# A family of steady two-phase generalized Forchheimer flows and their linear
stability analysis

Luan T. Hoang, Akif Ibragimov and Thinh T. Kieu† Department of Mathematics and
Statistics, Texas Tech University, Box 41042 Lubbock, TX 79409–1042, U.S.A.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>† Corresponding
author

###### Abstract.

We model multi-dimensional two-phase flows of incompressible fluids in porous
media using generalized Forchheimer equations and the capillary pressure.
Firstly, we find a family of steady state solutions whose saturation and
pressure are radially symmetric and velocities are rotation-invariant. Their
properties are investigated based on relations between the capillary pressure,
each phase’s relative permeability and Forchheimer polynomial. Secondly, we
analyze the linear stability of those steady states. The linearized system is
derived and reduced to a parabolic equation for the saturation. This equation
has a special structure depending on the steady states which we exploit to
prove two new forms of the lemma of growth of Landis-type in both bounded and
unbounded domains. Using these lemmas, qualitative properties of the solution
of the linearized equation are studied in details. In bounded domains, we show
that the solution decays exponentially in time. In unbounded domains, in
addition to their stability, the solution decays to zero as the spatial
variables tend to infinity. The Bernstein technique is also used in estimating
the velocities. All results have a clear physical interpretation.

Dedicated to the Memory of Evgenii Mikhailovich Landis (1921–1997)

###### Contents

  1. 1 Introduction
  2. 2 Special steady states
  3. 3 Linearization
  4. 4 Case of bounded domain
  5. 5 Case of unbounded domain
    1. 5.1 Maximum principle for unbounded domain
    2. 5.2 Lemma of growth in spatial variables
  6. A

##  1\. Introduction

In this paper, we study two-phase flows of incompressible fluids in porous
media with each phase subjected to a Forchheimer equation. Forchheimer
equations are often used by engineers to take into account the deviation from
Darcy’s law in case of high velocity, see e.g. [4, 20]. The standard
Forchheimer equations are two-term law with quadratic nonlinearity, three-term
law with cubic nonlinearity, and power law with a non-integer power less than
two (see again [4, 20]). These models are extended to the generalized
Forchheimer equation of the form

$g(|\mathbf{u}|)\mathbf{u}=-\nabla p,$ (1.1)

where $\mathbf{u}(\mathbf{x},t)$ is the velocity field, $p(\mathbf{x},t)$ is
the pressure, and $g(s)$ is a generalized polynomial of arbitrary order
(integer or non-integer) with positive coefficients. This equation was
intensively analyzed for single-phase flows from mathematical and applied
point of view in [3, 11, 12, 13, 15]. Its study for two-phase flows was later
initiated in [14]. Regarding two-phase flows in porous media, it is always a
challenging subject even for Darcy’s law. Their models involve a complicated
system of nonlinear partial differential equations (PDE) for pressures,
velocities, densities and saturations with many parameters such as porosity,
relative permeability functions and capillary pressure function. Current
analysis of two-phase Darcy flows in literature is mainly focused on the
existence of weak solutions [8, 7, 6] and their regularity [17, 18, 1, 2, 9].
However, questions about the stability and dynamics are not answered. The
nonlinearity of the relative permeabilities and capillary pressure and their
imprecise characteristics near the extreme values make it hard to analyze the
modeling PDE system. The two-phase generalized Forchheimer flows are even more
difficult due to the additional nonlinearity in the momentum equation. For
example, unlike the Darcy flows, there is no Kruzkov-Sukorjanski
transformation [17] to convert the system to a convenient form for the total
velocity. Therefore, new methods are needed for the Forchheimer flows. In
[14], we study the one-dimensional case using a novel approach. We will
develop the techniques in [14] further to investigate the multi-dimensional
case in this article.

We consider $n$-dimensional two-phase flows in porous media with constant
porosity $\phi$ between $0$ and $1$. Here the dimension $n$ is greater or
equal to $2$, even though in practice we only need $n=2,3$. Each position
$\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}$ in the medium is
considered to be occupied by two fluids called phase 1 (for example, water)
and phase 2 (for example, oil).

Saturation, density, velocity, and pressure for each $i$th-phase ($i=1,2$) are
$S_{i}\in[0,1]$, $\rho_{i}\geq 0$, $\bf u_{i}\in\mathbb{R}^{n}$ and
$p_{i}\in\mathbb{R}$, respectively. The saturation functions naturally satisfy

$S_{1}+S_{2}=1.$ (1.2)

Each phase’s velocity is assumed to obey the generalized Forchheimer equation:

$g_{i}(|\mathbf{u}_{i}|)\mathbf{u}_{i}=-\tilde{f}_{i}(S_{i})\nabla p_{i},\quad
i=1,2,$ (1.3)

where $\tilde{f}_{i}(S_{i})$ is the relative permeability for the $i$th phase,
and $g_{i}$ is of the form

$g_{i}(s)=a_{0}s^{\alpha_{0}}+a_{1}s^{\alpha_{1}}+\ldots+a_{N}s^{\alpha_{N}},\quad
s\geq 0,$ (1.4)

with $N\geq 0$, $a_{0}>0,$ $a_{1},\ldots a_{N}\geq 0$,
$\alpha_{0}=0<\alpha_{1}<\ldots\alpha_{N}$, all $\alpha_{1},\ldots\alpha_{N}$
are real numbers. The above $N$, $a_{j}$, $\alpha_{j}$ in (1.4) depend on each
$i$. We call $g_{i}(s)$ in (1.4) the Forchheimer polynomial of (1.3).

Conservation of mass commonly holds for each of the phases:

$\partial_{t}(\phi\rho_{i}S_{i})+{\rm div}(\rho_{i}\mathbf{u}_{i})=0,\quad
i=1,2.$ (1.5)

Due to incompressibility of the phases, i.e. $\rho_{i}=const.>0$, Eq. (1.5) is
reduced to

$\phi\partial_{t}S_{i}+{\rm div}\,\mathbf{u}_{i}=0,\quad i=1,2.$ (1.6)

Let $p_{c}$ be the capillary pressure between two phases, more specifically,

$p_{1}-p_{2}=p_{c}.$ (1.7)

Hereafterward, we denote $S=S_{1}$. The relative permeabilities and capillary
pressure are re-denoted as functions of $S$, that is,
$\tilde{f}_{1}(S_{1})=f_{1}(S)$, $\tilde{f}_{2}(S_{2})=f_{2}(S)$ and
$p_{c}=p_{c}(S)$. Then (1.3) and (1.7) become

$g_{i}(|{\mathbf{u}_{i}}|){\mathbf{u}_{i}}=-f_{i}(S)\nabla p_{i},\quad i=1,2,$
(1.8) $p_{1}-p_{2}=p_{c}(S).$ (1.9)

By scaling time, we can mathematically consider, without loss of generality,
$\phi=1$.

By (1.2) and (1.6):

$S_{t}=-{\rm div}\,{\bf u}_{1},\quad S_{t}={\rm div}\,{\bf u}_{2}.$ (1.10)

For $i=1,2$, define the function $\mathbf{G}_{i}({\bf u})=g_{i}(|{\bf u}|){\bf
u}$ for ${\bf u}\in\mathbb{R}^{n}$. Then by (1.8),

$\mathbf{G}_{i}({\bf u}_{i})=-f_{i}(S)\nabla p_{i},\quad\text{or,}\quad\nabla
p_{i}=-\frac{\mathbf{G}_{i}({\bf u}_{i})}{f_{i}(S)}.$ (1.11)

Taking gradient of the equation (1.9) we have

$\nabla p_{1}-\nabla p_{2}=p_{c}^{\prime}(S)\nabla S.$ (1.12)

Substituting (1.11) into (1.12) yields

$\frac{g_{2}(|\mathbf{u}_{2}|)\mathbf{u}_{2}}{f_{2}(S)}-\frac{g_{1}(|\mathbf{u}_{1}|)\mathbf{u}_{1}}{f_{1}(S)}=p_{c}^{\prime}(S)\nabla
S,$

hence

$F_{2}(S)g_{2}(|{\bf
u}_{2}|)\mathbf{u}_{2}-F_{1}(S)g_{1}(|\mathbf{u}_{1}|)\mathbf{u}_{1}=\nabla
S,$

where

$F_{i}(S)=\frac{1}{p_{c}^{\prime}(S)f_{i}(S)},\quad i=1,2.$ (1.13)

In summary we study the following PDE system for $\mathbf{x}\in\mathbb{R}^{n}$
and $t\in\mathbb{R}$:

$\displaystyle 0\leq S=S(\mathbf{x},t)\leq 1,$ (1.14a) $\displaystyle
S_{t}=-{\rm div}\,{\mathbf{u}}_{1},$ (1.14b) $\displaystyle S_{t}={\rm
div}\,{\mathbf{u}}_{2},$ (1.14c) $\displaystyle\nabla
S=F_{2}(S)\mathbf{G}_{2}(\mathbf{u}_{2})-F_{1}(S)\mathbf{G}_{1}(\mathbf{u}_{1}).$
(1.14d)

This paper is devoted to studying system (1.14). We will obtain a family of
non-constant steady states with particular geometric properties. Specifically,
the saturation and pressure are functions of $|\mathbf{x}|$, while each
phase’s velocity is $\mathbf{x}$ multiplied by a radial scalar function. Their
properties, particularly, the behavior as $|\mathbf{x}|\to\infty$, will be
obtained. For the stability study, we linearize system (1.14) at these steady
states. We deduce from this linearized system a parabolic equation for the
saturation. In bounded domains, we establish the lemma of growth in time and
prove the exponential decay of its solutions in sup-norm as time $t\to\infty$.
In unbounded domains, we prove the maximum principle and the stability.
Furthermore, we show that the solutions go to zero as the spatial variables
tend to infinity.

The paper is organized as follows. In section 2 we find the family of non-
constant steady states described above. Various sufficient conditions are
given for their existence in unbounded domains (Theorems 2.2). Their
asymptotic behavior as $|\mathbf{x}|\to\infty$ is studied in details. In
section 3, we linearize the originally system at the obtained steady states.
We derive a parabolic equation for the saturation which will become the focus
of our study. It is then converted to a convenient form for the study of sup-
norm of solutions. Such a conversion is possible thanks to the special
structure of the equation and of the steady states. Preliminary properties of
the coefficient functions of this linearized equation are presented. Section 4
is focused on the study of the linearized equation for saturation in bounded
domains. We prove the asymptotic stability results (Theorems 4.8 and 4.9) by
utilizing a variation of Landis’s lemma of growth in time variable (Lemma
4.3). The Bernstein’s a priori estimate technique is used in proving interior
continuous dependence of the velocities on the initial and boundary data
(Proposition 4.7). In section 5, we study the linearized equation in an
(unbounded) outer domain. The maximum principle (Theorem 5.2) is proved and
used to obtain the stability of the zero solution (Theorems 5.10 and 5.11,
part (ii)). We also prove a lemma of growth in the spatial variables (Lemma
5.5) by constructing particular barriers (super-solutions) using the specific
structure of the linearized equation for saturation (Lemma 5.4). Using this,
we prove a dichotomy theorem on the solution’s behavior (Lemma 5.6), and
ultimately show that the solution, on any finite time interval, decays to zero
as $|\mathbf{x}|\to\infty$. For time tending to infinity, we find an
increasing, continuous function $r(t)>0$ with $r(t)\to\infty$ as $t\to\infty$
such that along any curve $\mathbf{x}(t)$ with $|\mathbf{x}(t)|\geq r(t)$, the
solution goes to zero. (See Theorems 5.10 and 5.11, part (iii).) It is worth
mentioning that the asymptotic stability in sup-norm in section 4 and behavior
of the solution at spatial infinity have their own merits in the qualitative
theory of linear parabolic equations.

##  2\. Special steady states

In this section we find and study steady states which processes some symmetry.

Assume $p_{i}$ and $S$ are radial functions. We can write

$p_{i}({\bf x},t)=p_{i}(r,t),\quad S({\bf x},t)=S(r,t),\quad\text{where
}r=|\mathbf{x}|=\big{(}\sum_{i=1}^{n}x_{i}^{2}\big{)}^{1/2}.$ (2.1)

Denote ${\bf e}_{r}=\mathbf{x}/|\mathbf{x}|$. By (1.8),

$g_{i}(|\mathbf{u}_{i}|){\mathbf{u}_{i}}=-f_{i}(S)\frac{\partial
p_{i}}{\partial r}\cdot\frac{\bf x}{r}=-f_{i}(S)\frac{\partial p_{i}}{\partial
r}{\bf e}_{r}.$ (2.2)

Noting in (2.2) that $f_{i}(S)\frac{\partial p_{i}}{\partial r}$ is radial,
then clearly $|{\bf u}_{i}|$ is also radial and we have

${\bf u}_{i}=u_{ir}{\bf e}_{r},\quad\text{where }u_{ir}={\bf u}_{i}\cdot{\bf
e}_{r}=u_{ir}(r,t).$ (2.3)

Therefore

${\rm div}\,\mathbf{u}_{i}=\frac{1}{r^{n-1}}\frac{\partial}{\partial
r}(r^{n-1}u_{ir})$ (2.4)

and, from (1.14d),

$F_{2}(S)g_{2}(|\mathbf{u}_{2}|)\mathbf{u}_{2}-F_{1}(S)g_{1}(|\mathbf{u}_{1}|){\mathbf{u}_{1}}=\nabla
S=\frac{\partial S}{\partial r}{\bf e}_{r}.$ (2.5)

Taking the scalar product of both sides of (2.5) with ${\bf e}_{r}$ we obtain

$G_{2}(u_{2r})F_{2}(S)-G_{1}(u_{1r})F_{1}(S)=\frac{\partial S}{\partial r},$
(2.6)

where

$G_{i}(u)=g_{i}(|u|)u\quad\text{for }u\in\mathbb{R}.$ (2.7)

We will study $S(r,t)$ and $u_{i}(r,t)\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}u_{ir}$ ($i=1,2$) as functions of
independent variables $(r,t)\in(0,\infty)\times\mathbb{R}$. The system (1.14)
becomes

$\displaystyle 0\leq S\leq 1,$ (2.8a) $\displaystyle\frac{\partial S}{\partial
t}=-r^{1-n}\frac{\partial}{\partial r}(r^{n-1}u_{1}),$ (2.8b)
$\displaystyle\frac{\partial S}{\partial t}=r^{1-n}\frac{\partial}{\partial
r}(r^{n-1}u_{2}),$ (2.8c) $\displaystyle\frac{\partial S}{\partial
r}=G_{2}(u_{2})F_{2}(S)-G_{1}(u_{1})F_{1}(S).$ (2.8d)

We make basic assumptions on the relative permeabilities and capillary
pressure.

Assumption A.

$f_{1},f_{2}\in C([0,1])\cap C^{1}((0,1)),$ (2.9a) $f_{1}(0)=0,\quad
f_{2}(1)=0,$ (2.9b) $f_{1}^{\prime}(S)>0,\quad f_{2}^{\prime}(S)<0\text{ on
}(0,1).$ (2.9c)

Assumption B.

$p_{c}^{\prime}\in C^{1}((0,1)),\quad p^{\prime}_{c}(S)>0\text{ on }(0,1).$
(2.10)

We find steady state solutions $(S,u_{1},u_{2})=(S(r),u_{1}(r),u_{2}(r))$ for
system (2.8) in the domain $[r_{0},\infty)$ for a fixed $r_{0}>0$.

From (2.8b), we have $\frac{d}{dr}(r^{n-1}u_{i})=0$, hence

$u_{i}(r)=c_{i}r^{1-n},\quad\text{where }c_{i}=const.,\quad i=1,2.$ (2.11)

Substituting (2.11) into (1.14d) yields

$S^{\prime}=G_{2}(c_{2}r^{1-n})F_{2}(S)-G_{1}(c_{1}r^{1-n})F_{1}(S)\quad\text{for
}r>r_{0}.$ (2.12)

The rest of this section is devoted to studying the following initial value
problem with constraints:

$S^{\prime}=F(r,S(r))\quad\text{for }r>r_{0},\quad S(r_{0})=s_{0},\quad
0<S(r)<1.$ (2.13)

where $s_{0}$ is always a number in $(0,1)$ and

$F(r,S(r))=G_{2}(c_{2}r^{1-n})F_{2}(S)-G_{1}(c_{1}r^{1-n})F_{1}(S).$

First we state a standard local existence theorem.

######  Theorem 2.1.

There exist a maximal interval of existence $[r_{0},R_{\rm max})$, where
$R_{\rm max}\in(r_{0},\infty]$, and a unique solution $S\in
C^{1}([r_{0},R_{\rm max}))$ of (2.13) on $(r_{0},R_{\rm max})$. Moreover, if
$R_{\rm max}$ is finite then either

$\lim_{r\to R_{\rm max}^{-}}S(r)=0\quad\text{ or}\lim_{r\to R_{\rm
max}^{-}}S(r)=1.$ (2.14)

###### Proof.

Under Assumption B, $F(r,S)$ is continuous and locally Lipschitz for the
second variable for all $r\in(r_{0},\infty)$, $S\in(0,1)$. The existence of
the unique solution $S\in C^{1}([r_{0},R_{\rm max});(0,1))$ on the maximal
interval $[0,R_{\rm max})$ is classical.

Assume $R_{\max}<\infty$. For given
$0<\varepsilon\leq\varepsilon_{0}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\min\\{1/4,R_{\rm max}/2\\}$, let
$K=[r_{0},R_{\max}]\times[\varepsilon,1-\varepsilon]$. We claim that there is
$R_{\varepsilon}\geq r_{0}$ such that $(r,S(r))\notin K$ for all
$r\in(R_{\varepsilon},R_{\max})$. Suppose not, then there is the sequence
$r_{i}\to R_{\max}$ as $i\to\infty$ such that $(r_{i},S(r_{i}))\in K$ for all
$i$. Choose $N>0$ such that for all $i\geq N$,

$\\{(r,S):|r-r_{i}|\leq\varepsilon/2\text{ and
}|S-S(r_{i})|\leq\varepsilon/2\\}\subset K^{\prime},$

where
$K^{\prime}=[r_{0},R_{\max}+\varepsilon/2]\times[\varepsilon/2,1-\varepsilon/2]$.
According to the local Existence and Uniqueness theorem (Theorem 3.1 p. 18
in[10]) the solution starting at point $(r_{i},S(r_{i}))$ exists on the
interval $[r_{i},r_{i}+d)$, where
$d=\min\\{\frac{1}{L},\frac{\varepsilon}{2},\frac{\varepsilon}{2M}\\}$ with
$M=\max_{K^{\prime}}|F(r,S)|$ and $L$ being the Lipschitz constant for $F$ in
$K^{\prime}$. Note that $d$ is independent of $i$. Let $i$ be sufficiently
large such that $r_{i}+d>R_{\max}$, then solution $S(r)$ exists beyond
$R_{\max}$ which is a contradiction to maximality of $R_{\max}$. Hence our
claim is true. Now using the continuity of $S(r)$ we have

$\text{either }S(r)>1-\varepsilon,\forall r\in(R_{\varepsilon},R_{\max})\text{
or }S(r)<\varepsilon,\forall r\in(R_{\varepsilon},R_{\max}).$ (2.15)

In particular, for $\varepsilon=\varepsilon_{0}$ we have either (a)
$S(r)>1-\varepsilon_{0},\forall r\in(R_{\varepsilon_{0}},R_{\max})$, or (b)
$S(r)<\varepsilon_{0},\forall r\in(R_{\varepsilon_{0}},R_{\max})$. In case
(a), it is easy to see from (2.15) that for $0<\varepsilon<\varepsilon_{0}$,
$S(r)>1-\varepsilon,\forall r\in(R^{\prime}_{\varepsilon},R_{\max})$ where
$R^{\prime}_{\varepsilon}=\max\\{R_{\varepsilon_{0}},R_{\varepsilon}\\}$.
Thus, $\lim_{r\to R_{\max}^{-}}S(r)=1$. Similarly, for the case (b) we have
$\lim_{r\to R_{\max}^{-}}S(r)=0$. The proof is complete. ∎

Next, we are interested in the case $R_{\rm max}=\infty$. First, we find
sufficient conditions for that. We need to make the following assumptions on
the relative permeabilities and capillary pressure:

$\lim_{S\to 0}p^{\prime}_{c}(S)f_{1}(S)=\lim_{S\to
1}p^{\prime}_{c}(S)f_{2}(S)=+\infty.$ (2.16)

These are our interpretation of experimental data (c.f. [4]), especially of
those obtained in [5]. They cover certain scenarios of two-phase fluids in
reality.

