# Asymptotic stability for a free boundary tumor model with a periodic supply
of external nutrients

Yaodan Huang  School of Mathematics, Sun Yat-Sen University, Guangzhou,
510275, China<EMAIL_ADDRESS>
###### Abstract.

For tumor growth, the morphological instability provides a mechanism for
invasion via tumor fingering and fragmentation. This work considers the
asymptotic stability of a free boundary tumor model with a periodic supply of
external nutrients. The model consists of two elliptic equations describing
the concentration of nutrients and the distribution of the internal pressure
in the tumor tissue, respectively. The effect of the parameter $\mu$
representing a measure of mitosis on the morphological stability are taken
into account. It was recently established in [25] that there exists a critical
value $\mu_{\ast}$ such that the unique spherical periodic positive solution
is linearly stable for $0<\mu<\mu_{\ast}$ and linearly unstable for
$\mu>\mu_{\ast}$. In this paper, we further prove that the spherical periodic
positive solution is asymptotically stable for $0<\mu<\mu_{\ast}$ for the
fully nonlinear problem.

Corresponding author: Yaodan Huang

Keywords: Free boundary problems; Tumor growth; Periodic solution; Nonlinear
stability

2020 Mathematics Subject Classification: 35B10, 35B35, 35R35, 92C37

##  1\. Introduction

Within the past several decades a number of mathematical models have been
developed and studied that aimed at describing the evolution to carcinomas in
the form of free boundary problems. Choices have to be made for a variety of
different models to explore the mechanisms of tumor growth, such as several
types of cells (see, for instance, [32, 30]), the impact of inhibitors (cf.
[3]). Accordingly, many significant results have been obtained for rigorous
mathematical analysis and numerical simulation of such tumor models; see [6,
8, 10, 11, 15, 20, 21, 23, 24, 26, 27, 29, 31, 33, 35, 36, 37, 38, 39, 40] and
the references given there. Lowengrub et al. [28] provided a systematic survey
of tumor model studies.

Recent study indicates that human beings and many animals have regular living
and feeding activities, related to the biological rhythm, the concentration of
nutrients in their blood may change periodically over time [12]. Motivated by
this study, in this paper, we shall consider a free boundary tumor model
describing the growth of tumors with a periodic supply of external nutrients:

(1.1) $\displaystyle\Delta\sigma$ $\displaystyle=$ $\displaystyle\sigma\ \ \ \
\ \qquad\ \ \ \ \mbox{in}\ \Omega(t),\ \ t>0,$ (1.2) $\displaystyle-\Delta p$
$\displaystyle=$ $\displaystyle\mu(\sigma-\widetilde{\sigma})\ \ \ \ \
\mbox{in}\ \Omega(t),\ \ t>0,$ (1.3) $\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\phi(t)\ \ \ \qquad\ \ \mbox{on}\ \partial\Omega(t),\ \ t>0,$
(1.4) $\displaystyle p$ $\displaystyle=$ $\displaystyle\gamma\kappa\ \ \ \
\qquad\ \ \ \mbox{on}\ \partial\Omega(t),\ \ t>0,$ (1.5) $\displaystyle V_{n}$
$\displaystyle=$ $\displaystyle-\frac{\partial p}{\partial\vec{n}}\ \qquad\ \
\ \mbox{on}\ \partial\Omega(t),\ \ t>0.$

Here $\Omega(t)\subseteq\mathbb{R}^{3}$ is an apriorily unknown bounded domain
occupied by the tumor at time $t$, $\sigma=\sigma(x,t)$ and $p=p(x,t)$ are
unknown functions representing the nutrient concentration in the tumor region
and the pressure between tumor cells, respectively. $V_{n}$ is the velocity of
the free boundary in the direction $\vec{n}$ which is the unit outward normal.
$\mu$, $\widetilde{\sigma}$ and $\gamma$ are positive constants, among which
$\mu$ measures the aggressiveness of the tumor, $\widetilde{\sigma}$ is the
threshold nutrient concentration for apoptosis, and $\gamma$ represents the
surface tension coefficient of the tumor surface $\partial\Omega(t)$. $\kappa$
is the mean curvature of $\partial\Omega(t)$ whose sign is defined by the
convention that convex hypersurfaces are associated with positive mean
curvature; $\kappa=1/R(t)$ if $\Omega(t)$ is a ball of radius $R(t)$. Besides,
$\phi(t)$ accounts for the external concentration of nutrients which we shall
assume is a smooth, positive, periodic function with period $T$.

