{R}=\mathcal{A}^{-1}.$

Accordingly, the unique solution of (5.14) is given by

$v=\mathcal{A}^{-1}(\bar{f},\bar{\psi}),$

or returning to the function $u$ (see (5.13)), we obtain a unique solution to
the original problem (5.1)-(5.3) for $t\in[0,\tau]$:

$u(x,t)=\mathcal{V}(x,t)+\mathcal{A}^{-1}(\bar{f},\bar{\psi}).$

The estimate of the norm of $\mathcal{A}^{-1}$ follows from the corresponding
estimates in Lemma 5.3. Finally, inequality (2) is a simple consequence of
Lemmas 5.2 and 5.3, and relations (5.13), (5.15).

As a result, we have proved Theorem 2 for a small time interval $[0,\tau]$. In
order to obtain this result in the general case, i.e., for $t\in[0,T]$, all we
need is to extend the constructed solution on the interval
$[i\tau,(i+1)\tau],$ $i=1,2,...$ while the entire $[0,T]$ is exhausted. To
this end, we collect the technique from $\S 8$, Chapter 4 in [37] with the
arguments above in the case of $t\in[0,\tau]$. This allows us to have the
classical solution $u$ on $[0,T]$, which satisfies inequality (2). This
completes the proof of Theorem 2. ∎

######  Remark 5.3.

Actually, with nonessential modifications in the arguments, the solvability of
(5.1)-(5.3) in the weighted Hölder classes
$E_{s}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{\tau})$, $s\in\mathbb{R}$,
can be proved without requirement (5.6). Indeed, as it follows from the proof
of Theorem 2, one should obtain the results similar to Lemma 5.1 in
$E_{s}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\mathbb{R}}_{+,\tau})$. To this end,
it is enough to modify arguments of Sections 2-4 in [8].

###  5.4 Proof of Theorem 3

Performing simple calculations, we deduce that

$\|u_{0}\|_{\mathcal{C}_{\alpha}^{2+\alpha}(\bar{\Omega})}\leq
C\|u_{0}\|_{E_{s}^{2+\alpha}(\bar{\Omega})}\quad\text{and}\quad\|f\|_{\mathcal{C}_{\alpha}^{\alpha,\alpha/2}(\bar{\Omega}_{T})}\leq
C\|f\|_{E_{s}^{\alpha,\alpha/2}(\bar{\Omega}_{T})}.$

Thus, these relations and assumptions H1,H3 provide the one-valued classical
solvability of (5.1)-(5.3) in
$\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})$. Taking
into account this fact, we easily conclude that equality (5.8) follows
immediately from the estimate

$\int\limits_{0}^{y}u^{2}(x,t)dx+\int\limits_{0}^{t}u^{2}(0,s)ds\leq
C_{0}yt\rho(y)\,\text{for all}\,t\in[0,T],\,y\in(0,l^{\star}],$ (5.21)

with $l^{\star}=\min\\{1/2,l\\}$, and the bounded function
$\rho(y)\in\mathcal{C}([0,l])$ satisfying the equality

$\rho(y)=O(y^{2\gamma-1})$ (5.22)

for some $\gamma\in(1/2,1)$ and for small $y$. The positive constant $C_{0}$
in (5.21) depends only on $T,l$ and the corresponding norms of the
coefficients $a_{i}$, $i=0,1,2,$ and the functions $u_{0},$ $\psi$, $f$.

Therefore, here we are left to verify (5.21). To this end, we will exploit the
following strategy consisting in two main steps. In this first one, we prove
(5.21) for time $t\in[0,\tau]$ where

$\tau=\varepsilon^{\star}y^{\gamma}$ (5.23)

with some $\varepsilon^{\star}\in(0,\min\\{1,T(l^{\star})^{-\gamma}\\})$.

After that, we demonstrate how the obtained estimate can be extended to the
whole time interval $[\tau,T]$.