By (1.13) and (2.16), $F_{1}$ and $F_{2}$ can now be extended to functions of
class $C([0,1])\cap C^{1}((0,1))$ and satisfy

$F_{1}(0)=F_{1}(1)=F_{2}(0)=F_{2}(1)=0.$ (2.17)

Therefore the right hand side of (1.14d) is well-defined for all $S\in[0,1]$.
Note that

$\lim_{S\to 0^{+}}\frac{F_{1}(S)}{F_{2}(S)}=\lim_{S\to
1^{-}}\frac{F_{2}(S)}{F_{1}(S)}=\infty.$ (2.18)

The following additional conditions on $F_{1}$ and $F_{2}$ will be referred to
in our considerations:

$\limsup_{S\to 0^{+}}F_{1}^{\prime}(S)<\infty,$ (2.19) $\liminf_{S\to
1^{-}}F^{\prime}_{1}(S)>-\infty.$ (2.20) $\liminf_{S\to
1^{-}}F^{\prime}_{2}(S)>-\infty.$ (2.21) $\limsup_{S\to
0^{+}}F_{2}^{\prime}(S)<\infty,$ (2.22)

######  Theorem 2.2.

Assume (2.16) and $c_{1}^{2}+c_{2}^{2}>0$. Then $R_{\max}$ in Theorem 2.1 is
infinity, that is, the solution $S(r)$ of (2.13) exists on $[r_{0},\infty)$,
in the following cases

Case 1a. $c_{2}\leq 0<c_{1}$ and (2.19). Case 1b. $c_{1}=0>c_{2}$ and (2.22).

Case 2a. $c_{1}\leq 0<c_{2}$ and (2.21). Case 2b. $c_{2}=0>c_{1}$ and (2.20).

Case 3. $c_{1},c_{2}>0$ and (2.19), (2.21).

Case 4. $c_{1},c_{2}<0$.

###### Proof.

Suppose $R_{\max}<\infty$. We consider the following four cases.

Case 1. $c_{2}\leq 0\leq c_{1}$. We provide the proof of Case 1a, while Case
1b can be proved similarly. We have $F(r,S)<0$ for all $r\in[r_{0},R_{\max})$.
Thus $S^{\prime}<0$ for all $r\in[r_{0},R_{\max})$. By Theorem 2.1,

$\lim_{r\to R_{\rm max}^{-}}S(r)=0.$ (2.23)

Note that $G_{1}(c_{1}r^{1-n})$ and $G_{2}(c_{2}r^{1-n})$ are bounded, and
$G_{1}(c_{1}r^{1-n})$ is bounded below by a positive number on $[r_{0},R_{\rm
max}]$. Combining these facts with relation (2.18), we infer that there are
$\delta>0$ and $C_{1},C_{2}>0$ such that for $r\in[0,R_{\rm max})$ and
$S\in(0,\delta)$,

$-C_{1}F_{1}(S)\leq F(r,S)\leq-C_{2}F_{1}(S).$ (2.24)

By (2.23), there is $r_{1}\in(0,R_{\rm max})$ such that $S(r)<\delta$ for all
$r\in[r_{1},R_{\rm\max})$. Define $Y(r)=F_{1}(S(r))$. By (2.19), there are
$\tilde{r}\in(r_{1},R_{\max})$ and $C_{3}>0$

$F^{\prime}_{1}(S(r))<C_{3}\text{ for all }r\in(\tilde{r},R_{\rm max}).$
(2.25)

For $r\in(\tilde{r},R_{\max})$, using (2.24) we have

$Y^{\prime}(r)=F^{\prime}_{1}(S)S^{\prime}=F^{\prime}_{1}(S)F(r,S)\geq-
CF^{\prime}_{1}(S)F_{1}(S)>-C_{4}F_{1}(S)=-C_{4}Y(r),$ (2.26)

where $C>0$, $C_{4}=CC_{3}>0.$ Thus (2.26) gives

$Y(r)\geq Y(\tilde{r})e^{-C_{4}(r-\tilde{r})},\quad r\in[\tilde{r},R_{\max}).$
(2.27)

We have from (2.23) and (2.17) that

$\lim_{r\to R_{\max}^{-}}Y(r)=0.$ (2.28)

Let $r\to R_{\max}^{-}$ in (2.27) and using (2.28), we obtain $0\geq
Y(\tilde{r})e^{-C_{4}(R_{\max}-\tilde{r})}>0$ which is a contradiction.

Case 2. $c_{1}\leq 0\leq c_{2}$. Both Case 2a and 2b are proved similarly.
Consider Case 2a. Since $F(r,S)>0$ for all $r\in[r_{0},R_{\max})$,
$S^{\prime}>0$ for all $r\in[r_{0},R_{\max})$ therefore by Theorem 2.1,
$\lim_{r\to R_{\rm max}^{-}}S(r)=1$.

Let $X=1-S$. Then $\lim_{r\to R_{\rm max}^{-}}X(r)=0$ and

$X^{\prime}=-S^{\prime}=-F(r,1-X)=\tilde{F}(r,X)=G_{1}(c_{1}r^{1-n})\tilde{F}_{1}(X)-G_{2}(c_{2}r^{1-n})\tilde{F}_{2}(X),$
(2.29)

where $\tilde{F}_{i}(X)=F_{i}(1-X)$. Similar to the proof of Case 1a, there
are $\delta>0$ and $C_{1},C_{2}>0$ such that

$-C_{1}\tilde{F}_{2}(X)\leq\tilde{F}(r,X)\leq-C_{2}\tilde{F}_{2}(X),$ (2.30)

for all $r\in[r_{0},R_{\max}]$ and $X\in(0,\delta)$. Note that condition
(2.21) is equivalent to $\limsup_{X\to
0^{+}}\tilde{F}_{2}^{\prime}(X)<\infty$. Repeating the proof in Case 1a with
$\tilde{F}_{2}$ instead of $F_{1}$ leads to a contradiction.

Case 3. According to Theorem 2.1 we have two cases.

(i) Case $\lim_{r\to R_{\rm max}^{-}}S(r)=0$. By (2.18) there are constants
$C_{1},C_{2}>0$ and $\delta>0$ such that

$-C_{1}F_{1}(S)\leq F(r,S)\leq-C_{2}F_{1}(S).$

for all $r\in[r_{0},R_{\rm max}]$ and $S\in(0,\delta)$. Also, there is
$r_{1}\in(0,R_{\max})$ such that $S(r)<\delta$ for all $r\in(r_{1},R_{\rm
max})$. Then the exact argument for Case 1a yields a contradiction.

(ii) Case $\lim_{r\to R_{\rm max}^{-}}S(r)=1$. By (2.18), there $\delta>0$ and
$C_{1},C_{2}>0$ such that

$C_{1}F_{2}(S)\leq F(r,S)\leq C_{2}F_{2}(S)$

for all $r\in[r_{0},R_{\rm max}]$ and $S\in(1-\delta,1)$. Then the proof is
proceeded similar to Case 2a under condition (2.21) to obtain a contradiction.

Case 4. Again, according to Theorem 2.1 we have two cases.

(i) Case $\lim_{r\to R_{\rm max}^{-}}S(r)=0$. By (2.18), there are $\delta>0$
and $C_{1},C_{2}>0$ such that

$0<C_{1}F_{1}(S)\leq F(r,S)\leq C_{2}F_{1}(S)$

for all $r\in[r_{0},R_{\rm max}]$ and $S\in(0,\delta)$. Let $r_{1}$ be as in
Case 3(i). Then for $r\in(r_{1},R_{\rm max})$ we have $S^{\prime}(r)>0$, and
hence $S(r)\geq S(r_{1})>0$ which contradicts the fact $\lim_{r\to R_{\rm
max}^{-}}S(r)=0$.

(ii) Case $\lim_{r\to R_{\rm max}^{-}}S(r)=1$. By (2.18), there are $\delta>0$
and $C_{1},C_{2}>0$ such that

$-C_{1}F_{2}(S)\leq F(r,S)\leq-C_{2}F_{2}(S)<0$

for all $r\in[r_{0},R_{\rm max}]$ and $S\in(1-\delta,1)$. There is
$r_{1}\in(r_{0},R_{\max})$ such that $S(r)\in(1-\delta,1)$ for all
$r\in(r_{1},R_{\max})$. Thus $S^{\prime}(r)<0$ for all $r\in(r_{1},R_{\max})$
which gives $S(r)\leq S(r_{1})$. Letting $r\to R_{\max}$ yields $1\leq
S(r_{1})<1$. This is a contradiction.

From all the above contradictions, we must have $R_{\max}=\infty$ and the
proof is complete. ∎

To study $S(r)$ as $r\to\infty$, for the solution $S(r)$ in the Theorem 2.2 we
will need function $h(r)\in(0,1)$ such that

$G_{2}(c_{2}r^{1-n})F_{2}(h(r))-G_{1}(c_{1}r^{1-n})F_{1}(h(r))=0.$ (2.31)

To prove existence of such function consider $c_{1}c_{2}\neq 0$. Then (2.31)
is equivalent to

$\frac{f_{1}(h(r))}{f_{2}(h(r))}=\frac{c_{1}g_{1}(|c_{1}|r^{1-n})}{c_{2}g_{2}(|c_{2}|r^{1-n})}.$

Since $f\mathbin{\buildrel\rm def\over{\mathbin{=\kern-2.0pt=}}}f_{1}/f_{2}$
is strictly increasing and maps $(0,1)$ onto $(0,\infty)$, we can solve

$h(r)=f^{-1}\Big{(}\frac{c_{1}g_{1}(|c_{1}|r^{1-n})}{c_{2}g_{2}(|c_{2}|r^{1-n})}\Big{)}\quad\text{provided}\quad
c_{1}c_{2}>0.$ (2.32)

Note that

$\lim_{r\to\infty}h(r)=s^{*}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}f^{-1}\Big{(}\frac{c_{1}a_{1}^{0}}{c_{2}a_{2}^{0}}\Big{)}\in(0,1).$
(2.33)

Let $\xi(r)=r^{1-n}\in(0,\infty)$. We rewrite $h(r)$ as

$h(r)=f^{-1}\Big{(}Q(\xi(r))\Big{)}\quad\text{ where }\quad
Q(\xi)=\frac{c_{1}g_{1}(|c_{1}|\xi)}{c_{2}g_{2}(|c_{2}|\xi)}\quad\text{for
}\xi>0.$ (2.34)

######  Theorem 2.3.

If solution $S(r)$ of (2.13) exists in $[r_{0},\infty)$, then there exists
$R>r_{0}$ such that solution $S(r)$ is monotone on $(R,\infty)$, and,
consequently, $\lim_{r\to\infty}S(r)$ exists.

###### Proof.

If $c_{1}c_{2}\leq 0$ then all $r\geq r_{0}$ either $S^{\prime}\geq 0$ or
$S^{\prime}\leq 0$. Thus $S(r)$ is monotone on $[r_{0},\infty)$.

Consider the case $c_{1}c_{2}>0$. Then $h(r)$ in (2.34) exists. We rewrite
$Q(\xi)$ as

$Q(\xi)=\frac{c_{1}}{c_{2}}\cdot\frac{\sum_{i=0}^{m_{1}}a_{i}\xi^{\alpha_{i}}}{\sum_{i=0}^{m_{2}}b_{i}\xi^{\beta_{i}}}.$
(2.35)

where $a_{i},b_{i}>0$, $0=\alpha_{0}<\alpha_{1}<\cdots<\alpha_{m_{1}}$,
$0=\beta_{0}<\beta_{1}<\cdots<\beta_{m_{2}}$ .

If $Q^{\prime}\equiv 0$ then $h(r)\equiv s^{*}$ is an equilibrium. It is easy
to see that if $s_{0}>(<)s^{*}$ then $S(r)>(<)s^{*}$ for all $r$, hence $S(r)$
is monotone on $r\in[r_{0},\infty)$.

Now we consider $Q^{\prime}\neq 0$. A simple calculation gives

$\displaystyle Q^{\prime}(\xi)=\frac{c_{1}}{\xi
c_{2}}(\sum_{i=0}^{m_{2}}b_{i}\xi^{\beta_{i}})^{-2}\sum_{i=1}^{m_{3}}A_{i}\xi^{\gamma_{i}},$

where $m_{3}\geq 1$, $A_{i}\neq
0,0<\gamma_{1}<\gamma_{2}<\cdots<\gamma_{m_{3}}$. Note that $Q^{\prime}(\xi)$
has the same sign as $A_{1}$ for $\xi>0$ sufficiently small. Combining this
with the fact $f^{\prime}>0$, we have that $A_{1}h^{\prime}(r)<0$ for all
$r>R$, where $R>0$ is a sufficiently large number.

Claim 1. There is $\tilde{R}>R$ such that $S^{\prime}(r)\geq 0$ on
$(\tilde{R},\infty)$ or $S^{\prime}(r)\leq 0$ on $(\tilde{R},\infty)$.

Then the theorem’s statements obviously follow Claim 1.

To prove Claim 1 we consider the following cases.

Case 1: $A_{1}<0$. Then $h(r)$ is increasing in $[R,\infty)$ and, hence,
$h(r)<s^{*}$ for all $r\geq R$.

Case 1A: $S(r)\geq h(r)$ for all $r>R$. Then $S^{\prime}\geq 0$ for all $r>R$
or $S^{\prime}\leq 0$ for all $r>R$.

Case 1B: There exists $R_{1}>R$ such that $S(R_{1})<h(R_{1})$.

\+ Case 1B(i): $F(r,S)>0\Leftrightarrow S>h(r)$. Then $S^{\prime}>0$ if
$S(r)>h(r)$ and $S^{\prime}<0$ if $S(r)<h(r)$. It is easy to see that
$S(r)<h(R_{1})\leq h(r)$ for all $r>R_{1}$. Therefore $S^{\prime}(r)<0$ for
all $r>R_{1}$.

\+ Case 1B(ii): $F(r,S)<0\Leftrightarrow S>h(r)$. Then $S^{\prime}<0$ if
$S(r)>h(r)$ and $S^{\prime}>0$ if $S(r)<h(r)$.

Claim 2. $S(r)\leq h(r)$ for all $r\geq R_{1}$ and hence $S^{\prime}(r)\geq 0$
for all $r>R_{1}$.

Suppose Claim 2 is false. Then there is $R_{2}>R_{1}$ such that
$S(R_{2})>h(R_{2})$. There is $\tilde{r}\in(R_{1},R_{2})$ such that
$S(\tilde{r})=h(\tilde{r})$. Hence, $S$ is decreasing on $(\tilde{r},R_{2})$,
$S(R_{2})\leq S(\tilde{r})=h(\tilde{r})\leq h(R_{2})$. This is a
contradiction.

Case 2: $A_{1}>0$. Then $h(r)$ is decreasing in $[R,\infty)$ and $h(r)>s^{*}$
for all $r\geq R$ .

Case 2A: $S(r)\leq h(r)$ for all $r>R$. Then $S^{\prime}\leq 0$ for all $r>R$
or $S^{\prime}\geq 0$ for all $r>R$.

Case 2B: There exists $R_{1}>R$ such that $S(R_{1})>h(R_{1})$.

\+ Case 2B(i): $F(r,S)>0\Leftrightarrow S>h(r)$. Then $S^{\prime}>0$ if
$S(r)>h(r)$ and $S^{\prime}<0$ if $S(r)<h(r)$. Similar to Case 1B(i),
$h(r)<h(R_{1})<S(r)$ for all $r>R_{1}$. Therefore $S^{\prime}(r)>0$ for all
$r>R_{1}$.

\+ Case 2B(ii): $F(r,S)<0\Leftrightarrow S>h(r)$. Then $S^{\prime}<0$ if
$S(r)>h(r)$ and $S^{\prime}>0$ if $S(r)<h(r)$. Similar to Case 1B(ii),
$S(r)\geq h(r)$ for all $r\geq R_{1}$. Therefore $S^{\prime}(r)\leq 0$ for all
$r>R_{1}$.

From the above considerations, we see that Claim 1 holds true and the proof is
complete. ∎

Let $s_{\infty}=\lim_{r\to\infty}S(r)$ in Theorem 2.3. Note that
$s_{\infty}\in[0,1]$.

######  Lemma 2.4.

For $n=2$ and $c_{1}^{2}+c_{2}^{2}>0$, if $s_{\infty}$ is neither $0$ nor $1$
then $s_{\infty}$ must be $s^{*}$.

###### Proof.

Assume $s_{\infty}\neq 0,1$. We prove by contradiction. Suppose
$s_{\infty}\not=s^{*}$. Then

$c_{3}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}|F_{2}(s_{\infty})a_{2}^{0}c_{2}-F_{1}(s_{\infty})a_{1}^{0}c_{1}|>0.$
(2.36)

For any $R>r_{0}$, We write $S(r)=I_{1}(R)+I_{2}(R)$ where

$I_{1}(R)=s_{0}+\int_{r_{0}}^{R}F(z,S(z))dz\quad\text{and}\quad
I_{2}(R)=\int_{R}^{r}F(z,S(z))dz.$

For sufficiently large $R$ and $r>R$

$|I_{2}(R)|=\int_{R}^{r}F(z,S(z))dz\geq\frac{c_{3}}{2}\int_{R}^{r}z^{-1}dz=\frac{c_{3}}{2}(\ln
r-\ln R).$

Therefore

$|S(r)|\geq\frac{c_{3}}{2}(\ln r-\ln R)-I_{1}(R)\to\infty\text{ as
}r\to\infty.$

Thus $S(r)$ is unbounded which contradicts the fact $S(r)\in(0,1)$. Hence
$s_{\infty}=s^{*}$. ∎

Using Lemma 2.4 we can drastically reduce the range of $s_{\infty}$ in case
$n=2$.

######  Theorem 2.5.

Let $n=2$ and $c_{1}^{2}+c_{2}^{2}>0$. Suppose $S(r)$ is a solution of (2.13)
on $[r_{0},\infty)$.

(i) If $c_{1}\leq 0$ and $c_{2}\geq 0$ then $s_{\infty}=1$.

(ii) If $c_{1}\geq 0$ and $c_{2}\leq 0$ then $s_{\infty}=0$.

(iii) If $c_{1},c_{2}<0$ then $s_{\infty}=s^{*}$.

(iv) If $c_{1},c_{2}>0$ then $s_{\infty}\in\\{0,1,s^{*}\\}$.

###### Proof.

(i) In this case, $S^{\prime}(r)>0$ for all r, hence $S(r)>s_{0}$. This
implies $s_{\infty}\neq 0$. In addition, $s^{*}$ does not exist. Therefore, by
Lemma 2.4, $s_{\infty}$ must be $1$.

(ii) The proof is similar to that of (i).

(iii) We have $F(r,S)<0$ for $S<h(r)$ and $F(r,S)>0$ for $S<h(r)$. Thus, it is
easy to see that $s_{\infty}$ cannot be $0,1$. By Lemma 2.4, $s_{\infty}$ must
be $s^{*}$.

(iv) This is a direct consequence of Lemma 2.4. ∎

In general, we do not know the value of $s_{\infty}$ based on $s_{0}$.
However, in some particular cases, we can determine the range of $s_{\infty}$.

######  Example 2.6.

We consider the following special $g_{i}$’s:

$g_{i}(u)=a_{i}+b_{i}u^{\alpha}\quad\text{ where }a_{i}>0,\ b_{i}>0,\text{ for
}i=1,2\text{ and }\alpha>0.$ (2.37)

We have from (2.34) when $c_{1}c_{2}>0$ that

$Q^{\prime}(\xi)=\frac{c_{1}\Delta}{c_{2}(a_{2}+b_{2}|c_{2}|^{\alpha}\xi)^{2}}\text{
with }\Delta=a_{2}b_{1}|c_{1}|^{\alpha}-a_{1}b_{2}|c_{2}|^{\alpha}.$

We now detail the range of $s_{\infty}$ case by case.

Case $n>2$.

  * A.

$c_{1},c_{2}>0$.

    * A1.

$\Delta<0$.

(i) $s_{0}>s^{*}$. Then $s_{\infty}\in(s_{0},1]$.

(ii) $h(r_{0})\leq s_{0}\leq s^{*}$. Then $s_{\infty}\in[0,1]$.

(iii) $s_{0}<h(r_{0})$. Then $s_{\infty}\in[0,s_{0})$.

    * A2.

$\Delta>0$.

(i) $s_{0}>h(r_{0})$. Then $s_{\infty}\in(s_{0},1]$.

(ii) $s^{*}\leq s_{0}\leq h(r_{0})$. Then $s_{\infty}\in[0,1]$.

(iii) $s_{0}<s^{*}$. Then $s_{\infty}\in[0,s_{0})$.

    * A3.

$\Delta=0$.

(i) $s_{0}>s^{*}$. Then $s_{\infty}\in(s_{0},1]$.

(ii) $s_{0}=s^{*}$. Then $s_{\infty}=s^{*}$.

(iii) $s_{0}<s^{*}$. Then $s_{\infty}\in[0,s_{0})$.

  * B.

$c_{1},c_{2}<0$.

    * B1.

$\Delta<0$.

(i) $s_{0}>s^{*}$. Then $s_{\infty}\in(h(r_{0}),s_{0})$.

(ii) $h(r_{0})\leq s_{0}\leq s^{*}$. Then $s_{\infty}\in(h(r_{0}),s^{*}]$.

(iii) $s_{0}<h(r_{0})$. Then $s_{\infty}\in(s_{0},s^{*}]$.

    * B2.

$\Delta>0$.

(i) $s_{0}>h(r_{0})$. Then $s_{\infty}\in[s^{*},s_{0})$.

(ii) $s^{*}\leq s_{0}\leq h(r_{0})$. Then $s_{\infty}\in[s^{*},h(r_{0}))$.

(iii) $s_{0}<s^{*}$. Then $s_{\infty}\in(s_{0},h(r_{0}))$.

    * B3.