The problem (1.1)-(1.5) is a natural extension of that proposed by Bryne and
Chaplain in [3]: the action of a periodic supply of external nutrients is
considered, so that (1.3) holds. In this model, the quasi-stationary
approximation of the diffusion equation (1.1) combined with the boundary
condition (1.3) are used to describe the distribution of the nutrient
concentration. Accordingly, the problem (1.1)-(1.5) is called the _quasi-
stationary model_. In addition, assuming that the extracellular matrix is a
porous medium where Darcy’s law $\vec{V}=-\nabla p$ holds, the law of
conservation of mass $\mbox{div}\vec{V}=\mu(\sigma-\widetilde{\sigma})$ yields
the equation (1.2). Equation (1.5) is the classical Stefan condition for free
boundary $\partial\Omega(t)$. Finally, the cell-to-cell adhesiveness leads to
the relation (1.4) (see [2, 4, 19]). If the stationary diffusion equation
(1.1) is replaced by its non-stationary version

$c\sigma_{t}=\Delta\sigma-\sigma\qquad\qquad\ \mbox{in}\ \Omega(t),\ \ t>0,$

where the positive constant $c=T_{\mbox{diffusion}}/T_{\mbox{growth}}$ is the
ratio of the nutrient diffusion time scale, $T_{\mbox{diffusion}}\approx
1\mbox{min}$, to the tumor-cell doubling time scale, $T_{\mbox{growth}}\approx
1\mbox{day}$, so that $c\ll 1$ (see [3]), the corresponding problem of
(1.1)-(1.5) is called the _fully non-stationary model_.

If the external concentration of nutrients is assumed to be a constant, i.e.,
the boundary condition (1.3) is reduced to
$\sigma=\overline{\sigma}=\mbox{const.}$, rigorous mathematical analysis
including existence and stability theorems have been established. For the
fully non-stationary model, Friedman and Reitich [16] proved that under the
assumption $0<\widetilde{\sigma}<\overline{\sigma}$, there exists a unique
radially symmetric stationary solution $(\sigma_{S}(r),p_{S}(r),R_{S})$. They
also proved that there exists a family of symmetric-breaking bifurcation
branches of stationary solutions bifurcating from this unique radial
stationary solution $(\sigma_{S}(r),p_{S}(r),R_{S})$ in [18]. In the sequel,
it was proved in [1] that $(\sigma_{S}(r),p_{S}(r),R_{S})$ is asymptotically
stable with respect to non-radial perturbations for $\mu$ sufficiently small.
This work was refined by Friedman and Hu [13, 14], that is, they determined
the threshold value $\mu^{\ast}$ such that for $0<\mu<\mu^{\ast}$ the trivial
solution is asymptotically stable under small non-radial perturbations, while
for $\mu>\mu^{\ast}$ the trivial solution is unstable. Later on, Cui and
Escher extended the asymptotic stability result to the general case that the
nutrient consumption rate and the tumor cell proliferate rate are general
increasing functions. They found a positive critical value $\gamma^{\ast}$
(surface tension coefficient in (1.4)) for which the radial stationary
solution changes from instability to stability under non-radial perturbations
for the quasi-stationary model in [9] and for the fully non-stationary model
with small $c$ in [7], respectively.