Step 1: Choosing some arbitrarily $y\in(0,l^{\star}]$, we multiply equation
(5.1) by the solution $u$ and integrate over $(0,y)$. Standard calculations
(as integrating by parts in the terms related with
$\frac{\partial^{2}u}{\partial x^{2}}$ and $\frac{\partial u}{\partial x}$)
yield

$\displaystyle\frac{1}{2}\frac{d}{dt}\int\limits_{0}^{y}u^{2}(x,t)dx+\int\limits_{0}^{y}xa_{0}(x,t)\big{(}\frac{\partial
u}{\partial x}\big{)}^{2}dx+\frac{\delta_{2}}{2}u^{2}(0,t)$ (5.24)
$\displaystyle=\int\limits_{0}^{y}\bigg{[}a_{2}(x,t)+\frac{1}{2}\frac{\partial
a_{0}}{\partial x}(x,t)-\frac{1}{2}\frac{\partial a_{1}}{\partial
x}(x,t)\bigg{]}u^{2}(x,t)dx$ $\displaystyle+ya_{0}(y,t)u(y,t)\frac{\partial
u}{\partial x}(y,t)+\frac{1}{2}[a_{1}(y,t)-a_{2}(y,t)]u^{2}(y,t)$
$\displaystyle-\int\limits_{0}^{y}x\frac{\partial a_{0}}{\partial
x}(x,t)u(x,t)\frac{\partial u}{\partial
x}(x,t)dx+\int\limits_{0}^{y}f(x,t)u(x,t)dx.$

Here, we used the facts: conditions H6 and the smoothness of the solution
$u\in\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})$,
which provide the equalities:

$\displaystyle\underset{x\to 0}{\lim}xa_{0}(x,t)u(x,t)\frac{\partial
u}{\partial x}(x,t)=0,$ $\displaystyle\underset{x\to
0}{\lim}[a_{1}(x,t)-a_{0}(x,t)]u^{2}(x,t)=\delta_{2}u^{2}(0,t).$

Making use of H1 for the second term in the left-hand side of (5.24), and
treating the two last terms in the right-hand side via Cauchy inequality, we
arrive at

$\displaystyle\frac{1}{2}\frac{d}{dt}\int\limits_{0}^{y}u^{2}(x,t)dx+\delta_{0}\int\limits_{0}^{y}x\big{(}\frac{\partial
u}{\partial x}\big{)}^{2}dx+\frac{\delta_{2}}{2}u^{2}(0,t)$
$\displaystyle=\bigg{[}\underset{\bar{\Omega}_{T}}{\sup}\Bigg{|}a_{2}(x,t)+\frac{1}{2}\frac{\partial
a_{0}}{\partial x}(x,t)-\frac{1}{2}\frac{\partial a_{1}}{\partial
x}(x,t)\bigg{|}+\frac{1}{2}$
$\displaystyle+\frac{y}{\varepsilon}\underset{\bar{\Omega}_{T}}{\sup}\Bigg{|}\frac{\partial
a_{0}}{\partial
x}(x,t)\bigg{|}\bigg{]}\int\limits_{0}^{y}u^{2}(x,t)dx+y\,\underset{\bar{\Omega}_{T}}{\sup}|a_{0}(y,t)||u(y,t)|\bigg{|}\frac{\partial
u}{\partial x}(y,t)\bigg{|}$
$\displaystyle+\frac{1}{2}[\delta_{2}+|a(y,t)|]u^{2}(y,t)+\varepsilon\int\limits_{0}^{y}x\bigg{[}\frac{\partial
u}{\partial x}(x,t)\bigg{]}^{2}dx+\int\limits_{0}^{y}f^{2}(x,t)dx,$

or, choosing $\varepsilon=\frac{\delta_{0}}{2}$ and appealing to H1, H5, we
have

$\displaystyle\frac{d}{dt}\int\limits_{0}^{y}u^{2}(x,t)dx+\int\limits_{0}^{y}x\big{(}\frac{\partial
u}{\partial x}\big{)}^{2}dx+u^{2}(0,t)$ (5.25) $\displaystyle\leq
C_{0}^{1}\Big{\\{}y|u(y,t)|\bigg{|}\frac{\partial u}{\partial
x}(y,t)\bigg{|}+[1+y]u^{2}(y,t)$
$\displaystyle+\|f\|^{2}_{E_{s-1}^{\alpha,\alpha/2}(\bar{\Omega}_{T})}y^{2s-1}+\int\limits_{0}^{y}u^{2}(x,t)dx\Big{\\}},$

where the positive constant $C_{0}^{1}$ is defined with only the norms of the
coefficients $a_{i}$ and is independent of $y$ and $\tau$.