$\Delta=0$.

(i) $s_{0}>s^{*}$. Then $s_{\infty}\in[s^{*},s_{0})$.

(ii) $s_{0}=s^{*}$. Then $s_{\infty}=s^{*}$.

(iii) $s_{0}<s^{*}$. Then $s_{\infty}\in(s_{0},s^{*}]$.

  * C.

$c_{1}\leq 0<c_{2}$ or $c_{1}<0=c_{2}$. Then $s_{\infty}\in(s_{0},1]$.

  * D.

$c_{2}\leq 0<c_{1}$ or $c_{1}=0>c_{2}$. Then $s_{\infty}\in[0,s_{0})$.

Verifications of the cases above are presented in the Appendix.

Case $n=2$. We use the analysis in A, which is still valid for $n=2$, to
explicate the case $c_{1},c_{2}>0$ in Theorem 2.5. Let
$s_{m}=\min\\{h(r_{0}),s^{*}\\}$ and $s_{M}=\max\\{h(r_{0}),s^{*}\\}$.

  * (i)

$s_{0}>s_{M}$. Then $s_{\infty}=1$.

  * (ii)

$s_{m}\leq s_{0}\leq s_{M}$. Then $s_{\infty}\in\\{0,1,s^{*}\\}$.

  * (iii)

$s_{0}<s_{m}$. Then $s_{\infty}=0$.

##  3\. Linearization

We study the linear stability of a steady state solution
$(\mathbf{u}_{1}^{*}(\mathbf{x}),\mathbf{u}_{2}^{*}(\mathbf{x}),S_{*}(\mathbf{x}))$
of system (1.14). The formal linearizion of system (1.14) at
$(\mathbf{u}_{1}^{*}(\mathbf{x}),\mathbf{u}_{2}^{*}(\mathbf{x}),S_{*}(\mathbf{x}))$
is

$\displaystyle\sigma_{t}$ $\displaystyle=-{\rm div}\ {\mathbf{v}}_{1},$ (3.1a)
$\displaystyle\sigma_{t}$ $\displaystyle={\rm div}\ {\mathbf{v}}_{2},$ (3.1b)
$\displaystyle\nabla\sigma$
$\displaystyle=F_{2}(S_{*})\mathbf{G}^{\prime}_{2}({\mathbf{u}}_{2}^{*}){\mathbf{v}}_{2}+F^{\prime}_{2}(S_{*})\sigma\mathbf{G}_{2}({\mathbf{u}}_{2}^{*})-\Big{(}F_{1}(S_{*})\mathbf{G}^{\prime}_{1}({\mathbf{u}}_{1}^{*}){\mathbf{v}}_{1}+F^{\prime}_{1}(S_{*})\sigma\mathbf{G}_{1}({\mathbf{u}}_{1}^{*})\Big{)}.$
(3.1c)

Above, the unknowns are $\sigma(\mathbf{x},t)\in\mathbb{R}$,
$\mathbf{v}_{1}(\mathbf{x},t),\mathbf{v}_{2}(\mathbf{x},t)\in\mathbb{R}^{n}$.
A solution $(\sigma,\mathbf{v}_{1},\mathbf{v}_{2})$ of (3.1) is considered as
an approximation of the difference between a solution
$(S(\mathbf{x},t),\mathbf{u}_{1}(\mathbf{x},t),\mathbf{u}_{2}(\mathbf{x},t))$
of (1.14) and the steady state
$(\mathbf{u}_{1}^{*}(\mathbf{x}),\mathbf{u}_{2}^{*}(\mathbf{x}),S_{*}(\mathbf{x}))$
in (3.2). The system (3.1) is obtained by utilizing Taylor expansions in
(1.14) at $(\mathbf{u}_{1}^{*},\mathbf{u}_{2}^{*},S_{*})$ with respect to
variables $\mathbf{u}_{1},\mathbf{u}_{2},S$ and then neglecting non-linear
terms. In theory of ordinary differential equations, linearizion has direct
connections with the stability of steady states. In PDE theory, this is not
always the case. Nonetheless, in many scenarios, stability of the linearized
equations lead to the stability of the original ones. In this article we only
focus on the stability for the linearized system (3.1).

We consider, particularly, the steady states obtained in the previous section,
that is,

$\mathbf{u}_{1}^{*}(\mathbf{x})=c_{1}|\mathbf{x}|^{-n}\mathbf{x},\quad\mathbf{u}_{2}^{*}(\mathbf{x})=c_{2}|\mathbf{x}|^{-n}\mathbf{x},\quad
S_{*}(\mathbf{x})=\hat{S}(|\mathbf{x}|),$ (3.2)

where $c_{1},c_{2}$ are constants and $\hat{S}(r)$ is a solution of (2.13).

Let ${\mathbf{v}}={\mathbf{v}}_{1}+{\mathbf{v}}_{2}.$ Adding equation (3.1a)
to (3.1b) gives

${\rm div}\ {\mathbf{v}}=0.$ (3.3)

Assume ${\mathbf{v}}=\mathbf{V}(\mathbf{x},t)\in\mathbb{R}^{n}$, where
$\mathbf{V}(\mathbf{x},t)$ is a given function. We have

$\mathbf{v}_{1}=\mathbf{V}-\mathbf{v}_{2},$ (3.4)

hence (3.1c) provides

$\nabla\sigma=\sigma\mathbf{b}+\underline{\mathbf{B}}\mathbf{v}_{2}-\mathbf{c},$
(3.5)

where

$\displaystyle\underline{\mathbf{B}}$
$\displaystyle=\underline{\mathbf{B}}(\mathbf{x})=F_{2}(S_{*})\mathbf{G}^{\prime}_{2}({\mathbf{u}}_{2}^{*})+F_{1}(S_{*})\mathbf{G}^{\prime}_{1}({\mathbf{u}}_{1}^{*}),$
(3.6) $\displaystyle\mathbf{b}$
$\displaystyle=\mathbf{b}(\mathbf{x})=F^{\prime}_{2}(S_{*})\mathbf{G}_{2}({\mathbf{u}}_{2}^{*})-F^{\prime}_{1}(S_{*})\mathbf{G}_{1}({\mathbf{u}}_{1}^{*}),$
(3.7) $\displaystyle\mathbf{c}$
$\displaystyle=\mathbf{c}(\mathbf{x},t)=F_{1}(S_{*})\mathbf{G}^{\prime}_{1}({\mathbf{u}}_{1}^{*})\mathbf{V}(\mathbf{x},t).$
(3.8)

The $n\times n$ matrix $\underline{\mathbf{B}}$ is invertible (see Lemma 3.2
below), and we denote its inverse by

$\underline{\mathbf{A}}=\underline{\mathbf{A}}(\mathbf{x})=\underline{\mathbf{B}}^{-1}(\mathbf{x}).$
(3.9)

Solving for $\mathbf{v}_{2}$ from (3.5) we obtain

$\mathbf{v}_{2}=\underline{\mathbf{A}}(\nabla\sigma-\sigma\mathbf{b})+\underline{\mathbf{A}}\mathbf{c}.$
(3.10)

Substituting (3.10) into (3.1b) gives

$\displaystyle\sigma_{t}$
$\displaystyle=\nabla\cdot\Big{[}\underline{\mathbf{A}}(\nabla\sigma-\sigma\mathbf{b})\Big{]}+\nabla\cdot(\underline{\mathbf{A}}\mathbf{c}).$
(3.11)

Then (3.11), (3.4) and (3.10) is our linearized system for (1.14) at the
steady state
$(\mathbf{u}_{1}^{*}(\mathbf{x}),\mathbf{u}_{2}^{*}(\mathbf{x}),S_{*}(\mathbf{x}))$.

######  Remark 3.1.

In our approach, the total velocity $\bf V$ and hence the vector function $\bf
c$ are supposed to be known, whereas the phase velocities $\mathbf{v}_{i}$
($i=1,2$) are the unknowns. Therefore, our results below can be considered as
the qualitative study of the flow depending on the property of the total
velocity. Such restriction, however, is justified in practice or in case $\bf
V$, as a perturbation, itself is radial. In the latter consideration, by
(3.3), $\mathbf{V}=\mathbf{V}(t)$ is totally determined by its boundary
values.

We will focus on studying classical solutions of (3.11). For such purpose, the
maximum principle plays an important role. Although there is not an obvious
maximum principle for (3.11), we can convert it to an equation for which there
is one. We proceed as follows. Rewrite vector function
$\mathbf{b}(\mathbf{x})$ explicitly as

$\mathbf{b}(\mathbf{x})=\Big{(}F^{\prime}_{2}(S_{*}(\mathbf{x}))g_{2}(\frac{|c_{2}|}{|\mathbf{x}|^{n-1}})\frac{c_{2}}{|\mathbf{x}|^{n}}-F^{\prime}_{1}(S_{*}(\mathbf{x}))g_{1}(\frac{|c_{1}|}{|\mathbf{x}|^{n-1}})\frac{c_{1}}{|\mathbf{x}|^{n}}\Big{)}\mathbf{x}=\lambda(|\mathbf{x}|)\mathbf{x},$
(3.12)

where

$\lambda(r)=F^{\prime}_{2}(\hat{S}(r))g_{2}(\frac{|c_{2}|}{r^{n-1}})\frac{c_{2}}{r^{n}}-F^{\prime}_{1}(\hat{S}(r))g_{1}(\frac{|c_{1}|}{r^{n-1}})\frac{c_{1}}{r^{n}}.$
(3.13)

By defining

$\Lambda(\mathbf{x})=\frac{1}{2}\int_{r_{0}^{2}}^{|\mathbf{x}|^{2}}\lambda(\sqrt{\xi})d\xi=\int_{r_{0}}^{|\mathbf{x}|}r\lambda(r)dr,$
(3.14)

we have for $\mathbf{x}\neq 0$ that

$\mathbf{b}(\mathbf{x})=\nabla\Lambda(\mathbf{x}).$ (3.15)

Substituting this relation into (3.11) we obtain

$\displaystyle\sigma_{t}$
$\displaystyle=\nabla\cdot\Big{[}\underline{\mathbf{A}}(\nabla\sigma-\sigma\nabla\Lambda)\Big{]}+\nabla\cdot(\underline{\mathbf{A}}\mathbf{c})=\nabla\cdot\Big{[}e^{\Lambda}\underline{\mathbf{A}}\nabla(e^{-\Lambda}\sigma)\Big{]}+\nabla\cdot(\underline{\mathbf{A}}\mathbf{c})$
$\displaystyle=e^{\Lambda}\nabla\cdot\Big{[}\underline{\mathbf{A}}\nabla(e^{-\Lambda}\sigma)\Big{]}+e^{\Lambda}\nabla\Lambda\cdot\Big{[}\underline{\mathbf{A}}\nabla(e^{-\Lambda}\sigma)\Big{]}+\nabla\cdot(\underline{\mathbf{A}}\mathbf{c}).$

Let

$w(\mathbf{x},t)=e^{-\Lambda(\mathbf{x})}\sigma(\mathbf{x},t).$ (3.16)

Then $w$ satisfies

$w_{t}=e^{-\Lambda}\sigma_{t}=\nabla\cdot\Big{(}\underline{\mathbf{A}}\nabla
w\Big{)}+\nabla\Lambda\cdot\underline{\mathbf{A}}\nabla
w+e^{-\Lambda}\nabla\cdot(\underline{\mathbf{A}}\mathbf{c}).$ (3.17)

Using relation (3.15) again yields

$w_{t}-\nabla\cdot(\underline{\mathbf{A}}\nabla
w)-\mathbf{b}\cdot\underline{\mathbf{A}}\nabla
w=e^{-\Lambda}\nabla\cdot(\underline{\mathbf{A}}\mathbf{c}).$ (3.18)

For the velocities, we have from (3.10) and (3.16) that

$\displaystyle\mathbf{v}_{2}=\underline{\mathbf{A}}\big{[}\nabla(e^{\Lambda}w)-e^{\Lambda}w{\bf
b}\big{]}+\underline{\mathbf{A}}{\bf
c}=\underline{\mathbf{A}}\big{[}e^{\Lambda}\nabla
w+we^{\Lambda}\nabla\Lambda-e^{\Lambda}w{\bf
b}\big{]}+\underline{\mathbf{A}}{\bf c}.$

Thus,

$\mathbf{v}_{2}=e^{\Lambda}\underline{\mathbf{A}}\nabla
w+\underline{\mathbf{A}}{\bf c}.$ (3.19)

We will proceed by studying (3.18) first and then drawing conclusions for
$\sigma,\mathbf{v}_{1},\mathbf{v}_{2}$ via the relations (3.16), (3.19) and
(3.4).

In the following, we present some properties of $\underline{\mathbf{B}}$,
$\underline{\mathbf{A}}$ and $\bf b$. They have some structures and estimates
which are crucial for our next sections. These are based on the special form
of the steady state $(\mathbf{u}^{*}_{1},\mathbf{u}^{*}_{2},S_{*})$.

Denote by $\underline{\mathbf{I}}_{n}$ the $n\times n$ identity matrix.
Consider $c_{1}^{2}+c_{2}^{2}>0$ and $\mathbf{x}\neq 0$. We have for $i=1,2$
that

$\mathbf{G}^{\prime}_{i}(\mathbf{u}_{i}^{*})=g_{i}(|\mathbf{u}_{i}^{*}|)\underline{\mathbf{I}}_{n}+g^{\prime}_{i}(|\mathbf{u}_{i}^{*}|)\frac{\mathbf{u}_{i}^{*}(\mathbf{u}_{i}^{*})^{T}}{|\mathbf{u}_{i}^{*}|}=g_{i}(|c_{i}|\,|\mathbf{x}|^{1-n})\underline{\mathbf{I}}_{n}+g^{\prime}_{i}(|c_{i}|\,|\mathbf{x}|^{1-n})|c_{i}|\,|\mathbf{x}|^{-1-n}\mathbf{x}\mathbf{x}^{T}.$
(3.20)

Since these matrices are symmetric, so is $\underline{\mathbf{B}}$. For each
$i=1,2$ and arbitrary $\mathbf{z}\in\mathbb{R}^{n}$,

$\displaystyle\mathbf{z}^{T}\mathbf{G}^{\prime}_{i}(\mathbf{u}_{i}^{*})\mathbf{z}$
$\displaystyle=g_{i}(|c_{i}|\,|\mathbf{x}|^{1-n})|\mathbf{z}|^{2}+g^{\prime}_{i}(|c_{i}|\,|\mathbf{x}|^{1-n})|c_{i}|\,|\mathbf{x}|^{-1-n}|\mathbf{x}\cdot{\mathbf{z}}|^{2}.$

Define

$\displaystyle\beta$ $\displaystyle=\beta(\mathbf{x})\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))g_{i}(|c_{i}|\,|\mathbf{x}|^{1-n}),$
(3.21) $\displaystyle\gamma$
$\displaystyle=\gamma(\mathbf{x})\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))g^{\prime}_{i}(|c_{i}|\,|\mathbf{x}|^{1-n})|c_{i}|\,|\mathbf{x}|^{1-n}.$
(3.22)

Then

$\displaystyle\beta|\mathbf{z}|^{2}\leq\mathbf{z}^{T}\underline{\mathbf{B}}\mathbf{z}$
$\displaystyle\leq(\beta+\gamma)|\mathbf{z}|^{2}.$ (3.23)

The first inequality in (3.23) proves that
$\mathbf{z}^{T}\underline{\mathbf{B}}\mathbf{z}>0$ for all $\mathbf{z}\neq 0$.
Therefore, $\underline{\mathbf{B}}$ is positive definite and hence it is
invertible. Since $\underline{\mathbf{B}}$ is symmetric, so is its inverse
$\underline{\mathbf{A}}$. Thus, we have:

######  Lemma 3.2.

For any $c_{1}^{2}+c_{2}^{2}>0$ and $\mathbf{x}\neq 0$, matrices
$\underline{\mathbf{B}}(\mathbf{x})$ and $\underline{\mathbf{A}}(\mathbf{x})$
are symmetric, invertible and positive definite.

Since matrix $\underline{\mathbf{B}}$ is symmetric and positive definite, it
has positive eigenvalues
$\lambda_{1}(\underline{\mathbf{B}})\leq\lambda_{2}(\underline{\mathbf{B}})\leq\cdots\leq\lambda_{n}(\underline{\mathbf{B}})$.
We have

$\lambda_{1}(\underline{\mathbf{B}})=\min_{\mathbf{z}\neq
0}\frac{\mathbf{z}^{T}\underline{\mathbf{B}}\mathbf{z}}{|\mathbf{z}|^{2}}\quad\text{and}\quad\lambda_{n}(\underline{\mathbf{B}})=\max_{\mathbf{z}\neq
0}\frac{\mathbf{z}^{T}\underline{\mathbf{B}}\mathbf{z}}{|\mathbf{z}|^{2}}.$
(3.24)

It follows from (3.24) and (3.23) that

$\beta\leq\lambda_{1}(\underline{\mathbf{B}})\leq\lambda_{n}(\underline{\mathbf{B}})\leq\beta+\gamma.$
(3.25)

By the Spectral Theorem,

$\lambda_{1}(\underline{\mathbf{A}})=\frac{1}{\lambda_{n}(\underline{\mathbf{B}})}\geq\frac{1}{\beta+\gamma}\quad\text{and}\quad\lambda_{n}(\underline{\mathbf{A}})=\frac{1}{\lambda_{1}(\underline{\mathbf{B}})}\leq\frac{1}{\beta}.$
(3.26)

We now consider $0<r_{0}\leq|\mathbf{x}|<R_{\rm max}$. Let
$a_{0}^{(i)}=g_{i}(0)$ for $i=1,2$, and define

$\displaystyle d_{0}$ $\displaystyle=\min\\{a_{0}^{(1)},a_{0}^{(2)}\\},\quad
d_{1}=d_{1}(r_{0})=\sum_{i=1}^{2}g_{i}(|c_{i}|r_{0}^{1-n}),$ (3.27)
$\displaystyle d_{2}$
$\displaystyle=d_{2}(r_{0})=\sum_{i=1}^{2}g_{i}(|c_{i}|r_{0}^{1-n})|c_{i}|r_{0}^{1-n},\quad
d_{3}=d_{3}(r_{0})=\sum_{i=1}^{2}g^{\prime}_{i}(|c_{i}|r_{0}^{1-n})|c_{i}|r_{0}^{1-n},$
(3.28) $\displaystyle d_{4}$ $\displaystyle=d_{4}(r_{0})=d_{1}+d_{3}.$ (3.29)

Then

$d_{0}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))\leq\beta(\mathbf{x})\leq
d_{1}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))\quad\text{and}\quad\gamma(\mathbf{x})\leq
d_{3}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x})).$ (3.30)

By (3.23), (3.26) and (3.30),

$d_{0}|\mathbf{z}|^{2}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))\leq\mathbf{z}^{T}\underline{\mathbf{B}}(\mathbf{x})\mathbf{z}\leq
d_{4}|\mathbf{z}|^{2}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x})),$ (3.31)
$\frac{1}{d_{4}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))}\leq\lambda_{1}(\underline{\mathbf{A}})\leq\lambda_{n}(\underline{\mathbf{A}})\leq\frac{1}{d_{0}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))}.$
(3.32)

Applying (3.24) to matrix $\underline{\mathbf{A}}$, we have

$\mathbf{z}^{T}\underline{\mathbf{A}}(\mathbf{x})\mathbf{z}\geq\lambda_{1}(\underline{\mathbf{A}})|\mathbf{z}|^{2}\geq\frac{|{\bf
z}|^{2}}{d_{4}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))}\quad\forall\bf
z\in\mathbb{R}^{n}.$ (3.33)

Denote by $|\underline{\mathbf{A}}|$ and $\|\underline{\mathbf{A}}\|_{\rm op}$
the Euclidean and operator norms of matrix $\underline{\mathbf{A}}$,
respectively. Then

$|\underline{\mathbf{A}}|\leq c_{0}\|\underline{\mathbf{A}}\|_{\rm
op}=c_{0}\lambda_{n}(\underline{\mathbf{A}}),$ (3.34)

for some constant $c_{0}>0$. Thus,

$|\underline{\mathbf{A}}(\mathbf{x})|\leq\frac{c_{0}}{d_{0}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))}\quad\forall|\mathbf{x}|\in[r_{0},R_{\rm
max}).$ (3.35)

For the boundedness of $\mathbf{b}$, we have

$|\mathbf{b}(\mathbf{x})|\leq\sum_{i=1}^{2}\Big{[}|F^{\prime}_{i}(\hat{S}(|\mathbf{x}|))|g_{i}(|c_{i}||\mathbf{x}|^{1-n})|c_{i}||\mathbf{x}|^{1-n}\Big{]}\leq
d_{2}\sum_{i=1}^{2}|F^{\prime}_{i}(\hat{S}(|\mathbf{x}|))|\quad\forall|\mathbf{x}|\in[r_{0},R_{\rm
max}).$ (3.36)

From (3.14) and (3.13),

$\Lambda(\mathbf{x})=\int_{r_{0}}^{|\mathbf{x}|}r\lambda(r)dr=\int_{r_{0}}^{|\mathbf{x}|}\Big{[}F^{\prime}_{2}(\hat{S}(r))G_{2}(c_{2}r^{1-n})-F^{\prime}_{1}(\hat{S}(r))G_{1}(c_{1}r^{1-n})\Big{]}dr.$
(3.37)

Then

$|\Lambda(\mathbf{x})|\leq
d_{2}\int_{r_{0}}^{|\mathbf{x}|}\Big{[}|F^{\prime}_{1}(\hat{S}(r))|+|F^{\prime}_{2}(\hat{S}(r))|\Big{]}dr\quad\forall|\mathbf{x}|\in[r_{0},R_{\rm
max}).$ (3.38)

Also, matrix $\underline{\mathbf{B}}$ has the following special property:

$\underline{\mathbf{B}}(\mathbf{x})\mathbf{x}=\sum_{i=1}^{2}\Big{\\{}F_{i}(\hat{S}(|\mathbf{x}|))\left[g_{i}(|c_{i}||\mathbf{x}|^{1-n})+g^{\prime}_{i}(|c_{i}||\mathbf{x}|^{1-n})|c_{i}||\mathbf{x}|^{1-n}\right]\Big{\\}}\mathbf{x}=\phi(|\mathbf{x}|)\mathbf{x},$
(3.39)

where

$\phi(r)=\sum_{i=1}^{2}F_{i}(\hat{S}(r))\left[g_{i}(|c_{i}|r^{1-n})+g^{\prime}_{i}(|c_{i}|r^{1-n})|c_{i}|r^{1-n}\right].$
(3.40)

Since $g^{\prime}_{i}\geq 0$,

$\phi(r)\geq d_{0}[F_{1}(\hat{S}(r))+F_{2}(\hat{S}(r))]\quad\forall
r\in[r_{0},R_{\rm max}).$ (3.41)

Since $g_{i}(s)$ and $g_{i}^{\prime}(s)s$ are increasing on $[0,\infty)$, we
have

$\phi(r)\leq d_{4}[F_{1}(\hat{S}(r))+F_{2}(\hat{S}(r))]\quad\forall
r\in[r_{0},R_{\rm max}).$ (3.42)

We now discuss the regularity of the involved functions. For
$D\subset\mathbb{R}^{n}\times\mathbb{R}$, we define class
$C_{\mathbf{x}}^{m}(D)$ as the set of functions $f(\mathbf{x},t)\in C(D)$
whose partial derivatives with respect to $\mathbf{x}$ up to order $m$ are
continuous in $D$. The class $C_{t}^{m}$ is defined similarly and
$C_{\mathbf{x},t}^{m,m^{\prime}}=C_{\mathbf{x}}^{m}\cap C_{t}^{m^{\prime}}$.