In this paper, we study the quasi-stationary model (1.1)-(1.5) with the
nutrient supply given by a periodic function $\phi(t)$. We expect to determine
how the tumor will evolve as time goes to infinity under the effect of this
periodic external nutrient supply. To give a precise statement of our main
result, let us make some preparations. By a rescaling if necessary, we take
$\gamma=1$. It is not difficult to verify (see also [22]) that the radially
symmetric $T$-periodic positive solution of the problem (1.1)-(1.5), which we
denote by $(\sigma_{\ast}(r,t),p_{\ast}(r,t),R_{\ast}(t))$, is of the form

(1.6)
$\sigma_{\ast}(r,t)=\phi(t)\frac{R_{\ast}^{1/2}(t)}{I_{1/2}(R_{\ast}(t))}\frac{I_{1/2}(r)}{r^{1/2}},\
\ \ 0<r<R_{\ast}(t),$ (1.7)
$p_{\ast}(r,t)=-\mu\sigma_{\ast}(r,t)+\frac{1}{6}\mu\widetilde{\sigma}r^{2}+\frac{1}{R_{\ast}(t)}+\mu\phi(t)-\frac{1}{6}\mu\widetilde{\sigma}R_{\ast}^{2}(t),\
\ \ 0<r<R_{\ast}(t),$

and $R_{\ast}(t)$ satisfies

(1.8) $\begin{split}\frac{dR_{\ast}(t)}{dt}=\mu
R_{\ast}(t)\Big{\\{}\phi(t)P_{0}(R_{\ast}(t))-\frac{\widetilde{\sigma}}{3}\Big{\\}},\end{split}$

where $I_{n}(r)$ is the modified Bessel function of order $n$ and $P_{0}(r)$
is defined by (2.1). As was analyzed in [34], for
$\frac{1}{T}\int_{0}^{T}\phi(t)dt>\widetilde{\sigma}$, there exists a unique
$T$-periodic positive solution $R_{\ast}(t)$ for the equation (1.8).
Substituting the unique solution $R_{\ast}(t)$ into the expressions
(1.6)-(1.7), one finds that the radially symmetric $T$-periodic positive
solution of the problem (1.1)-(1.5) is uniquely determined.

Let us introduce the concept of linear stability/instability and asymptotic
stability of the periodic solution
$(\sigma_{\ast}(r,t),p_{\ast}(r,t),R_{\ast}(t))$. By _linear
stability/instability_ , we mean:
Linearize the system (1.1)-(1.5) at
$(\sigma_{\ast}(r,t),p_{\ast}(r,t),R_{\ast}(t))$ by writing

$\begin{split}\sigma(r,\theta,\varphi,t)=\sigma_{\ast}(r,t)+\varepsilon
w(r,\theta,\varphi,t),\ \ \ \ p(r,\theta,\varphi,t)=p_{\ast}(r,t)+\varepsilon
q(r,\theta,\varphi,t),\end{split}$ $\partial\Omega(t):\
r=R_{\ast}(t)+\varepsilon\rho(\theta,\varphi,t),$

and collect only the $\varepsilon$-order terms. Since the original problem is
translation invariant, we say that the periodic solution
$(\sigma_{\ast}(r,t),p_{\ast}(r,t),R_{\ast}(t))$ is linearly stable in the
sense that for any initial data $\rho|_{t=0}=\rho_{0}(\theta,\varphi)$ and
$w|_{t=0}=w_{0}(r,\theta,\varphi)$,

$\Big{|}\rho(\theta,\varphi,t)-\sum_{m=-1}^{1}a_{m}Y_{1,m}(\theta,\varphi)\Big{|}\leq
Ce^{-\delta t},\ \ \ \ t>\overline{t}$

for some constants $a_{m}$, $\delta>0$ and $\overline{t}>0$. The periodic
solution is said to be linearly unstable if it is not linearly stable.