In order to evaluate the first two terms in the right-hand side in this
inequality, we appeal to estimate (2) and end up with the relations

$\displaystyle y|u(y,t)|\bigg{|}\frac{\partial u}{\partial x}(y,t)\bigg{|}\leq
C_{1}y\mathfrak{F}(u_{0},f,\psi)\bigg{[}t\bigg{|}\frac{\partial u}{\partial
t}(y,t)\bigg{|}+u_{0}(y)\bigg{]}$ $\displaystyle\leq
C_{1}y[t+y^{s}]\mathfrak{F}_{s}^{2}(u_{0},f,\psi),$

and

$u^{2}(y,t)\leq C_{1}[t^{2}+y^{2s}]\mathfrak{F}^{2}_{s}(u_{0},f,\psi),$

where we set

$\mathfrak{F}_{s}^{2}(u_{0},f,\psi)=\bigg{[}\|u_{0}\|_{E_{s}^{2+\alpha}(\bar{\Omega})}+\|f\|_{E_{s-1}^{\alpha,\alpha/2}(\bar{\Omega}_{T})}+\|\psi\|_{\mathcal{C}^{1+\alpha,\frac{1+\alpha}{2}}(\Gamma_{1,T})}\bigg{]}.$

Then, taking into account these estimates, and coming to (5.25), we finally
obtain

$\displaystyle\frac{d}{dt}\int\limits_{0}^{y}u^{2}(x,t)dx+\int\limits_{0}^{y}x\big{(}\frac{\partial
u}{\partial x}\big{)}^{2}dx+u^{2}(0,t)$ $\displaystyle\leq
C_{0}^{1}\max\\{1,C_{1}\\}\mathfrak{F}_{s}^{2}(u_{0},f,\psi)[y^{2s-1}+yt+y^{s+1}+(t^{2}+y^{2s})(1+y)]$
$\displaystyle+C_{0}^{1}\int\limits_{0}^{y}u^{2}(x,t)dx,$

or, after integrating over $(0,t)$,

$\displaystyle\int\limits_{0}^{y}u^{2}(x,t)dx+\int\limits_{0}^{t}dz\int\limits_{0}^{y}x\big{(}\frac{\partial
u}{\partial x}\big{)}^{2}dx+\int\limits_{0}^{t}u^{2}(0,z)dz$
$\displaystyle\leq
C_{0}^{1}\int\limits_{0}^{t}dz\int\limits_{0}^{y}u^{2}(x,z)dx+C_{0}^{1}\max\\{1,C_{1}\\}\mathfrak{F}_{s}^{2}(u_{0},f,\psi)$
$\displaystyle\times(t[y^{2s-1}+yt+y^{s+1}+(t^{2}+y^{2s})(1+y)]+y^{2s+1}).$

After that, the Gronwall inequality and relation (5.23) entail

$\int\limits_{0}^{y}u^{2}(x,t)dx+\int\limits_{0}^{t}dz\int\limits_{0}^{y}x\big{(}\frac{\partial
u}{\partial x}\big{)}^{2}dx+\int\limits_{0}^{t}u^{2}(0,z)dz\leq
C_{0}ty\rho_{0}(y),$ (5.26)

where we set

$\displaystyle C_{0}$
$\displaystyle:=C_{0}^{1}\max\\{1,C_{1}\\}\mathfrak{F}_{s}^{2}(u_{0},f,\psi)e^{TC_{0}^{1}},$
(5.27) $\displaystyle\rho_{0}(y)$
$\displaystyle:=y^{2s-2}+y^{2s-1}+y^{2s}+y^{s}+2\varepsilon^{\star}y^{\gamma}+(\varepsilon^{\star})^{2}y^{2\gamma-1}.$

Since $1/2<\gamma<1$, $s>1+\alpha$ and $y<1/2$, we easily conclude that
$\rho_{0}(y)$ meets requirement (5.22), and estimate (5.26) leads to (5.21) if
$t\in[0,\tau]$.

Moreover, inequality (5.26) and the mean value theorem provides key estimate
which will be exploited in the second step of our arguments:

$u^{2}(0,t^{\star})\leq C_{0}y\rho_{0}(y)$ (5.28)

with some $t^{\star}\in(0,\tau)$. For simplicity consideration, we put
$t^{\star}=\tau/2$.