Note that

$\frac{\partial\underline{\mathbf{A}}}{\partial
x_{i}}=-\underline{\mathbf{A}}\frac{\partial\underline{\mathbf{B}}}{\partial
x_{i}}\underline{\mathbf{A}}.$ (3.43)

By definitions (3.6), (3.7), (3.8) and relation (3.43), we easily obtain:

######  Lemma 3.3.

Assume $F_{1},F_{2}\in C^{m}((0,1))$ for some $m\geq 1$. Let
$R\in(r_{0},R_{\rm max})$ and denote

$\mathcal{O}=\\{\mathbf{x}:r_{0}<|\mathbf{x}|<R\\}.$

(i) Then $\underline{\mathbf{B}},\underline{\mathbf{A}}\in
C^{m}(\bar{\mathcal{O}})$, $\mathbf{b}\in C^{m-1}(\bar{\mathcal{O}})$ and
$\Lambda\in C^{m}(\bar{\mathcal{O}})$.

(ii) If, in addition, $\mathbf{V}\in X(\mathcal{O}\times(0,\infty))$ then
$\mathbf{c}\in X(\mathcal{O}\times(0,\infty))$, where $X$ can be $C^{m}$ or
$C_{\mathbf{x}}^{m}$ or $C_{t}^{m}$.

##  4\. Case of bounded domain

In this section, we study the linear stability of the obtained steady flows in
section 2 on bounded domains. More specifically, we investigate the stability
of the trivial solution for the linearized system (3.1). The key instrument in
proving the asymptotic stability is a Landis-type lemma of growth (see [19]).
To prove such a lemma we use specific structures of the coefficients of
equation (3.18) to construct singular sub-parabolic functions. These are
motivated by the so-called $F_{s,\beta}$ functions introduced in [19].

Let $r_{0}>0$ be fixed throughout. We consider in this section an open,
bounded set $U$ in $\mathbb{R}^{n}\setminus\bar{B}_{r_{0}}$. We fix $R>0$ such
that $U\subset\mathscr{U}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}B_{R}\setminus\bar{B}_{r_{0}}$. Denote
$\Gamma=\partial U$, $D=U\times(0,\infty)$ and
$\mathscr{D}=\mathscr{U}\times(0,\infty)$.

We consider a steady state
$(u^{*}_{1}(\mathbf{x}),u^{*}_{2}(\mathbf{x}),S_{*}(\mathbf{x}))$ as in (3.2)
with $c_{1}^{2}+c_{2}^{2}>0$. Recall that (3.11), (3.4) and (3.10) is our
linearized system for (1.14). We study the equation for $\sigma(\mathbf{x},t)$
first. More specifically, we study the following initial-boundary value
problem (IBVP):

$\begin{cases}\sigma_{t}=\nabla\cdot\Big{[}\underline{\mathbf{A}}(\nabla\sigma-\sigma{\bf
b})\Big{]}+\nabla\cdot(\underline{\mathbf{A}}\mathbf{c})&\text{ on
}U\times(0,\infty),\\\ \sigma=g(\mathbf{x},t)&\text{ on
}\Gamma\times(0,\infty),\\\ \sigma=\sigma_{0}(\mathbf{x})&\text{ on
}U\times\\{t=0\\}.\end{cases}$ (4.1)

Regarding the initial and boundary data in (4.1), we always assume that

$\sigma_{0}\in C(\bar{U}),\ g\in C(\Gamma\times[0,\infty))\text{ and
}\sigma_{0}(\mathbf{x})=g(\mathbf{x},0)\text{ on }\Gamma.$ (4.2)

Assume that

$0<\underline{s}\leq\hat{S}(r)\leq\bar{s}<1\quad\forall
r\in[r_{0},R],\quad\text{where }\underline{s}\text{ and }\bar{s}\text{ are
constants.}$ (4.3)

Assumption (4.3) is valid for any solution $\hat{S}$ in Theorem 2.1 with
$R_{\rm max}>R$, in particular, when $R_{\rm max}=\infty$ as in Theorem 2.2.
Under constraint (4.3) and Assumptions A and B, we easily see the following
facts. Let

$\mu_{1}=\sum_{i=1}^{2}\max_{\underline{s}\leq
s\leq\bar{s}}F_{i}(s),\quad\mu_{2}=\sum_{i=1}^{2}\min_{\underline{s}\leq
s\leq\bar{s}}F_{i}(s),\quad\mu_{3}=\sum_{i=1}^{2}\max_{\underline{s}\leq
s\leq\bar{s}}|F^{\prime}_{i}(s)|.$ (4.4)

Then $\mu_{1}$, $\mu_{2}$ and $\mu_{3}$ are positive numbers.

From (3.33) and (4.3) follows that

${\bf z}^{T}\underline{\mathbf{A}}(\mathbf{x}){\bf z}\geq\frac{|{\bf
z}|^{2}}{C_{0}}\quad\forall\mathbf{x}\in\bar{\mathscr{U}},\
\mathbf{z}\in\mathbb{R}^{n},$ (4.5)

where $C_{0}=d_{4}\mu_{1}$.

From (3.35), (3.36) and (4.3), we get

$|\underline{\mathbf{A}}(\mathbf{x})|\leq\frac{c_{0}}{C_{1}}\quad\text{and}\quad\quad|\mathbf{b}(\mathbf{x})|\leq
C_{2}\quad\forall\mathbf{x}\in\bar{\mathscr{U}},$ (4.6)

where $c_{0}$ is in (3.34), $C_{1}=d_{0}\mu_{2}$ and $C_{2}=d_{2}\mu_{3}$.

For the smoothness, by Lemma 3.3,

$\underline{\mathbf{B}},\underline{\mathbf{A}}\in
C^{1}(\bar{\mathscr{U}})\quad\text{and}\quad\mathbf{b}\in
C(\bar{\mathscr{U}}).$ (4.7)

First, we consider the the existence of classical solutions of (4.1). We use
the known result from theory of linear parabolic equations in [16]. This will
require certain regularity of the coefficients of (4.1). Those requirements,
in turn, can be formulated in terms of functions $F_{1}$ and $F_{2}$, thanks
to Lemma 3.3.

Condition (E1). $F_{1},F_{2}\in C^{7}((0,1))$ and $V\in
C_{\mathbf{x}}^{6}(\bar{D})$; $V_{t}\in C_{\mathbf{x}}^{3}(\bar{D})$.

######  Theorem 4.1 ([16]).

Assume (E1), then there exists a unique solution $\sigma\in C(\bar{D})\cap
C^{2,1}_{\mathbf{x},t}(D)$ of problem (4.1).

Note that we did not attempt to optimize Condition (E1). As seen below, the
study of qualitative properties of solution $\sigma$ will require much less
stringent conditions than (E1).

Now we turn to the stability, asymptotic stability and structural stability
issues. Our main tool is the maximum principle. As discussed in the previous
section, we use the transformation (3.16) to convert the PDE in (4.1) to a
more convenient form (3.18). Define the differential operator on the left-hand
side of (3.18) by

$\mathcal{L}w=\partial_{t}w-\nabla\cdot(\underline{\mathbf{A}}\nabla w)-{\bf
b}\cdot\underline{\mathbf{A}}\nabla w.$ (4.8)

Corresponding to (4.1), the IBVP for $w(\mathbf{x},t)$ is

$\begin{cases}\mathcal{L}w=f_{0}&\text{in }U\times(0,\infty),\\\
w(\mathbf{x},0)=w_{0}(\mathbf{x})&\text{in }U,\\\
w(\mathbf{x},t)=G(\mathbf{x},t)&\text{on }\Gamma\times(0,\infty),\end{cases}$
(4.9)

where $w_{0}(\mathbf{x})$ and $G(\mathbf{x},t)$ are given initial data and
boundary data, respectively, and $f_{0}(\mathbf{x},t)$ is a known function. We
will obtain results for solution $w$ of (4.9) and then reformulate them in
terms of solution $\sigma$ of the original problem (4.1).

Since the existence and uniqueness issues are settled in Theorem 4.1, our main
focus now is the qualitative properties of solution $w$ of (4.9). For these,
we only need properties (4.5), (4.6), the special structure of equation (4.1),
and the assumption that the classical solution in $C(\bar{D})\cap
C^{2,1}_{\mathbf{x},t}(D)$ already exists. The fine properties of the
solutions obtained below have their own merit in the theory of linear
parabolic equations.

It follows from (4.5) and (4.6) that the maximum principle holds for any
classical solution of $\mathcal{L}w\leq(\geq)0$ in $D$. To obtain better
estimates for solutions, especially as $t\to\infty$, we use the following
barrier function. Define

$W(\mathbf{x},t)=\begin{cases}t^{-s}e^{-\frac{\varphi(\mathbf{x})}{t}}&\text{if
}t>0,\\\ 0&\text{if }t\leq 0,\end{cases}$ (4.10)

where the number $s>0$ and the function $\varphi(\mathbf{x})>0$ will be
decided later. Then

$\mathcal{L}W=t^{-s-2}e^{-\frac{\varphi}{t}}\Big{\\{}t\big{(}-s+\nabla\cdot(\underline{\mathbf{A}}\nabla\varphi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\varphi\big{)}+\varphi-(\underline{\mathbf{A}}\nabla\varphi)\cdot\nabla\varphi\Big{\\}}.$

Thus, $\mathcal{L}W\leq 0$ if

$s\geq\nabla\cdot(\underline{\mathbf{A}}\nabla\varphi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\varphi\quad\text{and}\quad\varphi\leq(\underline{\mathbf{A}}\nabla\varphi)\cdot\nabla\varphi.$
(4.11)

We will choose $\varphi$ to satisfy

$\underline{\mathbf{A}}\nabla\varphi=\kappa_{0}\mathbf{x},$ (4.12)

where $\kappa_{0}$ is a positive constant selected later. Equivalently, with
the use of (3.39),

$\nabla\varphi=\kappa_{0}\underline{\mathbf{A}}^{-1}\mathbf{x}=\kappa_{0}\underline{\mathbf{B}}\mathbf{x}=\kappa_{0}\phi(|\mathbf{x}|)\mathbf{x},$
(4.13)

where $\phi(r)$ is defined by (3.40). By (3.41), (4.3) and (4.4),

$\phi(r)\geq d_{0}\mu_{2}=C_{1}\quad\text{for }r_{0}\leq r\leq R.$ (4.14)

By (3.42), (4.3) and (4.4),

$\phi(r)\leq d_{4}\mu_{1}=C_{0}\quad\text{for }r_{0}\leq r\leq R.$ (4.15)

Define for $\mathbf{x}\in\bar{\mathscr{U}}$ the function

$\varphi(\mathbf{x})=\kappa_{0}\Big{(}\varphi_{0}+\int_{r_{0}}^{|\mathbf{x}|}r\phi(r)dr\Big{)},\quad\text{where
}\varphi_{0}=\frac{C_{0}r_{0}^{2}}{2}\text{ and
}\kappa_{0}=\frac{C_{0}}{2C_{1}}.$ (4.16)

Then $\varphi(\mathbf{x})$ satisfies both equations (4.12) and (4.13). We have
for $\mathbf{x}\in\bar{\mathscr{U}}$ that

$0<\varphi(\mathbf{x})\leq\kappa_{0}\Big{(}\varphi_{0}+C_{0}\int_{r_{0}}^{|\mathbf{x}|}rdr\Big{)}=\frac{\kappa_{0}C_{0}}{2}|\mathbf{x}|^{2}.$
(4.17)

Applying (4.12), (4.13), and then (3.31) and (4.4) we obtain

$(\underline{\mathbf{A}}\nabla\varphi)\cdot\nabla\varphi=\kappa_{0}^{2}\mathbf{x}^{T}\underline{\mathbf{B}}\mathbf{x}\geq
d_{0}\kappa_{0}^{2}\Big{(}\sum_{i=1}^{2}F_{i}(\hat{S}(|\mathbf{x}|))\Big{)}|\mathbf{x}|^{2}\geq
d_{0}\kappa_{0}^{2}\mu_{2}|\mathbf{x}|^{2}=\kappa_{0}^{2}C_{1}|\mathbf{x}|^{2}=\frac{\kappa_{0}C_{0}}{2}|\mathbf{x}|^{2}$
(4.18)

Then we have from (4.17) and (4.18) that
$\varphi\leq(\underline{\mathbf{A}}\nabla\varphi)\cdot\nabla\varphi$ in
$\mathscr{U}$, which is the second requirement in (4.11). On the other hand,
by (4.12) and (4.6),

$\nabla\cdot(\underline{\mathbf{A}}\nabla\varphi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\varphi=\kappa_{0}(\nabla\cdot\mathbf{x}+{\bf
b}\cdot\mathbf{x})\leq\kappa_{0}(n+C_{2}R).$ (4.19)

Select

$s=s_{R}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\kappa_{0}(n+C_{2}R).$ (4.20)

Then we have
$s\geq\nabla\cdot(\underline{\mathbf{A}}\nabla\varphi)+\mathbf{b}\cdot(\underline{\mathbf{A}}\nabla\varphi)$
in $\mathscr{U}$, which is the first requirement in (4.11). Thus, we obtain
$\mathcal{L}W\leq 0$ in $\mathscr{U}\times(0,\infty)$. For further references,
we formulate this as a lemma.

######  Lemma 4.2.

With parameter $s=s_{R}$ selected as in (4.20) and function $\varphi$ defined
by (4.16), the function $W(\mathbf{x},t)$ in (4.10) belongs to
$C_{\mathbf{x},t}^{2,1}(\mathscr{D})\cap C(\bar{\mathscr{D}})$ and satisfies
$\mathcal{L}W\leq 0$ in $\mathscr{D}$.

Above, the regularity of $W(\mathbf{x},t)$ follows the fact that
$\varphi(\mathbf{x})\geq\kappa_{0}\varphi_{0}>0$ for
$\mathbf{x}\in\bar{\mathscr{U}}$.

We now establish this section’s key lemma of growth. We fix $s=s_{R}$ by
(4.20) and also the following two parameters

$q=\frac{\kappa_{0}C_{0}}{2s}\quad\text{and}\quad\eta_{0}=\Big{(}\frac{r_{0}}{R}\Big{)}^{2s},$
(4.21)

and denote $D_{1}=U\times(0,qR^{2}]$.

######  Lemma 4.3 (Lemma of growth in time).

Assume $w(\mathbf{x},t)\in C_{\mathbf{x},t}^{2,1}(D_{1})\cap C(\bar{D}_{1})$.
If

$\mathcal{L}w\leq 0\text{ on }D_{1}\quad\text{ and }\quad w\leq 0\text{ on
}\Gamma\times(0,qR^{2}),$ (4.22)

then

$\max\\{0,\sup_{U}w(\mathbf{x},qR^{2})\\}\leq\frac{1}{1+\eta_{0}}\max\\{0,\sup_{U}w(\mathbf{x},0)\\}.$
(4.23)

###### Proof.

(i) Let $M=\max\\{0,\sup_{\bar{D}_{1}}w\\}$. By (4.22) and maximum principle,
we have

$M=\max\\{0,\sup_{\bar{U}}w(\mathbf{x},0)\\}.$ (4.24)

Let $W(\mathbf{x},t)$ be as in (4.10) and define the auxiliary function

$\tilde{W}(\mathbf{x},t)=M[1-\eta W(\mathbf{x},t)],$

where constant $\eta>0$ will be specified later. Our intention is to prove
that

$\tilde{W}(\mathbf{x},t)\geq w(\mathbf{x},t)\quad\text{for all
}(\mathbf{x},t)\in\bar{D}_{1}.$ (4.25)

By Lemma 4.2, $\mathcal{L}W\leq 0$ in $D_{1}$, hence,
$\mathcal{L}\tilde{W}\geq 0$ in $D_{1}.$ By maximum principle, it suffices to
show that

$\tilde{W}(\mathbf{x},t)\geq w(\mathbf{x},t)\quad\text{for all
}(\mathbf{x},t)\in\partial_{p}D_{1}=\big{[}\bar{U}\times\\{0\\}\big{]}\cup\big{[}\Gamma\times(0,qR^{2}]\big{]}.$
(4.26)

On the base $\bar{U}\times\left\\{0\right\\}$, function $W(\mathbf{x},0)$
vanishes, hence,

$\tilde{W}(\mathbf{x},0)=M\geq w(\mathbf{x},0).$

On the side boundary $\Gamma\times(0,qR^{2}]$, additional analysis is
required. First observe for $\mathbf{x}\in\bar{\mathscr{U}}$ that
$\varphi(\mathbf{x})\geq\kappa_{0}\varphi_{0}=\frac{\kappa_{0}C_{0}r_{0}^{2}}{2}$.
Therefore,

$\tilde{W}(\mathbf{x},t)=M\left[1-\eta
t^{-s}e^{-\frac{\varphi(\mathbf{x})}{t}}\right]\geq M\left[1-\eta
t^{-s}e^{-\frac{\kappa_{0}C_{0}r_{0}^{2}}{2t}}\right]\quad\text{in
}\bar{\mathscr{U}}\times[0,\infty).$ (4.27)

Let $h_{0}(t)=t^{-s}e^{-\frac{\kappa_{0}C_{0}r_{0}^{2}}{2t}}$ for $t\geq 0$.
By elementary calculations, the maximum of $h_{0}(t)$ is attained at
$t_{0}=\frac{\kappa_{0}C_{0}r_{0}^{2}}{2s}$. By letting

$\eta=\frac{1}{\max_{[0,\infty)}h_{0}(t)}=\frac{1}{h_{0}(t_{0})}=\Big{(}\frac{e\kappa_{0}C_{0}r_{0}^{2}}{2s}\Big{)}^{s},$
(4.28)

we get from (4.27) that $\tilde{W}(\mathbf{x},t)\geq M[1-\eta h_{0}(t_{0})]=0$
in $\bar{\mathscr{U}}\times[0,\infty)$. Particularly,

$\tilde{W}(\mathbf{x},t)\geq 0\geq w(\mathbf{x},t)\quad\text{on
}\Gamma\times(0,qR^{2}].$

Thus, the comparison in (4.26) holds and, therefore, (4.25) is proved.