By _asymptotic stability_ , we mean the following:
Given any initial data

(1.9) $\begin{split}\partial\Omega(0):\
r=R_{0}(\theta,\varphi)=R_{\ast}(0)+\varepsilon\rho_{0}(\theta,\varphi),\\\
\sigma|_{t=0}=\sigma_{0}(r,\theta,\varphi)=\sigma_{\ast}(r,0)+\varepsilon
w_{0}(r,\theta,\varphi),\end{split}$

if $|\varepsilon|$ is sufficiently small, then there exists a solution to
(1.1)-(1.5) for all $t>0$, and

$\partial\Omega(t)\ \ \ \mbox{behaves\ like}\ \ \ \partial
B_{R_{\ast}(t)}(a)=\big{\\{}x:|x-a|=R_{\ast}(t)\big{\\}},\ \ \ t>\overline{t}$

for some center $a$ and $\overline{t}>0$.

Set

(1.10)
$\mu_{\ast}=\frac{\int_{0}^{T}\frac{6}{R^{3}_{\ast}(t)}dt}{-\frac{\widetilde{\sigma}}{2}\int_{0}^{T}\left[R_{\ast}(t)\frac{I_{3}(R_{\ast}(t))}{I_{2}(R_{\ast}(t))}-R_{\ast}(t)\frac{I_{0}(R_{\ast}(t))}{I_{1}(R_{\ast}(t))}+2\right]dt}>0,\
\ \ (\mbox{two-space\ dimension}),$

or,

(1.11)
$\mu_{\ast}=\frac{\int_{0}^{T}\frac{4}{R^{3}_{\ast}(t)}dt}{\int_{0}^{T}\frac{\widetilde{\sigma}}{3}R^{2}_{\ast}(t)\left[P_{1}(R_{\ast}(t))-P_{2}(R_{\ast}(t))\right]dt}>0,\
\ \ \ (\mbox{three-space\ dimension}).$

It was recently established in [25] that the unique radially symmetric
$T$-periodic positive solution
$(\sigma_{\ast}(r,t),p_{\ast}(r,t),R_{\ast}(t))$ is linearly stable under non-
radial perturbations for $0<\mu<\mu_{\ast}$ and linearly unstable for
$\mu>\mu_{\ast}$ in the two-space dimensional case; see [22] for the similar
result in the three-space dimensional case. In this paper, we further study
the asymptotic stability for the fully nonlinear problem. Compared with the
linearized problem, the $O(\varepsilon^{2})$ terms cannot be dropped. On the
one hand, we need to find the necessary PDE estimates to estimate the
nonlinear error terms. On the other hand, since this system is translation
invariant in the coordinate space, the center of the limiting sphere is not
known in advance which depends on the perturbation of mode 1. To be specific,
the perturbation of mode 1 results in the translation of the origin with its
magnitude the same order as the perturbation, as well as the decay behavior in
time $t$. This is a challenge for this type of problems. We employ on a fixed
point theorem to find the correct translation of the origin which is technical
and carried out by Theorem 4.1.

Throughout the paper, $\mu_{\ast}$ is defined by (1.11). The main result of
this paper is stated as follows (see Theorem 5.1 for more explicit statement).

######  Theorem 1.1.

Let $\mu<\mu_{\ast}$, then the unique radially symmetric $T$-periodic solution
$(\sigma_{\ast}(r,t),p_{\ast}(r,t),R_{\ast}(t))$ of the system (1.1)-(1.5) is
asymptotically stable modulus translation, i.e., there exists a new center
$\varepsilon a^{\ast}(\varepsilon)$, where $a^{\ast}(\varepsilon)$ is a
bounded function of $\varepsilon$, such that $\partial\Omega(t)$ behaves like

$\partial B_{R_{\ast}(t)}(\varepsilon
a^{\ast}(\varepsilon))=\big{\\{}x:|x-\varepsilon
a^{\ast}(\varepsilon)|=R_{\ast}(t)\big{\\}},\ \ \ \ \ t>\overline{t}$

for some $\overline{t}>0$.

######  Remark 1.1.

Note that our method and result is also applicable to the two-space
dimensional case.

The structure of the rest of this paper is arranged as follows. In the next
section, we collect some preliminaries which are needed in the sequel. In
Section 3, we transform the nonlinear free boundary problem (1.1)-(1.5) into
an initial-boundary value problem defined on a ball, which we then rewrite as
a inhomogeneous linear system (4.1)-(4.5) in Section 4.