Step 2: At this point we proceed the technique which allows us to extend
estimate (5.21) to the interval $[\tau,T]$. First of all, we extend this
estimate from $[0,\tau]$ to $[\tau,3\tau/2]$. To this end, we introduce a new
variable $\bar{t}=t-\tau/2$, $\bar{t}\in[0,\tau],$ if $t\in[\tau/2,3\tau/2]$,
and denote

$\displaystyle\bar{a}_{i}$
$\displaystyle:=a_{i}(x,\bar{t}+\tau/2),\,i=0,1,2,\quad\quad\bar{b}_{j}:=b_{j}(x,\bar{t}+\tau/2),\,j=1,2,$
$\displaystyle\bar{\mathcal{L}}$
$\displaystyle:=x\bar{a}_{0}\frac{\partial^{2}}{\partial
x^{2}}+\bar{a}_{1}\frac{\partial}{\partial
x}+\bar{a}_{0},\qquad\bar{\mathcal{M}}:=\bar{b}_{1}\frac{\partial}{\partial
x}+\bar{b}_{2},$ $\displaystyle u$
$\displaystyle(x,\tau/2):=u_{\tau/2}(x),\qquad\qquad\bar{u}(x,\bar{t}):=u(x,\bar{t}+\tau/2),$
$\displaystyle\bar{f}$
$\displaystyle(x,\bar{t}):=f(x,\bar{t}+\tau/2),\qquad\quad\bar{\psi}(x,\bar{t}):=\psi(x,\bar{t}+\tau/2).$

It is apparent that the coefficients $\bar{a}_{i},$ $\bar{b}_{j}$ and the
functions $\bar{f},\bar{\psi}$ and $u_{\tau/2}(x)=\bar{u}(x,0)$ meet the
requirements of Theorem 2 and condition H6, and, besides, $\bar{f}$ and
$\bar{\psi}$ satisfy H5. Thus, recasting the arguments leading to estimate
(5.26) (with changing $u_{0}(x)$ by $u_{\tau/2}(x)$) in the case of the
problem

$\begin{cases}\frac{\partial\bar{u}}{\partial\bar{t}}-\bar{\mathcal{L}}\bar{u}=\bar{f}\quad\text{in}\quad\Omega_{\tau},\\\
\mathcal{M}\bar{u}=\bar{\psi}\quad\text{on}\quad\Gamma_{1,\tau},\\\
\bar{u}(x,0)=u_{\tau/2}(x)\quad\text{in}\quad\bar{\Omega},\end{cases}$

we conclude

$\int\limits_{0}^{y}\bar{u}^{2}(x,\bar{t})dx+\int\limits_{0}^{\bar{t}}dz\int\limits_{0}^{y}x\bigg{(}\frac{\partial\bar{u}}{\partial
x}\bigg{)}^{2}dx+\int\limits_{0}^{\bar{t}}\bar{u}^{2}(0,z)dz\leq
C_{0}\bar{t}y\bar{\rho}_{0}(y).$ (5.29)

Here (in virtue of (5.28)) the function

$\bar{\rho}_{0}(y)=y^{2s-2}+(1+(\varepsilon^{\star})^{2})y^{2\gamma-1}+(1+2\varepsilon^{\star})y^{\gamma}+2y^{2\gamma}$

satisfies (5.22). Besides, due to $y<1/2$ and $1/2<\gamma<1<s$, there holds

$\bar{\rho}_{0}(y)>\rho_{0}(y).$

Finally, collecting estimate (5.26) with (5.29) (where we come back from
$\bar{u}$ to $u$), we deduce inequality (5.21) for $t\in[0,3\tau/2]$, with
$\rho(y):=\bar{\rho}_{0}(y),$ and $C_{0}$ defining in (5.27). Other words, we
have extended estimate (5.21) from $[0,\tau]$ to $[\tau,3\tau/2]$. By the same
token, we repeat the procedure to continue the obtain estimate on the other
intervals until the entire $[0,T]$ is exhausted. Finally, keeping in mind that
$y$ was arbitrarily point in $[0,l^{\star}],$ estimate (5.21) holds for all
$y\in[0,l^{\star}]$ and $t\in[0,T]$. This completes the proof of Theorem 3.∎

###  5.5 Conservation Law

In this section we discuss the properties of the solution to problem
(5.1)-(5.3) which follow from Theorems 2 and 3. Namely, we describe sufficient
conditions on the given data in (5.1)-(5.3) which ensure the fulfillment of
the conservation law to the solution
$u\in\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})$:

$\frac{d}{dt}\int\limits_{\Omega}u(x,t)dx=0\quad\text{for all}\quad
t\in[0,T].$ (5.30)

First, we state additional assumptions on the coefficients and the right-hand
side of the model.