We now estimate $\tilde{W}(\mathbf{x},t)$. By (4.17), for $(\mathbf{x},t)\in
D$ we have

$\displaystyle\tilde{W}(\mathbf{x},t)\leq M\left[1-\eta
t^{-s}e^{-\frac{\kappa_{0}C_{0}|\mathbf{x}|^{2}}{2t}}\right]\leq M\left[1-\eta
t^{-s}e^{-\frac{\kappa_{0}C_{0}R^{2}}{2t}}\right].$

Let $h_{1}(t)=t^{-s}e^{-\frac{\kappa_{0}C_{0}R^{2}}{2t}}$ for $t>0$. Then
$t_{1}=\frac{\kappa_{0}C_{0}R^{2}}{2s}=qR^{2}$ is the critical point and

$h_{1}(t_{1})=(qR^{2})^{-s}e^{-\frac{\kappa_{0}C_{0}}{2q}}\geq\Big{(}\frac{2s}{e\kappa_{0}C_{0}R^{2}}\Big{)}^{s}.$

Letting $t=t_{1}$ in (4.25), we have

$w(\mathbf{x},t_{1})\leq\tilde{W}(\mathbf{x},t_{1})\leq
M\left[1-\eta\Big{(}\frac{2s}{e\kappa_{0}C_{0}R^{2}}\Big{)}^{s}\right]=M(1-\eta_{0})\leq\frac{M}{1+\eta_{0}},$
(4.29)

and, hence, (4.23) follows. ∎

Using Lemma 4.3, we show the decay, as $t\to\infty$, of solution
$w(\mathbf{x},t)$ of the IBVP (4.9) in the homogeneous case, i.e., when
$f_{0}\equiv 0$ and $G\equiv 0$.

######  Proposition 4.4 (Homogeneous problem).

Assume $w(\mathbf{x},t)\in C_{\mathbf{x},t}^{2,1}(D)\cap C(\bar{D})$ satisfies

$\mathcal{L}w=0\text{ in }D\quad\text{and}\quad w=0\text{ on
}\Gamma\times(0,\infty).$ (4.30)

Then

$-e^{-\eta_{1}t}\inf_{U}|w(\mathbf{x},0)|\leq
w(\mathbf{x},t)\leq(1+\eta_{0})e^{-\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|\quad\forall(\mathbf{x},t)\in
D,$ (4.31)

where $\eta_{1}=\frac{\ln(1+\eta_{0})}{qR^{2}}$.

###### Proof.

Let $k\in\mathbb{N}$. Applying Lemma 4.3 with $D_{1}$ being replaced by
$U\times(T_{k-1},T_{k}]$ gives

$\max\\{0,\sup_{U}w(\mathbf{x},kqR^{2})\\}\leq\frac{1}{1+\eta_{0}}\max\\{0,\sup_{U}w(\mathbf{x},(k-1)qR^{2})\\}.$

By induction in $k$, we obtain

$\max\\{0,\sup_{U}w(\mathbf{x},kqR^{2})\\}\leq\frac{1}{(1+\eta_{0})^{k}}\max\\{0,\sup_{U}w(\mathbf{x},0)\\}.$
(4.32)

Now applying (4.32) to function $-w$ instead of $w$, we obtain

$\min\\{0,\inf_{U}w(\mathbf{x},kqR^{2})\\}\geq\frac{1}{(1+\eta_{0})^{k}}\min\\{0,\inf_{U}w(\mathbf{x},0)\\}.$
(4.33)

For any $t>0$, there is an integer $k\geq 0$ such that $t\in(T_{k},T_{k+1}]$
where $T_{k}=kqT^{2}$. By (4.30) and maximum principle for domain
$U\times(T_{k},T_{k+1}]$, and then using (4.32) we have

$\displaystyle w(\mathbf{x},t)$
$\displaystyle\leq\max\\{0,\sup_{U}w(\mathbf{x},T_{k})\\}\leq(1+\eta_{0})^{-k}\max\\{0,\sup_{U}w(\mathbf{x},0)\\}$
$\displaystyle=(1+\eta_{0})e^{-\eta_{1}T_{k+1}}\sup_{U}|w(\mathbf{x},0)|\leq(1+\eta_{0})e^{-\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|.$
(4.34)

Similarly, using (4.33) instead of (4.32) we have

$\displaystyle w(\mathbf{x},t)$
$\displaystyle\geq\min\\{0,\inf_{U}w(\mathbf{x},T_{k})\\}\geq(1+\eta_{0})^{-k}\min\\{0,\inf_{U}w(\mathbf{x},0)\\}$
$\displaystyle\geq-e^{-\eta_{1}T_{k}}\inf_{U}|w(\mathbf{x},0)|\geq-e^{-\eta_{1}t}\inf_{U}|w(\mathbf{x},0)|.$
(4.35)

Therefore, (4.31) follows (4) and (4). ∎

Next, we consider the non-homogeneous case for the IBVP (4.9). Similar to
(4.2), we always consider

$w_{0}\in C(\bar{U}),\ G\in C(\Gamma\times[0,\infty))\text{ and
}w_{0}(\mathbf{x})=G(\mathbf{x},0)\text{ on }\Gamma.$ (4.36)

######  Proposition 4.5 (Non-homogeneous problem).

Assume $f_{0}\in C(\bar{D})$ and

$\Delta_{1}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sup_{U\times(0,\infty)}|f_{0}(\mathbf{x},t)|+\sup_{\Gamma\times(0,\infty)}|G(\mathbf{x},t)|<\infty$
(4.37)

There is a positive constant $C$ such that if $w(\mathbf{x},t)\in
C_{\mathbf{x},t}^{2,1}(D)\cap C(\bar{D})$ is a solution of (4.9), then

$|w(\mathbf{x},t)|\leq
C\big{[}e^{-\eta_{1}t}\sup_{U}|w_{0}(\mathbf{x})|+\Delta_{1}\big{]}\quad\forall(\mathbf{x},t)\in
D,$ (4.38)

where $\eta_{1}>0$ is defined in Proposition 4.4.

###### Proof.

Denote $T_{k}=kqR^{2}$ for any integer $k\geq 0$. Let $k\in\mathbb{N}$ and

$v_{k}(\mathbf{x},t)=w(\mathbf{x},t)-\Delta_{1}(t-T_{k-1}+1)\quad\text{ for
}(\mathbf{x},t)\in\bar{U}\times[T_{k-1},T_{k}].$ (4.39)

Then $v_{k}$ satisfies

$\mathcal{L}v_{k}=\mathcal{L}w-\Delta_{1}\mathcal{=}f_{0}-\Delta_{1}\leq
0\quad\text{in }U\times(T_{k-1},T_{k}],$

and

$v_{k}(\mathbf{x},t)\leq 0\text{ on }\Gamma\times(T_{k-1},T_{k}).$

Applying Lemma 4.3 to function $v_{k}$, we have

$\max\\{0,\sup_{U}v_{k}(\mathbf{x},T_{k})\\}\leq\frac{1}{1+\eta_{0}}\max\\{0,\sup_{U}v_{k}(\mathbf{x},T_{k-1})\\}.$
(4.40)

Note that $v_{k}(\mathbf{x},T_{k})=w(\mathbf{x},T_{k})-\Delta_{1}(qR^{2}+1)$
and $v_{k}(\mathbf{x},T_{k-1})=w(\mathbf{x},T_{k-1})-\Delta_{1}\leq
w(\mathbf{x},T_{k-1})$. Hence,

$\displaystyle\max\\{0,\sup_{U}w(\mathbf{x},T_{k})\\}$
$\displaystyle\leq\max\\{0,\sup_{U}v_{k}(\mathbf{x},T_{k})\\}+\Delta_{1}(qR^{2}+1)$
$\displaystyle\leq\frac{1}{1+\eta_{0}}\max\\{0,\sup_{U}w(\mathbf{x},T_{k-1})\\}+\Delta_{1}(qR^{2}+1).$

Iterating this inequality gives

$\displaystyle\max\\{0,\sup_{U}w(\mathbf{x},T_{k})\\}$
$\displaystyle\leq\frac{1}{(1+\eta_{0})^{k}}\max\\{0,\sup_{U}w(\mathbf{x},0)\\}+\Delta_{1}(qR^{2}+1)\sum_{j=0}^{k-1}\frac{1}{(1+\eta_{0})^{j}}$
(4.41)
$\displaystyle\leq\frac{1}{(1+\eta_{0})^{k}}\max\\{0,\sup_{U}w(\mathbf{x},0)\\}+\frac{\Delta_{1}(1+qR^{2})(1+\eta_{0})}{\eta_{0}}.$

By using the relation (4.39) between $v_{k}(\mathbf{x},t)$ and
$w(\mathbf{x},t)$, maximum principle for function $v_{k}(\mathbf{x},t)$, and
estimate (4.41), we have for any $t\in[T_{k-1},T_{k}]$ with $k\geq 1$ that

$\displaystyle w(\mathbf{x},t)$ $\displaystyle\leq
v_{k}(\mathbf{x},t)+\Delta_{1}(1+qR^{2})\leq\max\\{0,\sup_{U}w(\mathbf{x},T_{k-1})\\}+\Delta_{1}(1+qR^{2})$
$\displaystyle\leq(1+\eta_{0})^{-k+1}\max\\{0,\sup_{U}w(\mathbf{x},0)\\}+\frac{\Delta_{1}(1+qR^{2})(1+\eta_{0})}{\eta_{0}}+\Delta_{1}(1+qR^{2})$
$\displaystyle\leq(1+\eta_{0})^{-\frac{t}{qR^{2}}+1}\sup_{U}|w(\mathbf{x},0)|+\frac{2\Delta_{1}(1+qR^{2})(1+\eta_{0})}{\eta_{0}}.$

Therefore,

$w(\mathbf{x},t)\leq
C\big{[}e^{-\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}\big{]}.$ (4.42)

Similarly, we obtain the same estimate for $(-w)$ and hence, (4.38) follows. ∎

For the asymptotic behavior of $w(\mathbf{x},t)$ as $t\to\infty$, we have the
following.

######  Corollary 4.6.

Assume $f_{0}\in C(\bar{D})$ is bounded and

$\Delta_{2}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\limsup_{t\to\infty}\left[\sup_{\mathbf{x}\in
U}|f_{0}(\mathbf{x},t))|+\sup_{\mathbf{x}\in\Gamma}|G(\mathbf{x},t)|\right]<\infty.$
(4.43)

There exists $C=C(\eta_{0},q,R,M)>0$ such that if $w(\mathbf{x},t)\in
C_{\mathbf{x},t}^{2,1}(D)\cap C(\bar{D})$ solves (4.9), then

$\limsup_{t\to\infty}\left[\sup_{\mathbf{x}\in U}|w(\mathbf{x},t)|\right]\leq
C\Delta_{2}.$ (4.44)

###### Proof.

Note that

$\sup_{U}|w_{0}(\mathbf{x})|+\sup_{D}|f_{0}(\mathbf{x},t)|+\sup_{\Gamma\times(0,\infty)}|G(\mathbf{x},t)|<\infty.$

Then by Proposition 4.5, $w(\mathbf{x},t)$ is bounded on $\bar{D}$. Let
$\varepsilon>0$. From our assumption there is $t_{0}>0$ such that

$\sup_{U\times[t_{0},\infty)}|f_{0}(\mathbf{x},t))|+\sup_{\Gamma\times[t_{0},\infty)}|G(\mathbf{x},t)|<\Delta_{2}+\varepsilon.$

Applying Lemma 4.5 to the domain $U\times(t_{0},\infty)$ we obtain

$|w(\mathbf{x},t)|\leq C[e^{-\eta_{1}(t-t_{0})}\sup_{\mathbf{x}\in
U}|w(\mathbf{x},t_{0})|+\Delta_{2}+\varepsilon].$ (4.45)

Therefore, passing $t\to\infty$ and then $\varepsilon\to 0$ in (4.45) yields
(4.44). ∎

Next, we estimate $|\nabla w(\mathbf{x},t)|$ by using Bernstein’s technique
(c.f. [16]).

######  Proposition 4.7.

Assume $f_{0}\in C(\bar{D})$, $\nabla f_{0}\in C(D)$, (4.37) and

$\Delta_{3}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sup_{D}|\nabla f_{0}|<\infty.$ (4.46)

For any $U^{\prime}\Subset U$ there is $\tilde{M}>0$ such that if
$w(\mathbf{x},t)\in C_{\mathbf{x},t}^{2,1}(D)\cap C(\bar{D})$ is a solution of
(4.9) that also satisfies $w\in C_{\mathbf{x}}^{3}(D)$ and $w_{t}\in
C_{\mathbf{x}}^{1}(D)$, then

$|\nabla
w(\mathbf{x},t)|\leq\tilde{M}\Big{[}1+\frac{1}{\sqrt{t}}\Big{]}\Big{[}e^{-\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}+\sqrt{\Delta}_{3}\Big{]}\quad\forall(\mathbf{x},t)\in
U^{\prime}\times(0,\infty).$ (4.47)

###### Proof.

Note that $\nabla w\in C_{\mathbf{x},t}^{2,1}(D)$. By using finite covering of
compact set $U^{\prime}$, it suffices to prove (4.47) for $\mathbf{x}$ in some
ball inside $U$. Consider a ball
$B_{{\delta}}(\mathbf{x}_{*})=\\{\mathbf{x}:|\mathbf{x}-\mathbf{x}^{*}|\leq\delta\\}\Subset
U$ with some $\mathbf{x}_{*}\in U$ and $\delta>0$. Let $t_{0}>0$, define in
the cylinder $G_{\delta}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}B_{\delta}(\mathbf{x}_{*})\times(t_{0},1+t_{0}]$
the following auxiliary function

$\tilde{w}(\mathbf{x},t)=\tau\Phi(\mathbf{x})|\nabla
w|^{2}+Nw^{2}+N_{1}(1+t_{0}-t),$ (4.48)

where

$\tau=t-t_{0}\in(0,1],\quad\Phi(\mathbf{x})=(\delta^{2}-|\mathbf{x}-\mathbf{x}_{*}|^{2})^{2}.$
(4.49)

The numbers $N,N_{1}\geq 0$ will be chosen later. We rewrite the operator
$\mathcal{L}$ as

$\mathcal{L}w=w_{t}-\sum_{i,j=1}^{n}a_{ij}(\mathbf{x})\partial_{i}\partial_{j}w-\mathbf{\tilde{b}}\cdot\nabla
w,$ (4.50)

where
$\mathbf{\tilde{b}}(\mathbf{x})=(\tilde{b}_{1},\tilde{b}_{2},\ldots,\tilde{b}_{n})\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\nabla\cdot\underline{\mathbf{A}}+\underline{\mathbf{A}}\mathbf{b}$.
Then following the calculations in Theorem 1 of section 2 on page 450 in [16]
we have

$\displaystyle\mathcal{L}\tilde{w}$ $\displaystyle\leq
2\tau\Phi\Big{\\{}\sum_{i,j,k=1}^{n}\frac{\partial{a_{ij}}}{\partial{x_{k}}}\frac{\partial{w}}{\partial{x_{k}}}\frac{\partial^{2}{w}}{\partial{x_{i}}\partial{x_{j}}}+\sum_{i,k=1}^{n}\frac{\partial{\tilde{b}_{i}}}{\partial{x_{k}}}\frac{\partial{w}}{\partial{x_{k}}}\frac{\partial{w}}{\partial{x_{i}}}-\sum_{i,j,k=1}^{n}a_{ij}\frac{\partial^{2}{w}}{\partial{x_{k}}\partial{x_{i}}}\frac{\partial^{2}{w}}{\partial{x_{k}}\partial{x_{j}}}\Big{\\}}$
(4.51) $\displaystyle\quad-(\tau\mathcal{L}(\Phi)-\Phi)|\nabla
w|^{2}-4\tau\sum_{i,j,k=1}^{n}a_{ij}\frac{\partial{\Phi}}{\partial{x_{i}}}\frac{\partial{w}}{\partial{x_{k}}}\frac{\partial^{2}{w}}{\partial{x_{k}}\partial{x_{j}}}-2N\sum_{i,j=1}^{n}a_{ij}\frac{\partial{w}}{\partial{x_{i}}}\frac{\partial{w}}{\partial{x_{j}}}$
$\displaystyle\quad-2\tau\Phi\sum_{k=1}^{n}\frac{\partial f_{0}}{\partial
x_{k}}-2Nwf_{0}-N_{1}.$

We estimate the right-hand side of (4.51) term by term. Let $\varepsilon>0$.
The numbers $K_{i}$, for $i=1,2,3\ldots$, used in the calculations below are
all positive and independent of $w$. We denote the matrix of second
derivatives of $w$ by $\nabla^{2}w$, and denote its Euclidean norm by
$|\nabla^{2}w|$. Note that $\underline{\mathbf{A}}$, $\mathbf{b}$ and
$\mathbf{\tilde{b}}$ are bounded in $B_{\delta}(\mathbf{x}^{*})$. This and
Cauchy-Schwarz inequality imply

$\displaystyle
2\tau\Phi\sum_{i,j,k=1}^{n}\frac{\partial{a_{ij}}}{\partial{x_{k}}}\frac{\partial{w}}{\partial{x_{k}}}\frac{\partial^{2}{w}}{\partial{x_{i}}\partial{x_{j}}}\leq
2C\tau\Phi|\nabla w||\nabla^{2}w|^{2}\leq\varepsilon^{-1}K_{1}|\nabla
w|^{2}+2\varepsilon\tau\Phi|\nabla^{2}w|^{2},$
$\displaystyle-(\tau\mathcal{L}(\Phi)-\Phi)|\nabla
w|^{2}+2\tau\Phi\sum_{i,k=1}^{n}\frac{\partial{\tilde{b}_{i}}}{\partial{x_{k}}}\frac{\partial{w}}{\partial{x_{k}}}\frac{\partial{w}}{\partial{x_{i}}}\leq
K_{2}|\nabla w|^{2}.$

Since $\underline{\mathbf{A}}$ is positive definite,

$\sum_{i,j,k=1}^{n}a_{ij}\frac{\partial^{2}{w}}{\partial{x_{k}}\partial{x_{i}}}\frac{\partial^{2}{w}}{\partial{x_{k}}\partial{x_{j}}}\geq
K_{3}|\nabla^{2}w|^{2},\quad\sum_{i,j=1}^{n}a_{ij}\frac{\partial{w}}{\partial{x_{i}}}\frac{\partial{w}}{\partial{x_{j}}}\geq
K_{3}|\nabla w|^{2}.$

Also, we have

$-4\tau\sum_{i,j,k=1}^{n}a_{ij}\frac{\partial{\Phi}}{\partial{x_{i}}}\frac{\partial{w}}{\partial{x_{k}}}\frac{\partial^{2}{w}}{\partial{x_{k}}\partial{x_{j}}}\leq\varepsilon^{-1}K_{4}|\nabla
w|^{2}+2\varepsilon\tau|\nabla\Phi|^{2}|\nabla^{2}w|^{2},$
$-2\tau\Phi\sum_{k=1}^{n}\frac{\partial f_{0}}{\partial x_{k}}\leq
K_{5}\Delta_{3},$

and by using estimate (4.38) for $w$,

$-2Nwf_{0}\leq
K_{6}\Delta_{1}N\big{[}e^{-\eta_{1}t_{0}}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}\big{]}.$

Combining the above estimates, we obtain from (4.51) that

$\displaystyle\mathcal{L}\tilde{w}$ $\displaystyle\leq
2\tau\Phi\Big{(}2\varepsilon+\varepsilon\frac{|\nabla\Phi|^{2}}{\Phi}-K_{3}\Big{)}|\nabla^{2}w|^{2}+\Big{(}K_{2}+\varepsilon^{-1}(K_{1}+K_{4})-2NK_{3}\Big{)}|\nabla
w|^{2}$
$\displaystyle\quad+K_{5}\Delta_{3}+K_{6}\Delta_{1}N\big{[}e^{-\eta_{1}t_{0}}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}\big{]}-N_{1}.$

Since $|\nabla\Phi|^{2}/\Phi\leq 16\delta^{2}$, we have

$\displaystyle\mathcal{L}\tilde{w}$ $\displaystyle\leq
2\tau\Phi\Big{(}K_{7}\varepsilon-
K_{3}\Big{)}|\nabla^{2}w|^{2}+\Big{(}K_{2}+K_{8}\varepsilon^{-1}-2NK_{3}\Big{)}|\nabla
w|^{2}$ (4.52)
$\displaystyle\quad+(K_{5}+K_{6}N)\big{[}\Delta_{3}+\Delta_{1}e^{-\eta_{1}t_{0}}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}^{2}\big{]}-N_{1}.$

In (4.52), choose $\varepsilon=K_{3}/K_{7}$ and
$N=[K_{2}+K_{8}\varepsilon^{-1}]/(2K_{3})$, then take

$N_{1}=(K_{5}+K_{6}N)(\Delta_{1}e^{-\eta_{1}t_{0}}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}^{2}+\Delta_{3}).$

We find that $\mathcal{L}\tilde{w}\leq 0$ in $G_{\delta}$. Applying the
maximum principle gives

$\max_{\bar{G}_{\delta}}\tilde{w}=\max\big{\\{}\tilde{w}(\mathbf{x},t):(\mathbf{x},t)\in
B_{\delta}(\mathbf{x}_{*})\times\\{t_{0}\\}\cup\partial
B_{\delta}(\mathbf{x}_{*})\times[t_{0},t_{0}+1]\big{\\}}.$ (4.53)

Note that $\tau\Phi(\mathbf{x})=0$ when $t=t_{0}$ or $\mathbf{x}\in\partial
B_{\delta}(\mathbf{x}_{*})$. Hence (4.53) implies,

$\max_{\bar{G}_{\delta}}\tilde{w}\leq
N\max_{B_{\delta}(\mathbf{x}_{*})}w^{2}(\mathbf{x},t_{0})+N\max_{\partial
B_{\delta}(\mathbf{x}_{*})\times[t_{0},t_{0}+1]}w^{2}(\mathbf{x},t)+N_{1}.$
(4.54)

Using estimate (4.38) for the first two terms on the right-hand side of (4.54)
we obtain

$\displaystyle\max_{\bar{G}_{\delta}}\tilde{w}$ $\displaystyle\leq
2K_{9}N\Big{[}e^{-\eta_{1}t_{0}}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}\Big{]}^{2}+N_{1}\leq
K_{10}\Big{[}e^{-2\eta_{1}t_{0}}\sup_{U}|w(\mathbf{x},0)|^{2}+\Delta_{1}^{2}+\Delta_{3}\Big{]}$
$\displaystyle\leq
C\Big{[}e^{-2\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|^{2}+\Delta_{1}^{2}+\Delta_{3}\Big{]}.$

Now, we consider $\mathbf{x}\in B_{\delta/2}(\mathbf{x}_{*})$. If $t\in(0,1]$
let $t_{0}=t/2$, then $t=2t_{0}\in[t_{0},1+t_{0}]$ and hence

$\frac{t}{2}|\nabla
w(\mathbf{x},t)|^{2}\min_{B_{\delta/2}(\mathbf{x}_{*})}\Phi(\mathbf{x})\leq(t-t_{0})\Phi(\mathbf{x})|\nabla
w(\mathbf{x},t)|^{2}\\\ \leq\tilde{w}(\mathbf{x},t)\leq
C\Big{[}e^{-2\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}^{2}+\Delta_{3}\Big{]}.$
(4.55)

If $t>1$ let $t_{0}=t-1/2$, then $t\in[t_{0},1+t_{0}]$ and hence

$\frac{1}{2}|\nabla
w(\mathbf{x},t)|^{2}\min_{B_{\delta/2}(\mathbf{x}_{*})}\Phi(\mathbf{x})\leq(t-t_{0})\Phi(\mathbf{x})|\nabla
w(\mathbf{x},t)|^{2}\\\ \leq\tilde{w}(\mathbf{x},t)\leq
C\Big{[}e^{-2\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}^{2}+\Delta_{3}\Big{]}.$
(4.56)

Since $\min_{B_{\delta/2}(\mathbf{x}_{*})}\Phi(\mathbf{x})>0$, it follows
(4.55) and (4.56) that

$|\nabla w(\mathbf{x},t)|\leq
M(\delta)\Big{[}1+\frac{1}{\sqrt{t}}\Big{]}\Big{[}e^{-\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}+\sqrt{\Delta_{3}}\Big{]}$
(4.57)

for $\mathbf{x}\in B_{\delta/2}(\mathbf{x}_{*})$ and $t>0$. Then using a
finite covering of $U^{\prime}$, we obtain (4.47) from (4.57). ∎

We return to the IBVP (4.1) for $\sigma(\mathbf{x},t)$ now. Recall that the
existence and uniqueness of the solution $\sigma$ were already addressed in
Theorem 4.1.