H7: Let
$a_{0}\in\mathcal{C}_{\alpha}^{\alpha,\alpha/2}(\bar{\Omega}_{\tau})\cap\mathcal{C}([0,T],\mathcal{C}^{2}(\bar{\Omega})),$
and the following equalities hold:

$\displaystyle x\frac{\partial^{2}a_{0}}{\partial x^{2}}-\frac{\partial
a_{1}}{\partial x}+2\frac{\partial a_{0}}{\partial x}+a_{2}=0\quad\text{for
each }(x,t)\in\bar{\Omega}_{T},$ $\displaystyle
b_{1}(x,t)=xa_{0}(x,t),\,b_{2}(x,t)=-a_{0}(x,t)-x\frac{\partial
a_{0}}{\partial x}(x,t)+a_{1}(x,t)\,\text{on}\,\Gamma_{1,T},$
$\displaystyle\psi(l,t)=-\int\limits_{0}^{l}f(x,t)dx,\quad t\in[0,T].$

######  Theorem 4.

Let $T>0$ be fixed and assumptions of Theorem 3 and H7 hold. Then the
classical solution
$u\in\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})$
satisfies equality (5.30) for all $t\in[0,T]$.

###### Proof.

First, we recall that Theorem 3 ensures the existence of the classical
solution
$u\in\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})$
which vanishes at $x=0$ for all $t\in[0,T]$. Taking into account this fact,
and integrating equation (5.1) over $\Omega$, we deduce

$\displaystyle\frac{d}{dt}\int\limits_{0}^{l}u(x,t)dx$
$\displaystyle=\int\limits_{0}^{l}\Big{[}\frac{\partial^{2}}{\partial
x^{2}}(xa_{0})-\frac{\partial a_{1}}{\partial
x}+a_{2}\Big{]}u(x,t)dx+\int\limits_{0}^{l}f(x,t)dx$
$\displaystyle+\underset{x\to l}{\lim}\bigg{[}xa_{0}\frac{\partial u}{\partial
x}-u\frac{\partial}{\partial x}(xa_{0})+a_{1}u\bigg{]}.$

Then, taking advantage of H7 to the terms in the right-hand side of this
equality, we arrive at the desired relation (5.30). This completes the proof
of this claim. ∎

######  Remark 5.4.

As it follows from the proof of Theorem 4, the last three equalities in H7 can
be changed by the more general condition:

$\int\limits_{0}^{l}f(x,t)dx+\underset{x\to
l}{\lim}\bigg{[}xa_{0}\frac{\partial u}{\partial x}-u\frac{\partial}{\partial
x}(xa_{0})+a_{1}u\bigg{]}=0.$

###  5.6 A priori estimates in the Sobolev spaces

Here we obtain a priori estimates in $L^{2}$ norms of solution to problem
(5.1)-(5.3). We start with state additional assumption on the coefficients of
equation (5.1).

H8: We require that

$a_{1}(0,t)-a_{0}(0,t)\geq 0\quad\text{and}\quad\frac{\partial a_{0}}{\partial
x}(0,t)<0\quad\text{for all}\quad t\in[0,T].$

######  Lemma 5.4.

Let $T>0$ be fixed, assumptions H1,H8 hold. Moreover, we assume that
$a_{0}\in\mathcal{C}([0,T],\mathcal{C}^{2}(\bar{\Omega})),$
$a_{1}\in\mathcal{C}([0,T],\mathcal{C}^{1}(\bar{\Omega})),$ $b_{1}$ and
$b_{2}$ meet requirements H7. If a solution $u$ of problem (5.1)-(5.2) with
homogenous condition (5.3) satisfies the equality

$\underset{x\to 0}{\lim}xa_{0}u\frac{\partial u}{\partial x}=0\quad\text{for
each}\quad t\in[0,T],$ (5.31)

then the following estimate is fulfilled

$\displaystyle\|u\|_{\mathcal{C}([0,T],L^{2}(\Omega))}+\|x^{1/2}\partial
u/\partial x\|_{L^{2}(\Omega_{T})}+\|u(0,\cdot)\|_{L^{2}(0,T)}$
$\displaystyle\leq C[\|u_{0}\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(\Omega_{T})}]$

with the positive constant $C$ depending only on $T,$ $\Omega$ and
corresponding norms of the coefficients $a_{i}$.