######  Theorem 4.8.

Assume (E1) and

$\Delta_{4}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sup_{D}(|\mathbf{V}(\mathbf{x},t)|+|\nabla\mathbf{V}(\mathbf{x},t)|)+\sup_{\Gamma\times[0,\infty)}|g(\mathbf{x},t)|<\infty.$
(4.58)

Then the solution $\sigma(\mathbf{x},t)$ of the IBVP (4.1) satisfies

$\sup_{\mathbf{x}\in U}|\sigma(\mathbf{x},t)|\leq
C\Big{[}e^{-\eta_{1}t}\sup_{U}|\sigma_{0}(\mathbf{x})|+\Delta_{4}\Big{]}\quad\text{for
all }t>0.$ (4.59)

Moreover,

$\limsup_{t\to\infty}\left[\sup_{\mathbf{x}\in
U}|\sigma(\mathbf{x},t)|\right]\leq C\Delta_{5},$ (4.60)

where

$\Delta_{5}=\limsup_{t\to\infty}\Big{[}\sup_{\mathbf{x}\in
U}(|\mathbf{V}(\mathbf{x},t)|+|\nabla\mathbf{V}(\mathbf{x},t)|)+\sup_{\mathbf{x}\in\Gamma}|g(\mathbf{x},t)|\Big{]}.$
(4.61)

###### Proof.

Let $w(\mathbf{x},t)=\sigma(\mathbf{x},t)e^{-\Lambda(\mathbf{x})}$,
$f_{0}(\mathbf{x},t)=e^{-\Lambda(\mathbf{x})}\nabla\cdot(\underline{\mathbf{A}}(\mathbf{x})\mathbf{c}(\mathbf{x},t))$,
$G(\mathbf{x},t)=e^{-\Lambda(\mathbf{x})}g(\mathbf{x},t)$ and
$w_{0}(\mathbf{x})=e^{-\Lambda(\mathbf{x})}\sigma_{0}(\mathbf{x})$. Then
$w(\mathbf{x},t)$ solves (4.9). We observe from (3.38) that

$|\Lambda(\mathbf{x})|\leq
d_{2}\mu_{3}(R-r_{0})\quad\forall\mathbf{x}\in\mathscr{U}.$ (4.62)

Combining with the boundedness of
$\|\underline{\mathbf{A}}\|_{C^{1}(\mathscr{U})}$, we have

$|f_{0}(\mathbf{x},t)|\leq
C(|\mathbf{V}(\mathbf{x},t)|+|\nabla\mathbf{V}(\mathbf{x},t)|)\quad\forall(\mathbf{x},t)\in
D.$ (4.63)

Thanks to these relations, the assumptions in Proposition 4.5 hold, thus, the
assertions (4.59) and (4.60) follow directly from (4.38) and (4.44). ∎

For the velocities, we have the following result.

######  Theorem 4.9.

Assume (E1) and

$\Delta_{6}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sup_{D}(|\mathbf{V}(\mathbf{x},t)|+|\nabla\mathbf{V}(\mathbf{x},t)|+|\nabla^{2}\mathbf{V}(\mathbf{x},t)|)<\infty\text{
and }\Delta_{7}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sup_{\Gamma\times[0,\infty)}|g(\mathbf{x},t)|<\infty.$
(4.64)

Then for any $U^{\prime}\Subset U$, there is a positive number $\tilde{M}$
such that for $i=1,2$, and $t>0$,

$\sup_{\mathbf{x}\in
U^{\prime}}|\mathbf{v}_{i}(\mathbf{x},t)|\leq\tilde{M}\Big{(}1+\frac{1}{\sqrt{t}}\Big{)}\Big{[}e^{-\eta_{1}t}\sup_{U}|\sigma_{0}(\mathbf{x})|+\Delta_{6}+\sqrt{\Delta}_{6}+\Delta_{7}\Big{]}.$
(4.65)

Consequently, if

$\lim_{t\to\infty}\Big{\\{}\sup_{\mathbf{x}\in
U}(|\mathbf{V}(\mathbf{x},t)|+|\nabla\mathbf{V}(\mathbf{x},t)|+|\nabla^{2}\mathbf{V}(\mathbf{x},t)|)+\sup_{\mathbf{x}\in\Gamma}|g(\mathbf{x},t)|\Big{\\}}=0,$
(4.66)

then for any $\mathbf{x}\in U$,

$\lim_{t\to\infty}\mathbf{v}_{1}(\mathbf{x},t)=\lim_{t\to\infty}\mathbf{v}_{2}(\mathbf{x},t)=0.$
(4.67)

###### Proof.

Note that solution $\sigma(\mathbf{x},t)$ of (4.1) satisfies $\sigma\in
C_{\mathbf{x}}^{3}(D)$ and $\sigma_{t}\in C_{\mathbf{x}}^{2}(D)$. Let
$w,f_{0},G,w_{0}$ be the same as in Theorem 4.8. Using the estimate of $\nabla
w$ in Lemma 4.7 and formula (3.19), we easily obtain estimate (4.65) for
$\mathbf{v}_{2}$. Then the estimate for $\mathbf{v}_{1}$ follows this and
(3.4). The proof of (4.67) is similar to that of (4.44). We take
$U^{\prime}=B_{\delta}(\mathbf{x})$ such that $U^{\prime}\Subset U$. For
$T>0$, let

$\Delta_{6,T}=\sup_{U\times[T,\infty)}(|\mathbf{V}(\mathbf{x},t)|+|\nabla\mathbf{V}(\mathbf{x},t)|+|\nabla^{2}\mathbf{V}(\mathbf{x},t)|)\text{
and }\Delta_{7,T}=\sup_{\Gamma\times[T,\infty)}|g(\mathbf{x},t)|.$

Use (4.65) for all $t>T$ and $\Delta_{6,T}$, $\Delta_{7,T}$ in place of
$\Delta_{6}$, $\Delta_{7}$. Then let $T\to\infty$ noting that $\Delta_{6,T}\to
0$ and $\Delta_{7,T}\to 0$. ∎

######  Remark 4.10.

The key ingredient of the above asymptotic results is Lemma 4.3, the lemma of
growth in time. It is worth mentioning that this result can be extended to
more general parabolic equations in more general domains $D$ in
$\mathbb{R}^{n+1}$ rather than just cylindrical-in-time domains
$D=U\times(0,\infty)$.

##  5\. Case of unbounded domain

We will analyze the linear stability of the steady flows from section 2 in an
unbounded, outer domain $U=\mathbb{R}^{n}\setminus\bar{\Omega}$, where
$\Omega$ is a simply connected, open, bounded set containing the origin. To
emphasize the ideas and techniques, we consider the simple case
$\Omega=B_{r_{0}}$ for some $r_{0}>0$.

For $R>r>0$, denote $\mathcal{O}_{r}=\mathbb{R}^{n}\setminus\bar{B}_{r}$,
$\mathcal{O}_{r,R}=B_{R}\setminus\bar{B}_{r}$, and denote their closures by
$\bar{\mathcal{O}}_{r}$ and $\bar{\mathcal{O}}_{r,R}$, respectively. Then
$U=\mathcal{O}_{r_{0}}$. Let $\Gamma=\partial
U=\\{\mathbf{x}:|\mathbf{x}|=r_{0}\\}$ and $D=U\times(0,\infty)$.

For $T>0$ we denote $U_{T}=U\times(0,T]$, then its closure is
$\bar{U}_{T}=\bar{U}\times[0,T]$ and its parabolic boundary is
$\partial_{p}U_{T}=[\bar{U}\times\\{0\\}]\cup[\Gamma\times(0,T]]$.

Same as in section 4, we consider a steady state
$(u^{*}_{1}(\mathbf{x}),u^{*}_{2}(\mathbf{x}),S_{*}(\mathbf{x}))$ in (3.2)
with $c_{1}^{2}+c_{2}^{2}>0$ and $\hat{S}(r)$ exists for all $r\geq r_{0}$. We
assume throughout this section that

$0<\underline{s}\leq\hat{S}(r)\leq\bar{s}<1\quad\forall r\geq r_{0},\text{
where }\underline{s},\bar{s}=const.$ (5.1)

For instance, in one of the cases in Theorem 2.2 if the limit
$s_{\infty}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\lim_{r\to\infty}\hat{S}(r)$, which exists
according to Theorem 2.3, belongs to the interval $(0,1)$ then (5.1) holds.

The problems of our interest are (4.1) and its transformed form (4.9).

Let $\mu_{i}$, for $i=1,2,3$, and $C_{j}$, for $j=0,1,2$, be defined as in
section 4 (see (4.4), (4.5) and (4.6)). Thanks to condition (5.1), which plays
the role of (4.3) in section 4, the main properties (4.5), (4.6) and (4.7)
still hold with $\mathscr{U}=\mathcal{O}_{r_{0},R}$ being replaced by
$\mathscr{U}=U=\mathcal{O}_{r_{0}}$.

###  5.1. Maximum principle for unbounded domain

We establish the maximum principle for equation $\mathcal{L}w=0$ in the domain
$U$ with operator $\mathcal{L}$ defined by (4.8). For $T>0$, we construct a
barrier function $W(\mathbf{x},t)$ of the form:

$W(\mathbf{x},t)\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}(T-t)^{-s}e^{\frac{\varphi(\mathbf{x})}{T-t}}\quad\text{for
}(\mathbf{x},t)\in\mathcal{O}_{r_{0},R}\times(0,T),$ (5.2)

where constant $s>0$ and function $\varphi(\mathbf{x})>0$ will be decided
later. Elementary calculations give

$\mathcal{L}W=(T-t)^{-s-2}e^{\frac{\varphi}{T-t}}\Big{\\{}(T-t)\big{(}s-\nabla\cdot(\underline{\mathbf{A}}\nabla\varphi)-{\bf
b}\cdot\underline{\mathbf{A}}\nabla\varphi\big{)}+\varphi-(\underline{\mathbf{A}}\nabla\varphi)\cdot\nabla\varphi\Big{\\}}.$

Then $\mathcal{L}W\geq 0$ if

$s\geq\nabla\cdot(A\nabla\varphi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\varphi\quad\text{and}\quad\varphi\geq(\underline{\mathbf{A}}\nabla\varphi)\cdot\nabla\varphi.$
(5.3)

Similar to section 4, we choose

$\varphi(\mathbf{x})=\kappa_{1}\Big{(}\varphi_{1}+\int_{r_{0}}^{|\mathbf{x}|}r\phi(r)dr\Big{)},\quad\text{where
}\varphi_{1}=\frac{C_{1}r_{0}^{2}}{2}>0\text{ and
}\kappa_{1}=\frac{C_{1}}{2C_{0}},$ (5.4)

and function $\phi$ is defined by (3.40). As in Lemma 4.2, we have

$\underline{\mathbf{A}}\nabla\varphi=\kappa_{1}\mathbf{x}\quad\text{and}\quad\nabla\varphi=\kappa_{1}\phi(|\mathbf{x}|)\mathbf{x}.$
(5.5)

By (3.41), $\phi(r)\geq d_{0}\mu_{2}=C_{1}>0$. Then

$\varphi(\mathbf{x})\geq\kappa_{1}\Big{(}\varphi_{1}+C_{1}\int_{r_{0}}^{|\mathbf{x}|}rdr\Big{)}=\frac{\kappa_{1}C_{1}}{2}|\mathbf{x}|^{2}.$
(5.6)

Also, we see from (5.5) and (3.31) that

$(\underline{\mathbf{A}}\nabla\varphi)\cdot\nabla\varphi=\kappa_{1}^{2}\mathbf{x}^{T}\underline{\mathbf{B}}\mathbf{x}\leq
d_{4}\kappa_{1}^{2}|\mathbf{x}|^{2}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))\leq\kappa_{1}^{2}d_{4}\mu_{1}|\mathbf{x}|^{2}=\kappa_{1}^{2}C_{0}|\mathbf{x}|^{2}=\frac{\kappa_{1}C_{1}}{2}|\mathbf{x}|^{2}.$
(5.7)

Then we have from (5.6) and (5.7) that

$\varphi(\mathbf{x})\geq(A\nabla\varphi)\cdot\nabla\varphi.$ (5.8)

By (4.6) and (5.5), we have

$\nabla\cdot(\underline{\mathbf{A}}\nabla\varphi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\varphi\leq\kappa_{1}(n+C_{2}|\mathbf{x}|)\leq
C_{3}(1+|\mathbf{x}|),\quad\text{where }C_{3}=\kappa_{1}(n+C_{2}).$

Select

$s=s_{R}\mathbin{\buildrel\rm def\over{\mathbin{=\kern-2.0pt=}}}C_{3}(1+R),$
(5.9)

then

$s\geq\nabla\cdot(\underline{\mathbf{A}}\nabla\varphi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\varphi\quad\text{in
}\mathcal{O}_{r_{0},R}.$ (5.10)

Therefore $\mathcal{L}W\geq 0$ in $\mathcal{O}_{r_{0},R}\times(0,T)$. We
summarize the above arguments in the following lemma.

######  Lemma 5.1.

Let $T>0$, $R>r_{0}$ and let the function $\varphi$ be defined by (5.4). Then
for $s=s_{R}$ in (5.9), the function $W(\mathbf{x},t)$ in (5.2) belongs to
$C_{\mathbf{x},t}^{2,1}(D)\cap C(\bar{D})$ and satisfies $\mathcal{L}W\geq 0$
in $\mathcal{O}_{r_{0},R}\times(0,T)$.

Using the above barrier function $W(x,t)$, we have the following maximum
principle.

######  Theorem 5.2.

Let $T>0$ and $w(\mathbf{x},t)$ be a bounded function in
$C^{2,1}_{\mathbf{x},t}(U_{T})\cap C(\bar{U}_{T})$ that solves
$\mathcal{L}w=f_{0}$ in $U_{T}$, where $f_{0}\in C(\bar{U}_{T})$. Then

$\sup_{\bar{U}_{T}}|w(\mathbf{x},t)|\leq\sup_{\partial_{p}U_{T}}|w(\mathbf{x},t)|+(T+1)\sup_{\bar{U}_{T}}|f_{0}|.$
(5.11)

###### Proof.

Given any $(\mathbf{x}_{0},t_{0})\in U\times(0,T)$. Let $\delta>0$ such that
$t_{0}<T-\delta$. Let $M=\sup_{\bar{U}_{T}}|w(\mathbf{x},t)|$ and
$N=\sup_{\bar{U}_{T}}|f_{0}|$ which are finite numbers. Let $\mu>0$ be
arbitrary. Select $R>0$ sufficiently large such that

$T^{-C_{3}(1+R)}e^{\frac{\kappa_{1}C_{1}R^{2}}{2T}}>M/\mu.$ (5.12)

Denote $\mathcal{C}=\mathcal{O}_{r_{0},R}\times(0,T-\delta]$. Then
$(\mathbf{x}_{0},t_{0})\in\mathcal{C}$. Let $W(\mathbf{x},t)$ be as in Lemma
5.1. We define the auxiliary function

$u(\mathbf{x},t)=w(\mathbf{x},t)-N(t+1)-\mu
W(\mathbf{x},t),\quad(\mathbf{x},t)\in\mathcal{C}.$ (5.13)

We have $u\in C^{2,1}_{\mathbf{x},t}(\mathcal{C})\cap C(\mathcal{C})$ and,
thanks to Lemma 5.1, function $u$ satisfies

$\mathcal{L}u=f_{0}-N-\mu\mathcal{L}W\leq 0\quad\text{in }\mathcal{C}.$

By the maximum principle,

$\max_{\bar{\mathcal{C}}}u=\max_{\partial_{p}\mathcal{C}}u.$ (5.14)

Let us evaluate $u(x,t)$ on the parabolic boundary $\partial_{p}\mathcal{C}$.
For any $\mathbf{x}\in\mathcal{O}_{r_{0},R}$,

$u(\mathbf{x},0)\leq w(\mathbf{x},0)-\mu W(\mathbf{x},0)=w(\mathbf{x},0)-\mu
T^{-s}e^{\frac{\varphi(\mathbf{x})}{T}}\leq w(\mathbf{x},0).$ (5.15)

For $|\mathbf{x}|=r_{0}$ and $0\leq t\leq T-\delta$,

$u(\mathbf{x},t)\leq w(\mathbf{x},t)-\mu W(\mathbf{x},t)\leq w(\mathbf{x},t).$
(5.16)

For $|\mathbf{x}|=R$ and $0\leq t\leq T-\delta$, we have from (5.6), (5.9) and
(5.12) that

$u(\mathbf{x},t)\leq
w(\mathbf{x},t)-\mu(T-t)^{-s}e^{\frac{\varphi(\mathbf{x})}{T-t}}\leq M-\mu
T^{-C_{3}(1+R)}e^{\frac{\kappa_{1}C_{1}R^{2}}{2T}}\leq 0.$ (5.17)

Hence, we have from (5.14), (5.15),(5.16) and (5.17) that

$\max_{\bar{\mathcal{C}}}u(\mathbf{x},t)\leq\max\\{0,\sup_{U}w(\mathbf{x},0),\sup_{\Gamma\times[0,T]}w(\mathbf{x},t)\\}.$
(5.18)

In particular, it follows from (5.18) that

$u(\mathbf{x}_{0},t_{0})\leq\max\\{0,\sup_{\partial_{p}U_{T}}w\\}.$ (5.19)

Now, letting $\mu\to 0$ in (5.13) yields

$w(\mathbf{x}_{0},t_{0})-N(t_{0}+1)\leq\max\\{0,\sup_{\partial_{p}U_{T}}w\\}\leq\sup_{\partial_{p}U_{T}}|w|.$

Hence,

$w(\mathbf{x}_{0},t_{0})\leq\sup_{\partial_{p}U_{T}}|w|+N(T+1).$

Repeating the above arguments for $(-w)$ gives

$|w(\mathbf{x}_{0},t_{0})|\leq\sup_{\partial_{p}U_{T}}|w|+N(T+1)$ (5.20)

for any $(\mathbf{x}_{0},t_{0})\in U\times(0,T)$. Therefore, (5.11) follows. ∎

We study the following IBVP (4.9) for $w(\mathbf{x},t)$.

Condition (E2). $F_{1},F_{2}\in C^{7}((0,1))$, $w_{0}\in C(\bar{U})$, $G\in
C(\Gamma\times[0,\infty))$ and $G(\mathbf{x},0)=w_{0}(\mathbf{x})$ on
$\Gamma$.

######  Theorem 5.3.

Assume (E2), $f_{0}\in C_{\mathbf{x}}^{5}(\bar{D})$, $\partial_{t}f_{0}\in
C_{\mathbf{x}}^{3}(\bar{D})$. Suppose $w_{0}(\mathbf{x})$, $G(\mathbf{x},t)$
and $f_{0}(\mathbf{x},t)$ are bounded functions. Then,

(i) There exists a solution $w(\mathbf{x},t)\in C^{2,1}_{\mathbf{x},t}(D)\cap
C(\bar{D})$ of (4.9) .

(ii) This solution is unique in class of locally (in time) bounded solutions,
i.e., the class of solutions $w(\mathbf{x},t)$ such that

$\sup_{U\times[0,T]}|w(\mathbf{x},t)|<\infty\quad\text{for any }T>0.$ (5.21)

(iii) Furthermore, for $(\mathbf{x},t)\in D$,

$|w(\mathbf{x},t)|\leq\Delta_{8}+\Delta_{9}(t+1),$ (5.22)

where

$\Delta_{8}=\max\\{\sup_{U}|w_{0}(\mathbf{x})|,\sup_{\Gamma\times[0,\infty)}|G(\mathbf{x},t)|\\}\quad\text{and}\quad\Delta_{9}=\sup_{D}|f_{0}|.$
(5.23)

###### Proof.

We rewrite equation in the non-divergent form

$\mathcal{L}w=\frac{\partial w}{\partial
t}-\sum_{i,j=1}^{n}a_{ij}\frac{\partial^{2}w}{\partial x_{i}\partial
x_{j}}-\sum_{i,j=1}^{n}\big{[}(a_{ij})_{x_{i}}+a_{ij}b_{i}\big{]}\frac{\partial
w}{\partial x_{j}}=0.$

Thanks to Theorem 4 p.474 of [16] and the maximum principle in Theorem 5.2,
one can prove (i), (ii) and (iii) using similar arguments presented in Theorem
4.6 of [14]. We omit the details. ∎

###  5.2. Lemma of growth in spatial variables

We now study the behavior of the solutions as $|\mathbf{x}|\to\infty$. This
requires a different type of lemma of growth and a new barrier function.

Let $R>0$ and $\ell\geq R+r_{0}$. Denote

$\mathcal{O}_{R}(\ell)=\mathcal{O}_{\ell-R,\ell+R}=\\{\mathbf{x}\in\mathbb{R}^{n}:||\mathbf{x}|-\ell|<R\\}\quad\text{and}\quad\mathcal{S}_{\ell}=\\{\mathbf{x}\in\mathbb{R}^{n}:|\mathbf{x}|=\ell\\}.$
(5.24)

Define the barrier function

$\mathcal{W}(\mathbf{x},t)=\frac{1}{(t+1)^{s}}e^{-\frac{\psi(\mathbf{x})}{t+1}}\quad\text{for
}|\mathbf{x}|\geq r_{0},\ t\geq 0,$ (5.25)

where parameter $s>0$ and function $\psi>0$. Then

$\mathcal{L}\mathcal{W}=(t+1)^{-s-2}e^{-\frac{\psi(\mathbf{x})}{t+1}}\Big{\\{}(t+1)\big{[}-s+\nabla\cdot(\underline{\mathbf{A}}\nabla\psi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\psi\big{]}+\psi-(\underline{\mathbf{A}}\nabla\psi)\cdot\nabla\psi\Big{\\}}.$
(5.26)

Hence, $\mathcal{L}\mathcal{W}\leq 0$ if

$s\geq\nabla\cdot(\underline{\mathbf{A}}\nabla\psi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\psi\quad\text{and}\quad\psi\leq(\underline{\mathbf{A}}\nabla\psi)\cdot\nabla\psi.$
(5.27)

Denote ${\boldsymbol{\xi}}(\mathbf{x})=\ell\mathbf{x}/|\mathbf{x}|$. We will
choose $\psi$ such that

$\underline{\mathbf{A}}\nabla\psi=\kappa_{2}(\mathbf{x}-\boldsymbol{\xi})\quad\text{for
some }\kappa_{2}>0.$

By (3.39) and (3.40),

$\nabla\psi=\kappa_{2}\underline{\mathbf{A}}^{-1}(\mathbf{x}-\boldsymbol{\xi})=\kappa_{2}\underline{\mathbf{B}}\mathbf{x}(|\mathbf{x}|-\ell)/|\mathbf{x}|=\kappa_{2}\phi(|\mathbf{x}|)(|\mathbf{x}|-\ell)\mathbf{x}/|\mathbf{x}|.$
(5.28)

Select

$\psi(\mathbf{x})=\kappa_{2}\int_{\ell}^{|\mathbf{x}|}(r-\ell)\phi(r)dr,\quad\text{where
}\kappa_{2}=\frac{C_{0}}{2C_{1}}$ (5.29)

and function $\phi$ is defined by (3.40). For all
$\mathbf{x}\in\mathcal{O}_{R}(\ell)$, we have from (3.42) that

$\psi(\mathbf{x})\leq\kappa_{2}C_{0}\int_{\ell}^{|\mathbf{x}|}(r-\ell)dr=\frac{\kappa_{2}C_{0}}{2}(|\mathbf{x}|-\ell)^{2}.$
(5.30)

By (3.31),

$\displaystyle(\underline{\mathbf{A}}\nabla\psi)\cdot\nabla\psi$
$\displaystyle=\kappa_{2}^{2}(\mathbf{x}-\boldsymbol{\xi})^{T}\underline{\mathbf{B}}(\mathbf{x})(\mathbf{x}-\boldsymbol{\xi})\geq
d_{0}\kappa_{2}^{2}|\mathbf{x}-\boldsymbol{\xi}|^{2}\sum_{j=1}^{2}F_{j}(S_{*}(\mathbf{x}))$
$\displaystyle\geq\kappa_{2}^{2}C_{1}(|\mathbf{x}|-\ell)^{2}=\frac{\kappa_{2}C_{0}}{2}(|\mathbf{x}|-\ell)^{2}.$

Hence this and (5.30) give
$\psi\leq(\underline{\mathbf{A}}\nabla\psi)\cdot\nabla\psi$, that is, the
second condition in (5.27). Also,

$\displaystyle\nabla\cdot(\underline{\mathbf{A}}\nabla\psi)+{\bf
b}\cdot(\underline{\mathbf{A}}\nabla\psi)$
$\displaystyle=\kappa_{2}\Big{[}\nabla\cdot(\mathbf{x}-\boldsymbol{\xi})+{\bf
b}\cdot(\mathbf{x}-\boldsymbol{\xi})\Big{]}=\kappa_{2}\Big{[}n-(n-1)\frac{\ell}{|\mathbf{x}|}+{\bf
b}\cdot(\mathbf{x}-\boldsymbol{\xi})\Big{]}$
$\displaystyle\leq\kappa_{2}(n+|{\bf b}|R).$

Then by (4.6),

$\nabla\cdot(\underline{\mathbf{A}}\nabla\psi)+{\bf
b}\cdot(\underline{\mathbf{A}}\nabla\psi)\leq\kappa_{2}(n+C_{2}R)\leq
C_{3}(1+R),$ (5.31)

where $C_{3}=\kappa_{2}(n+C_{2})$. By selecting

$s=s_{R}\mathbin{\buildrel\rm def\over{\mathbin{=\kern-2.0pt=}}}C_{3}(1+R),$
(5.32)

we have $s\geq\nabla\cdot(\underline{\mathbf{A}}\nabla\psi)+{\bf
b}\cdot(\underline{\mathbf{A}}\nabla\psi)$ which is the first condition in
(5.27). Therefore $\mathcal{L}W\leq 0$ in
$\mathcal{O}_{R}(\ell)\times(0,\infty)$. We have proved:

######  Lemma 5.4.

Given any $R>0$ and $\ell\geq R+r_{0}$. Let $s=s_{R}$ be defined by (5.32) and
the function $\psi$ be defined by (5.29). Then the function
$\mathcal{W}(\mathbf{x},t)$ in (5.25) belongs to
$C_{\mathbf{x},t}^{2,1}(D)\cap C(\bar{D})$ and satisfies
$\mathcal{L}\mathcal{W}\leq 0$ on $\mathcal{O}_{R}(\ell)\times(0,\infty)$.

Next is the lemma of growth in the spatial variables.

######  Lemma 5.5.

Given $T>0$, let

$\displaystyle R$ $\displaystyle=R(T)=C_{4}(1+T),$ (5.33)
$\displaystyle\eta_{0}$
$\displaystyle=\eta_{0}(T)=\Big{(}1-\frac{1}{2^{C_{5}(T+1)}}\Big{)}\frac{1}{(T+1)^{2C_{5}(T+1)}},$
(5.34)

where $C_{4}=\max\\{1,\frac{8C_{3}}{\kappa_{2}eC_{0}}\\}$ and
$C_{5}=C_{3}C_{4}$. Suppose $w(\mathbf{x},t)\in
C_{\mathbf{x},t}^{2,1}(U_{T})\cap C(\bar{U}_{T})$ satisfies $\mathcal{L}w\leq
0$ on $U_{T}$ and $w(\mathbf{x},0)\leq 0$ on $\bar{U}$. Let $\ell$ be any
number such that $\ell\geq R+r_{0}$, then

$\max\big{\\{}0,\sup_{\mathcal{S}_{\ell}\times[0,T]}w(\mathbf{x},t)\big{\\}}\leq\frac{1}{1+\eta_{0}}\max\big{\\{}0,\sup_{\bar{\mathcal{O}}_{R}(\ell)\times[0,T]}w(\mathbf{x},t)\big{\\}}.$
(5.35)

###### Proof.

Denote

$\displaystyle M_{\ell}$
$\displaystyle=\max\big{\\{}0,\sup_{\bar{\mathcal{O}}_{R}(\ell)\times[0,T]}w(\mathbf{x},t)|\big{\\}}\quad\text{and}\quad
m_{\ell}=\max\big{\\{}0,\sup_{\mathcal{S}_{\ell}\times[0,T]}w(\mathbf{x},t)\big{\\}}.$

Let $\mathcal{W}$ be defined as in Lemma 5.4. Let $\eta>0$ chosen later and
define

$\widetilde{W}(\mathbf{x},t)=M_{\ell}(1-\mathcal{W}(\mathbf{x},t)+\eta),$

then $\mathcal{L}\widetilde{\mathcal{W}}\geq 0$ in
$\mathcal{O}_{R}(\ell)\times(0,T]$. We have

$\widetilde{W}(\mathbf{x},0)=M_{\ell}(1-e^{-\psi(\mathbf{x})}+\eta)\geq 0\geq
w(\mathbf{x},0).$ (5.36)

By (5.30), $\psi(\mathbf{x})\leq\kappa_{2}C_{0}R^{2}/2$ when
$|\mathbf{x}|=\ell\pm R$, hence

$\widetilde{W}(\mathbf{x},t)|_{|\mathbf{x}|=\ell\pm R}\geq
M_{\ell}\Big{(}1-(t+1)^{-s}e^{-\frac{\kappa_{2}C_{0}R^{2}}{2(t+1)}}+\eta\Big{)}.$
(5.37)

Let $f(z)=z^{-s}e^{-\frac{\kappa_{2}C_{0}R^{2}}{2z}}$ for $z\geq 0$. Select
$\eta=\max_{[0,\infty)}f(z)$. Elementary calculations show
$\eta=(\frac{2s}{\kappa_{2}eC_{0}R^{2}})^{s}$. Then $t\in[0,T]$, it follows
(5.37) that

$\widetilde{W}(\mathbf{x},t)|_{|\mathbf{x}|=\ell\pm R}\geq
M_{\ell}\geq\max\\{0,w(\mathbf{x},t)|_{|\mathbf{x}|=\ell\pm R}\\}.$ (5.38)

From (5.36), (5.38) and maximum principle we obtain

$\widetilde{W}(\mathbf{x},t)\geq w(\mathbf{x},t)\quad\text{on
}\bar{\mathcal{O}}_{R}(\ell)\times(0,T).$

Particularly,

$\widetilde{W}(\mathbf{x},t)\geq w(\mathbf{x},t)\quad\text{on
}\mathcal{S}_{\ell}\times(0,T).$ (5.39)

Moreover, since $\psi(\mathbf{x})=0$ when $|\mathbf{x}|=\ell$,
$\mathcal{W}(\mathbf{x},t)\geq\frac{1}{(T+1)^{s}}$ thus

$\widetilde{W}(\mathbf{x},t)|_{|\mathbf{x}|=\ell}\leq
M_{\ell}\Big{[}1-\frac{1}{(T+1)^{s}}+\eta\Big{]}.$ (5.40)

Since $R\geq 1$, we easily estimate

$\eta=\Big{[}\frac{2C_{3}(1+R)}{\kappa_{2}eC_{0}R^{2}}\Big{]}^{C_{3}(1+R)}\leq\Big{(}\frac{4C_{3}R}{\kappa_{2}eC_{0}R^{2}}\Big{)}^{C_{3}(1+R)}\leq\Big{(}\frac{C_{4}}{2R}\Big{)}^{C_{3}(1+R)}.$

Hence

$\displaystyle\frac{1}{(T+1)^{s}}-\eta$
$\displaystyle\geq\frac{1}{(T+1)^{C_{3}(1+R)}}-\Big{(}\frac{C_{4}}{2R}\Big{)}^{C_{3}(1+R)}=\Big{(}1-\frac{1}{2^{C_{3}(R+1)}}\Big{)}\frac{1}{(T+1)^{C_{3}(1+R)}}$
(5.41)
$\displaystyle\geq\Big{(}1-\frac{1}{2^{C_{3}R}}\Big{)}\frac{1}{(T+1)^{2C_{3}R}}=\Big{(}1-\frac{1}{2^{C_{5}(T+1)}}\Big{)}\frac{1}{(T+1)^{2C_{5}(T+1)}}=\eta_{0}.$

From (5.39), (5.40) and (5.41) we obtain $(1-\eta_{0})M_{\ell}\geq m_{\ell}$,
thus, $M_{\ell}\geq\frac{m_{\ell}}{1-\eta_{0}}\geq(1+\eta_{0})m_{\ell}$, which
gives (5.35). ∎

######  Lemma 5.6.

Let $T>0$ and $R$, $\eta_{0}$ and $w(\mathbf{x},t)$ be as in Lemma 5.5. For
$i\geq 1$, let

$\bar{m}_{i}=\max\big{\\{}0,\sup_{\mathcal{S}_{r_{0}+iR}\times[0,T]}w(\mathbf{x},t)\big{\\}}.$
(5.42)

Part A (Dichotomy for one cylinder). Then for any $i\geq 1$, we have either of
the following cases.

  * (a)

If $\bar{m}_{i+1}\geq\bar{m}_{i-1}$, then
$\bar{m}_{i+1}\geq(1+\eta_{0})\bar{m}_{i}$.

  * (b)

If $\bar{m}_{i-1}\geq\bar{m}_{i+1}$, then
$\bar{m}_{i-1}\geq(1+\eta_{0})\bar{m}_{i}$.

Part B (Dichotomy for many cylinders). For any $k\geq 0$, we have the
following two possibilities:

  * (i)

There is $i_{0}\geq k+1$ such that
$\bar{m}_{i_{0}+j}\geq(1+\eta_{0})^{j}\bar{m}_{i_{0}}$ for all $j\geq 0$.

  * (ii)

For all $j\geq 0$, $\bar{m}_{k+j}\leq(1+\eta_{0})^{-j}\bar{m}_{k}$.

###### Proof.

Part A. By maximum principle,

$\displaystyle\sup_{\bar{\mathcal{O}}_{R}(r_{0}+iR)\times[0,T]}w(\mathbf{x},t)$
$\displaystyle\leq\max\big{\\{}\sup_{\mathcal{S}_{r_{0}+(i\pm
1)R}\times[0,T]}w(\mathbf{x},t),\sup_{\bar{\mathcal{O}}_{R}(r_{0}+iR)}w(\mathbf{x},0)\big{\\}}$
$\displaystyle\leq\max\big{\\{}\sup_{\mathcal{S}_{r_{0}+(i\pm
1)R}\times[0,T]}w(\mathbf{x},t),0\big{\\}}\leq\max\\{\bar{m}_{i-1},\bar{m}_{i+1}\\}.$

Hence,

$\sup_{\bar{\mathcal{O}}_{R}(r_{0}+iR)\times[0,T]}w(\mathbf{x},t)\leq\max\\{\bar{m}_{i-1},\bar{m}_{i+1}\\}.$
(5.43)

Let $\ell=r_{0}+iR$. Applying Lemma 5.5 and (5.43), we obtain

$\bar{m}_{i}\leq\frac{1}{1+\eta_{0}}\max\big{\\{}0,\sup_{\bar{\mathcal{O}}_{R}(r_{0}+iR)\times[0,T]}w(\mathbf{x},t)\big{\\}}\leq\frac{1}{1+\eta_{0}}\max\\{\bar{m}_{i-1},\bar{m}_{i+1}\\}.$

Then the statements (a) and (b) obviously follow.

Part B. For $i<j$, define the cylinder

$\mathcal{C}_{i,j}=\mathcal{O}_{r_{0}+iR,r_{0}+jR}\times(0,T)=\\{(\mathbf{x},t):r_{0}+iR<|\mathbf{x}|<r_{0}+jR,\
t\in(0,T)\\}.$

We say that (a) and (b) above are two cases for cylinder
$\mathcal{C}_{i-1,i+1}$.

Let $k\geq 0$. By Part A, we have either of the following two cases.

Case 1. There is $i_{0}\geq k$ such that Case (a) holds for
$\mathcal{C}_{i_{0},i_{0}+2}$, that is,

$\bar{m}_{i_{0}+2}\geq\bar{m}_{i_{0}}\quad\text{ and
}\quad\bar{m}_{i_{0}+2}\geq(1+\eta_{0})\bar{m}_{i_{0}+1}.$ (5.44)

Then applying Part A to $\mathcal{C}_{i_{0}+1,i_{0}+3}$ we have either

$\text{Case (a) holds for $\mathcal{C}_{i_{0}+1,i_{0}+3}$, which gives
}\bar{m}_{i_{0}+3}\geq\bar{m}_{i_{0}+1}\text{ and
}\bar{m}_{i_{0}+3}\geq(1+\eta_{0})\bar{m}_{i_{0}+2},$ (5.45)

or

$\text{Case (b) holds for $\mathcal{C}_{i_{0}+1,i_{0}+3}$, which gives
}\bar{m}_{i_{0}+1}\geq\bar{m}_{i_{0}+3}\text{ and
}\bar{m}_{i_{0}+1}\geq(1+\eta_{0})\bar{m}_{i_{0}+2}.$ (5.46)

Observe that (5.44) and (5.46) hold simultaneously if only if

$\bar{m}_{i_{0}}=\bar{m}_{i_{0}+1}=\bar{m}_{i_{0}+2}=\bar{m}_{i_{0}+3}=0,$
(5.47)

which is a special case of (5.45). Hence we always have Case (a) for the next
cylinder $\mathcal{C}_{i_{0}+1,i_{0}+3}$. Then by induction, Case (a) holds
for the cylinders $\mathcal{C}_{i_{0}+j-1,i_{0}+j+1}$ for all $j\geq 1$. Thus,

$\bar{m}_{i_{0}+j+1}\geq(1+\eta_{0})\bar{m}_{i_{0}+j}\geq(1+\eta_{0})^{2}\bar{m}_{i_{0}+j-1}\geq\ldots\geq(1+\eta_{0})^{j}\bar{m}_{i_{0}+1}.$
(5.48)

Re-indexing $i_{0}+1$ by $i_{0}$ in (5.48), we obtain (i).

Case 2. For all $i\geq k$, Case (b) holds for $\mathcal{C}_{i,i+2}$, that is,
$\bar{m}_{i}\geq(1+\eta_{0})\bar{m}_{i+1}$ for all $i\geq k$. Therefore,

$\bar{m}_{k}\geq(1+\eta_{0})\bar{m}_{k+1}\geq(1+\eta_{0})^{2}\bar{m}_{k+2}\geq\ldots\geq(1+\eta_{0})^{j}\bar{m}_{k+j},$
(5.49)

which implies (ii). ∎

Using the above dichotomy, we obtain the behavior of a sub-solution $w$ as
$|\mathbf{x}|\to\infty$.

######  Proposition 5.7.

Assume $w\in C_{\mathbf{x},t}^{2,1}(U_{T})\cap C(\bar{U}_{T})$ satisfies
$w(\mathbf{x},0)\leq 0$ in $U$, $\mathcal{L}w\leq 0$ on $U_{T}$, and
$w(\mathbf{x},t)$ is bounded on $\bar{U}_{T}$. Then

$\limsup_{r\to\infty}(\sup_{\mathcal{S}_{r}\times[0,T]}w(\mathbf{x},t))\leq
0.$ (5.50)

###### Proof.

Let $\bar{m}_{i}$ be defined as in Lemma 5.6.