###### Proof.

We multiply equation (5.1) by $u(x,t)$ and integrate over $\Omega=[0,l]$.
Taking into account H1 and the smoothness of $a_{i},$ we have

$\displaystyle\frac{1}{2}\frac{d}{dt}\int\limits_{0}^{l}u^{2}(x,t)dx+\delta_{0}\int\limits_{0}^{l}x\big{(}\frac{\partial
u}{\partial x}\big{)}^{2}dx+L_{0}(u)$ $\displaystyle\leq-
L_{l}(u)+\int_{0}^{l}f^{2}dx+\frac{1}{2}\int\limits_{0}^{l}\bigg{[}2-\frac{\partial
a_{1}}{\partial x}+\frac{\partial^{2}(xa_{0})}{\partial
x^{2}}\bigg{]}u^{2}dx,$

where we put

$\displaystyle L_{0}(u)$ $\displaystyle=\underset{x\to
0}{\lim}\bigg{[}xa_{0}u\frac{\partial u}{\partial
x}+\frac{1}{2}\bigg{(}a_{1}-a_{0}-x\frac{\partial a_{0}}{\partial
x}\bigg{)}u^{2}\bigg{]},$ $\displaystyle L_{l}(u)$
$\displaystyle=\underset{x\to l}{\lim}\bigg{[}-xa_{0}u\frac{\partial
u}{\partial x}-\frac{1}{2}\bigg{(}a_{1}-a_{0}-x\frac{\partial a_{0}}{\partial
x}\bigg{)}u^{2}\bigg{]}.$

Assumptions on the coefficients $b_{1}$, $b_{2}$ and conditions H8 with (5.31)
provide inequalities:

$L_{0}(u)\geq 0\quad\text{and}\quad-L_{l}(u)<0.$

Appealing to these relations and conditions on $a_{i}$, we end up with the
estimate

$\frac{d}{dt}\int\limits_{0}^{l}u^{2}(x,t)dx+\int\limits_{0}^{l}x\big{(}\frac{\partial
u}{\partial x}\big{)}^{2}dx+L_{0}(u)\leq
C\bigg{[}\int_{0}^{l}f^{2}dx+\int\limits_{0}^{l}u^{2}dx\bigg{]},$

where the positive $C$ depends only on the norms of $a_{i}$, $i=0,1,2.$

Finally, Gronwall inequality arrives at the desired estimate. ∎

######  Remark 5.5.

It is apparent that the classical solution constructed in Theorem 2 satisfies
assumption (5.31) of Lemma 5.1.

###  5.7 Conclusions of Theorems 2-4 and Lemma 5.4

Coming to original problem (1.1)–(1.3), we conclude that this problem can be
rewritten in the form of (5.1)-(5.3) with the following coefficients and the
right-hand sides:

$\displaystyle u(x,t)$ $\displaystyle=p(x,t),\quad f(x,t)=\psi(x,t)=0,\quad
u_{0}(x)=p_{0}(x),$ $\displaystyle a_{0}(x,t)$
$\displaystyle=\frac{R_{0}+1-R_{0}x}{2N},\,a_{1}(x,t)=\frac{R_{0}+1-R_{0}x}{N}-x(R_{0}-1-R_{0}x),$
$\displaystyle a_{2}(x,t)$
$\displaystyle=-\frac{R_{0}}{N}-R_{0}+1+2R_{0}x,\quad
b_{1}(x,t)=x\frac{R_{0}+1-R_{0}x}{2N},$ $\displaystyle b_{2}(x,t)$
$\displaystyle=-a_{0}(x,t)-x\frac{\partial a_{0}}{\partial
x}(x,t)+a_{1}(x,t).$

These representations tell us that all assumptions of Theorems 2, 3 and 4,
Lemma 5.4 hold, and we can assert the following.

######  Theorem 5.

Let $T>0$ be fixed and $p_{0}$ meet requirements of Theorem 2. Then original
problem (1.1)–(1.3) has a unique classical solution
$p\in\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})$
satisfying estimates:

$\displaystyle\|p\|_{\mathcal{C}_{\alpha}^{2+\alpha,\frac{2+\alpha}{2}}(\bar{\Omega}_{T})}\leq
C\|p_{0}\|_{\mathcal{C}_{\alpha}^{2+\alpha}(\bar{\Omega})},$
$\displaystyle\|p\|_{\mathcal{C}([0,T],L^{2}(\Omega))}+\|\sqrt{x}\partial
p/\partial x\|_{L^{2}(\Omega_{T})}+\|p(0,\cdot)\|_{L^{2}(0,T)}\leq
C\|p_{0}\|_{L^{2}(\Omega)}.$

If, in addition, $p_{0}$ meets assumptions of Theorem 3, then the solution $p$
vanishes at $x=0$ for all $t\in[0,T]$ and the conservation law to this
solution holds, i.e.

$\frac{d}{dt}\int\limits_{\Omega}p(x,t)dx=0\quad\text{for all}\quad
t\in[0,T].$

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