Case 1: There are infinitely many $i$ such that $\bar{m}_{i}=0$. Then there is
a sequence $\\{i_{l}\\}$ increasing to $\infty$ as $l\to\infty$ such that
$\bar{m}_{i_{l}}=0$ for all $l\geq 1$. Then by maximum principle for cylinder
$\mathcal{C}_{i_{l},i_{l+1}}$ we have $w(\mathbf{x},t)\leq 0$ on
$\mathcal{C}_{i_{l},i_{l+1}}$ for all $l\geq 1$. Therefore
$w(\mathbf{x},t)\leq 0$ in $\\{|\mathbf{x}|\geq r_{0}+i_{1}R\\}\times[0,T]$.
This gives (5.50).

Case 2: There are only finitely many $i$ such that $\bar{m}_{i}=0$. Then there
is $N>0$ such that $\bar{m}_{i}>0$ for all $i\geq N$. We apply part B of Lemma
5.6 to $k=N$. If (i) holds, then there is $i_{0}\geq N+1$ such that
$\bar{m}_{i_{0}+j}\geq(1+\eta_{0})^{j}\bar{m}_{i_{0}}>0$ for all $j\geq 0$;
thus, $\lim_{j\to\infty}\bar{m}_{i_{0}+j}=\infty$ which contradicts
$w(\mathbf{x},t)$ being bounded on $U_{T}$. Hence we must have (ii), that is,
for all $j\geq 0$, $\bar{m}_{N+j}\leq(1+\eta_{0})^{-j}\bar{m}_{N}$. Therefore,
$\lim_{j\to\infty}\bar{m}_{N+j}=0$ which, in combining with (5.43), proves
(5.50). ∎

As for solutions of the IBVP (4.9) in a finite time interval, we have the
following.

######  Theorem 5.8.

Let $w\in C_{\mathbf{x},t}^{2,1}(U_{T})\cap C(\bar{U}_{T})$ be a bounded
solution of (4.9) on $U_{T}$ with $f_{0}\in C(\bar{U}_{T})$. If

$\lim_{|\mathbf{x}|\to\infty}w_{0}(\mathbf{x})=0,$ (5.51)
$\lim_{|\mathbf{x}|\to\infty}\sup_{0\leq t\leq T}|f_{0}(\mathbf{x},t))|=0,$
(5.52)

then

$\lim_{r\to\infty}\Big{(}\sup_{\mathcal{S}_{r}\times[0,T]}|w(\mathbf{x},t)|\Big{)}=0.$
(5.53)

###### Proof.

Note that $w_{0}\in C(\bar{U})$, $G\in C(\Gamma\times[0,T])$. By Theorem 5.2,
$w(\mathbf{x},t)$ is bounded on $\bar{U}_{T}$. Let $\varepsilon$ be an
arbitrary positive number. There is $\tilde{r}_{0}>0$ such that for
$|\mathbf{x}|>\tilde{r}_{0}$ we have

$|w_{0}(\mathbf{x})|<\varepsilon\quad\text{ and }\quad\sup_{0\leq t\leq
T}|f_{0}(\mathbf{x},t)|<\varepsilon.$ (5.54)

Let $\tilde{w}=\pm w-\varepsilon(t+1)$ then $\tilde{w}$ is bounded on
$\bar{U}_{T}$ and $\mathcal{L}\tilde{w}<0$ on
$\mathcal{O}_{\tilde{r}_{0}}\times(0,T]$, and $\tilde{w}(\mathbf{x},0)\leq 0$
on $\bar{\mathcal{O}}_{\tilde{r}_{0}}$. Applying Proposition 5.7 to
$\tilde{w}$ with $r_{0}$ being replaced by $\tilde{r}_{0}$ gives

$\limsup_{r\to\infty}(\sup_{\mathcal{S}_{r}\times[0,T]}\tilde{w}(\mathbf{x},t))\leq
0.$

This implies

$\limsup_{r\to\infty}(\sup_{\mathcal{S}_{r}\times[0,T]}[\pm
w(\mathbf{x},t)])\leq\varepsilon(T+1).$

Therefore,

$\limsup_{r\to\infty}(\sup_{\mathcal{S}_{r}\times[0,T]}|w(\mathbf{x},t)|)\leq\varepsilon(T+1).$

Letting $\varepsilon\to 0$ we obtain (5.53). ∎

We now consider problem (4.9) for all $t>0$ under condition (5.51). Although
it is not known whether $\lim_{t\to\infty}w(\mathbf{x},t)$ exists for each
$\mathbf{x}$, we prove in the corollary below that such limit is zero along
some curve $\mathbf{x}(t)$ which goes to infinity as $t\to\infty$.

######  Corollary 5.9.

Let $w(\mathbf{x},t)\in C_{\mathbf{x},t}^{2,1}(D)\cap C(\bar{D})$ be a bounded
solution of (4.9) on $D$ with $f_{0}\in C(\bar{D})$. Assume $w_{0}\in
C(\bar{U})$ satisfies (5.51), $G\in C(\Gamma\times[0,\infty))$ is bounded, and
(5.52) holds for each $T>0$. Then there exists an increasing, continuous
function $r(t)>0$ satisfying $\lim_{t\to\infty}r(t)=\infty$ such that

$\lim_{t\to\infty}\Big{(}\sup_{\mathbf{x}\in\bar{\mathcal{O}}_{r(t)}}|w(\mathbf{x},t)|\Big{)}=0.$
(5.55)

###### Proof.

By Theorem 5.8, there exists a strictly increasing sequence
$\\{r_{k}\\}_{k=1}^{\infty}$ of positive numbers such that
$\lim_{k\to\infty}r_{k}=\infty$ and

$\sup_{\\{\mathbf{x}:|\mathbf{x}|\geq
r_{k}\\}\times[0,k]}|w(\mathbf{x},t)|<\frac{1}{k}.$ (5.56)

Let $r(t)$ be the piecewise linear function passing through the points
$(k,r_{k+1})$ then $r(t)$ is increasing and $r(t)\to\infty$ as $t\to\infty$.
By (5.56), for each $k$ we have

$\sup\\{|w(\mathbf{x},t)|:k\leq t\leq k+1,|\mathbf{x}|\geq
r(t)\\}\leq\sup_{\\{\mathbf{x}:|\mathbf{x}|\geq
r_{k+1}\\}\times[0,k+1]}|w(\mathbf{x},t)|<\frac{1}{k+1}.$

Taking $k\to\infty$ we obtain (5.55). ∎

We now return to the IBVP (4.1) for $\sigma$. We will use the transformation
$\sigma=we^{\Lambda}$. To compare $\sigma$ and $w$, we need to estimate
$\Lambda(\mathbf{x})$. Recall from (3.37) that

$\displaystyle\Lambda(\mathbf{x})$
$\displaystyle=\int_{r_{0}}^{|\mathbf{x}|}\tilde{F}(r)dr,\text{ where
}\tilde{F}(r)=F^{\prime}_{2}(\hat{S}(r))g_{2}(\frac{|c_{2}|}{r^{n-1}})\frac{c_{2}}{r^{n-1}}-F^{\prime}_{1}(\hat{S}(r))g_{1}(\frac{|c_{1}|}{r^{n-1}})\frac{c_{1}}{r^{n-1}}.$

For $R$ sufficiently large and $r\geq R$, we have $|\tilde{F}(r)|\leq
Cr^{1-n}$. Then we have in the case $n\geq 3$ that $|\tilde{F}(r)|\leq
Cr^{-2}$, hence $|\Lambda(\mathbf{x})|\leq C_{6}$ for all $|\mathbf{x}|\geq
r_{0}$, and

$0<C_{7}^{-1}\leq e^{\Lambda(\mathbf{x})}\leq
C_{7}\quad\forall|\mathbf{x}|\geq r_{0}.$ (5.57)

######  Theorem 5.10.

Let $n\geq 3$. Assume (E1) and

$\Delta_{10}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\max\\{\sup_{U}|\sigma_{0}(\mathbf{x})|,\sup_{\Gamma\times[0,\infty)}|g(\mathbf{x},t)|\\}<\infty,$
(5.58) $\Delta_{11}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sup_{D}|\nabla\cdot(\underline{\mathbf{A}}(\mathbf{x})\mathbf{c}(\mathbf{x},t))|<\infty.$
(5.59)

Then,

(i) There exists a solution $\sigma(\mathbf{x},t)\in
C^{2,1}_{\mathbf{x},t}(D)\cap C(\bar{D})$ of problem (4.1). This solution is
unique in class of solutions $\sigma(\mathbf{x},t)$ that satisfy

$\sup_{U\times[0,T]}|\sigma(\mathbf{x},t)|<\infty\quad\text{for any }T>0.$
(5.60)

(ii) There is $C>0$ such that for $(\mathbf{x},t)\in D$,

$|\sigma(\mathbf{x},t)|\leq C\big{[}\Delta_{10}+\Delta_{11}(t+1)\big{]}.$
(5.61)

(iii) In addition, if

$\lim_{|\mathbf{x}|\to\infty}\sigma_{0}(\mathbf{x})=0\quad\text{and}\quad\lim_{|\mathbf{x}|\to\infty}\sup_{0\leq
t\leq
T}|\nabla\cdot(\underline{\mathbf{A}}(\mathbf{x})\mathbf{c}(\mathbf{x},t))|=0\text{
for each }T>0,$ (5.62)

then

$\lim_{r\to\infty}\Big{(}\sup_{\mathcal{S}_{r}\times[0,T]}|\sigma(\mathbf{x},t)|\Big{)}=0\quad\text{for
any }T>0,$ (5.63)

and furthermore, there is a continuous, increasing function $r(t)>0$ with
$\lim_{t\to\infty}r(t)=\infty$ such that

$\lim_{t\to\infty}\Big{(}\sup_{\mathbf{x}\in\bar{\mathcal{O}}_{r(t)}}|\sigma(\mathbf{x},t)|\Big{)}=0.$
(5.64)

###### Proof.

Let $w_{0}(\mathbf{x})=\sigma_{0}(\mathbf{x})e^{-\Lambda(\mathbf{x})}$,
$G(\mathbf{x},t)=g(\mathbf{x},t)e^{-\Lambda(\mathbf{x})}$ and
$f_{0}(x,t)=e^{-\Lambda(\mathbf{x})}|\nabla\cdot(\underline{\mathbf{A}}(\mathbf{x})\mathbf{c}(\mathbf{x},t))|$.
Thanks to (5.57) and (5.58), we have

$\max\\{\sup_{U}|w_{0}(\mathbf{x})|,\sup_{\Gamma\times[0,\infty)}|w(\mathbf{x},t|\\}\leq
C\Delta_{10},$ $\sup_{D}|f_{0}|\leq C\Delta_{11}.$

Then statements in (i), (ii) and (iii) follow directly from Theorems 5.3 and
5.8, and Corollary 5.9 for problem (4.9), the relation
$\sigma(\mathbf{x},t)=w(\mathbf{x},t)e^{\Lambda(\mathbf{x})}$ and the
boundedness of $e^{\Lambda(\mathbf{x})}$ in (5.57). We omit the details. ∎

As a consequence of (5.64), for any continuous curve $\mathbf{x}(t)$ with
$|\mathbf{x}(t)|\geq r(t)$, one has

$\lim_{t\to\infty}\sigma(\mathbf{x}(t),t)=0.$ (5.65)

The case $n=2$ is treated next with some restriction on the steady state.

######  Theorem 5.11.

Let $n=2$ and $\hat{S}(r)$ be a solution of (2.13) with $c_{1},c_{2}<0$.
Assume (E1) and

$\Delta_{12}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\max\\{\sup_{U}e^{-\Lambda(\mathbf{x})}|\sigma_{0}(\mathbf{x})|,\sup_{\Gamma\times[0,\infty)}|g(\mathbf{x},t)|\\}<\infty,$
(5.66) $\Delta_{13}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sup_{D}e^{-\Lambda{(\mathbf{x})}}|\nabla\cdot(\underline{\mathbf{A}}(\mathbf{x})\mathbf{c}(\mathbf{x},t))|<\infty.$
(5.67)

Then the following statements hold true.

(i) There exists a solution $\sigma(\mathbf{x},t)\in
C^{2,1}_{\mathbf{x},t}(D)\cap C(\bar{D})$ of problem (4.1). This solution is
unique in class of solutions $\sigma(\mathbf{x},t)$ that satisfy

$\sup_{U\times[0,T]}e^{-\Lambda(\mathbf{x})}|\sigma(\mathbf{x},t)|<\infty\quad\text{for
any }T>0.$ (5.68)

(ii) There is $C>0$ such that for $(\mathbf{x},t)\in D$,

$|\sigma(\mathbf{x},t)|\leq C\big{[}\Delta_{12}+\Delta_{13}(t+1)\big{]}.$

(iii)  Statement (iii) of Theorem 5.10 holds true if condition (5.62) is
replaced by

$\lim_{|\mathbf{x}|\to\infty}e^{-\Lambda(\mathbf{x})}\sigma_{0}(\mathbf{x})=0\quad\text{and}\quad\lim_{|\mathbf{x}|\to\infty}\sup_{0\leq
t\leq
T}e^{-\Lambda(\mathbf{x})}|\nabla\cdot(\underline{\mathbf{A}}(\mathbf{x})\mathbf{c}(\mathbf{x},t))|=0\text{
for each }T>0.$ (5.69)

###### Proof.

According to Theorem 2.5, $\lim_{r\to\infty}\hat{S}(r)=s^{*}\in(0,1),$ where
$s^{*}$ is defined in (2.33). The proof consists of two steps.

Step 1. We show that statements (i)–(iii) hold true under the following
condition

$F_{2}^{\prime}(s^{*})a_{2}^{0}c_{2}-F_{1}^{\prime}(s^{*})a_{1}^{0}c_{1}<0.$
(5.70)

Let
$c_{4}=-(F_{2}^{\prime}(s^{*})a_{2}^{0}c_{2}-F_{1}^{\prime}(s^{*})a_{1}^{0}c_{1})>0$.
We have for any $R>r_{0}$ and $|\mathbf{x}|>R$ that

$\Lambda(\mathbf{x})=\int_{r_{0}}^{R}\tilde{F}(r)dr+\int_{R}^{|\mathbf{x}|}\tilde{F}(r)dr=I_{1}(R)+I_{2}(R).$

For sufficiently large $R_{0}>r_{0}$, we have for $|\mathbf{x}|>R_{0}$ that

$I_{2}(R_{0})\leq\frac{1}{2}\int_{R_{0}}^{|\mathbf{x}|}\big{(}F^{\prime}_{2}(\hat{S}(r))a_{2}^{0}c_{2}-F^{\prime}_{1}(\hat{S}(r))a_{1}^{0}c_{1}\big{)}r^{-1}dr\leq-\frac{1}{4}\int_{R_{0}}^{|\mathbf{x}|}c_{4}r^{-1}d\xi\leq
0.$

Obviously, $I_{1}(R_{0})$ is finite. This gives $e^{\Lambda(\mathbf{x})}\leq
C_{8}<\infty$ for all $|\mathbf{x}|\geq r_{0}$. Thus,

$|\sigma|\leq C_{9}|w|\quad\text{with constant }C_{9}>0.$ (5.71)

Setting $w(\mathbf{x},t)=\sigma(\mathbf{x},t)e^{-\Lambda(\mathbf{x})}$, we
have $\mathcal{L}w=f_{0}$, where $f_{0}$ is as in Theorem 5.10. Then (i)–(iii)
easily follow Theorems 5.3, 5.8, Corollary 5.9 and relation (5.71).

Step 2. Now, it suffices to show that condition (5.70) is satisfied with
$c_{1},c_{2}<0$. On the one hand, we have from (2.33) that

$\frac{a_{1}^{0}c_{1}}{a_{2}^{0}c_{2}}=f(s^{*})=\frac{f_{1}}{f_{2}}(s^{*})=\frac{F_{2}(s^{*})}{F_{1}(s^{*})}.$

Then
$a_{1}^{0}c_{1}F_{1}(s^{*})=a_{2}^{0}c_{2}F_{2}(s^{*})\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\mathcal{A}\neq 0$. On the other hand,

$\displaystyle
F_{2}^{\prime}(s^{*})a_{2}^{0}c_{2}-F_{1}^{\prime}(s^{*})a_{1}^{0}c_{1}=\mathcal{A}\Big{[}\frac{F_{2}^{\prime}(s^{*})}{F_{2}(s^{*})}-\frac{F_{1}^{\prime}(s^{*})}{F_{1}(s^{*})}\Big{]}=\mathcal{A}\frac{F_{1}(s^{*})}{F_{2}(s^{*})}\Big{(}\frac{F_{2}}{F_{1}}\Big{)}^{\prime}(s^{*}).$

The assumptions on $f_{1}$ and $f_{2}$ provide
$(F_{2}/F_{1})^{\prime}(s^{*})=(f_{1}/f_{2})^{\prime}(s^{*})>0$ and
$F_{1}(s^{*}),F_{2}(s^{*})>0$. Since $c_{1},c_{2}<0$, we have $\mathcal{A}<0$
and, hence,
$F_{2}^{\prime}(s^{*})a_{2}^{0}c_{2}-F_{1}^{\prime}(s^{*})a_{1}^{0}c_{1}<0$.
The proof is complete. ∎

######  Remark 5.12.

Similar to Theorem 4.9, we can use Bernstein’s technique to estimate
$\mathbf{v}_{1}(\mathbf{x},t)$ and $\mathbf{v}_{2}(\mathbf{x},t)$ uniformly in
$\mathbf{x}\in U^{\prime}\Subset U$. We do not provide details here.

## Appendix A

We give proof to the statements on the range of $s_{\infty}$ in Example 2.6.
Recall that $s_{\infty}\in[0,1]$.

In the case $\Delta=0$ of A and B, $h(r)\equiv s^{*}$ is the equilibrium and
the conclusions are clear. Also, for C and D, $S(r)$ is monotone and the
statements easily follow. We focus on the remaining cases.

A. $c_{1},c_{2}>0$. Note that $F(r,S)>0$ iff $S>h(r)$, hence $S^{\prime}(r)>0$
iff $S(r)>h(r)$.

  * •

$\Delta<0$. Then $h(r)$ increases and $h(r)<s^{*}$ for all $r$. Consider
$s_{0}>s^{*}$. Then $S(r)>s^{*}>h(r)$ for all $r$. It follows that $S(r)$ is
strictly increasing which implies $s_{\infty}>s_{0}$. Now, consider
$s_{0}<h(r_{0})$. Then $S(r)<h(r)$ for all $r$, thus $S(r)$ is strictly
decreasing and, therefore, $s_{\infty}<s_{0}$.

  * •

$\Delta>0$. In this case, $h(r)$ is decreasing, and $h(r)>s^{*}$ for all $r$.
Then the arguments are the same as in the case $\Delta<0$.

B. $c_{1},c_{2}<0$. Observe that $F(r,S)>0$ iff $S<h(r)$, hence
$S^{\prime}(r)>0$ iff $S(r)<h(r)$.

  * •

$\Delta<0$. Then $h(r)$ is increasing and $h(r)<s^{*}$ for all $r$.

We prove (iii) first when $s_{0}<h(r_{0})$. Exactly the same as Claim 2 in the
proof of Theorem 2.3, we have $S(r)\leq h(r)<s^{*}$ for all $r$. Thus $S(r)$
is increasing on $[r_{0},\infty)$. Hence $s_{\infty}\in[s_{0},s^{*}]$. Since
$S(r)$ is strictly increasing for $r$ near $r_{0}$, we have
$s_{\infty}>s_{0}$.

We prove (ii). Consider the subcase $h(r_{0})<s_{0}\leq s^{*}$. Then there
exists $r_{1}>r_{0}$ such that $S(r)>h(r)$ for $r<r_{1}$ and
$S(r_{1})=h(r_{1})$. Similar arguments to (iii), we have $S(r_{1})\leq
S(r)\leq h(r)$ for all $r<r_{1}$. Hence $s_{\infty}\leq s^{*}$ and
$s_{\infty}\geq h(r_{1})>h(r_{0})$.

In the particular case $s_{0}=h(r_{0})$, one can show that $h(r_{0})\leq
S(r)\leq h(r)$ for all $r>r_{0}$. If $S(r)\equiv h(r)$ then
$s_{\infty}=s^{*}$. Otherwise, there is $r_{1}>r_{0}$ and such that
$h(r_{0})\leq S(r_{1})<h(r_{1})$. Similar to (iii) with $r_{0}$ playing the
role of $r_{1}$, we have $s_{\infty}\in(S(r_{1}),s^{*}]$. Hence
$s_{0}\in(h(r_{0}),s^{*}]$.

Finally, we prove (i) when $s_{0}>s^{*}$. Clearly, $S(r)<s_{0}$ for all
$r>r_{0}$. If $s_{0}>S(r)>s^{*}$ for all $r>r_{0}$ then we have $S(r)$
strictly deceasing and $s_{\infty}\in[s^{*},s_{0})$. Otherwise, there is
$r_{1}$ such that $S(r_{1})=s^{*}$. Then using (ii) we obtain
$s_{\infty}\in(h(r_{0}),s_{*}]$.

  * •

$\Delta>0$. Then $h(r)$ is decreasing, and $h(r)>s^{*}$ for all $r$. The proof
is similar to the case $\Delta<0$.

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