The method we shall use is a fixed point argument:

(i) For given inhomogeneous terms $(f^{i},b^{j})$, solve the inhomogeneous
linear system (4.1)-(4.5) by using the spherical harmonic expansion;

(ii) define new inhomogeneous terms $(\widetilde{f}^{i},\widetilde{b}^{j})$
(see (4.18));

(iii) establish the existence of a fixed point for the mapping:

$S:(f^{i},b^{j})\rightarrow(\widetilde{f}^{i},\widetilde{b}^{j}).$

Furthermore, in Section 4, we establish decay estimates for each mono-mode
system obtained from the spherical harmonic expansion. In order to derive
decay estimates for mode 1 terms, we need to translate the origin $x=0$ to
$x=\varepsilon a^{\ast}(\varepsilon)$, which is carried out by Theorem 4.1.
Section 5 aims at showing the mapping $S$ admits a fixed point, thereby
completing the proof of the asymptotic stability for the fully nonlinear
problem.

##  2\. Preliminaries

In this section, we collect some preliminaries which are needed in the sequel.

The function $P_{n}(r)$ introduced by Friedman and Hu [13] is defined by

(2.1) $P_{n}(r)=\frac{I_{n+3/2}(r)}{rI_{n+1/2}(r)},\ \ \ \ n=0,1,2,3,\cdots$

where $I_{m}(r)$ is the modified Bessel function, $m\geq 0$ and $r>0$. Recall
[13, 26] that

(2.2) $P_{0}(r)=\frac{1}{r}\coth r-\frac{1}{r^{2}},$ (2.3)
$P_{n}(r)=\frac{1}{r^{2}P_{n+1}(r)+2n+3},$ (2.4) $P_{n}(r)>P_{n+1}(r),$ (2.5)
$\frac{d}{dr}\Big{(}\frac{I_{n+1/2}(r)}{r^{1/2}}\Big{)}=\frac{I_{n+3/2}(r)+\frac{n}{r}I_{n+1/2}(r)}{r^{1/2}}.$

######  Lemma 2.1.

For $n\geq 1$ and $r>0$,

(2.6) $rP_{n}(r)<rP_{0}(r)\leq 1.$

Proof. By (2.2), we get

$\frac{d}{dr}(rP_{0}(r))=\frac{d}{dr}\Big{(}\coth
r-\frac{1}{r}\Big{)}=\frac{(e^{2r}-1)^{2}-4r^{2}e^{2r}}{r^{2}(e^{2r}-1)^{2}}.$

Denote the numerator by $F(r)$. We compute

$F(0)=0\ \ \ \ \ \mbox{and}\ \ \ \
\frac{dF(r)}{dr}=4e^{2r}(e^{2r}-1-2r-2r^{2})>0,$

then $rP_{0}(r)$ is an increasing function, and

$rP_{0}(r)\leq\lim_{r\rightarrow\infty}rP_{0}(r)=1.$

It follows from (2.4) that (2.6) holds. Therefore, our proof is complete.
$\Box$

######  Lemma 2.2.

(see [22, Lemma 4.1]) The following relations hold:

(2.7) $\frac{\partial\sigma_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}=\phi(t)R_{\ast}(t)P_{0}(R_{\ast}(t)),$ (2.8)
$\frac{\partial p_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}=-\frac{dR_{\ast}(t)}{dt},$ (2.9)
$\frac{\partial^{2}p_{\ast}}{\partial
r^{2}}\Big{|}_{r=R_{\ast}(t)}=-\mu\phi(t)R^{2}_{\ast}(t)P_{0}(R_{\ast}(t))P_{1}(R_{\ast}(t))-\frac{1}{R_{\ast}(t)}\frac{dR_{\ast}(t)}{dt}.$

As established in [5], we have the following local existence theorem:

######  Theorem 2.1.

If

(2.10) $(\sigma_{0},R_{0})\in C^{1+\gamma}(\overline{\Omega}(0))\times
C^{4+\alpha}(\partial\Omega(0))\ \ \ \mbox{and}\ \ \sigma_{0}=\phi(0)\ \ \
\mbox{on}\ \ \partial\Omega(0)$

for some $\alpha$, $\gamma$ $\in(0,1)$, then there exists a unique solution
$(\sigma,p,R)$ of (1.1)-(1.5) for $t\in[0,T]$ with some $T>0$, and

$\sigma\in
C^{1+\gamma,(1+\gamma)/2}\Big{(}\bigcup_{t\in[0,T]}\overline{\Omega}(t)\times\\{t\\}\Big{)}\cap
C^{2+2\alpha/3,1+\alpha/3}\Big{(}\bigcup_{t\in[t_{0},T]}\overline{\Omega}(t)\times\\{t\\}\Big{)}\
\ \ \mbox{for\ any}\ t_{0}>0,$ $p\in
C^{2+\alpha,\alpha/3}\Big{(}\bigcup_{t\in[0,T]}\overline{\Omega}(t)\times\\{t\\}\Big{)},\
\ \ R\in C^{4+\alpha,1+\alpha/3}.$

The following fixed point theorem ([26]) will be crucial in the proof of
Theorem 4.1:

######  Theorem 2.2.

Let $(X,\|\cdot\|)$ be a Banach space and let $\overline{B}_{K}(a_{0})$ denote
the closed ball in $X$ with center $a_{0}$ and radius $K$. Let $F$ be a
mapping from $\overline{B}_{K}(a_{0})$ into $X$ and

$F(x)=F_{1}(x)+\varepsilon G(x),$

such that

(i) $F_{1}^{\prime}(x)$ and $G^{\prime}(x)$ are both continuous for
$x\in\overline{B}_{K}(a_{0})$,

(ii) $F_{1}(a_{0})=0$ and the operator $F^{\prime}_{1}(a_{0})$ is invertible.

Then for small $|\varepsilon|$, the equation $F(x)=0$ admits a unique solution
$x$ in $\overline{B}_{K}(a_{0})$.

##  3\. Transformation

In this section, we transform the nonlinear free boundary problem into a
nonlinear perturbation in a spherical region.

Let us assume the solution of the system (1.1)-(1.5) is of the form

$\begin{split}\partial\Omega(t):\
r&=R_{\ast}(t)+\varepsilon\rho(\theta,\varphi,t),\\\
\sigma(r,\theta,\varphi,t)&=\sigma_{\ast}(r,t)+\varepsilon
w(r,\theta,\varphi,t),\\\ p(r,\theta,\varphi,t)&=p_{\ast}(r,t)+\varepsilon
q(r,\theta,\varphi,t).\end{split}$

Recall [23] that

$\vec{n}=\frac{1}{\sqrt{1+|\varepsilon\nabla_{\omega}\rho|^{2}/(R_{\ast}(t)+\varepsilon\rho)^{2}}}\Big{(}\vec{e}_{r}-\frac{\varepsilon}{R_{\ast}(t)+\varepsilon\rho}\frac{\partial\rho}{\partial\theta}\vec{e}_{\theta}-\frac{\varepsilon}{(R_{\ast}(t)+\varepsilon\rho)\sin\theta}\frac{\partial\rho}{\partial\varphi}\vec{e}_{\varphi}\Big{)},$

and

$\nabla=\vec{e}_{r}\frac{\partial}{\partial
r}+\vec{e}_{\theta}\frac{1}{r}\frac{\partial}{\partial\theta}+\vec{e}_{\varphi}\frac{1}{r\sin\theta}\frac{\partial}{\partial\varphi}=\vec{e}_{r}\frac{\partial}{\partial
r}+\frac{1}{r}\nabla_{\omega},$

where
$\nabla_{\omega}=\vec{e}_{\theta}\frac{\partial}{\partial\theta}+\vec{e}_{\varphi}\frac{1}{\sin\theta}\frac{\partial}{\partial\varphi}$.

The system (1.1)-(1.5) can then be written in terms of $(w,q,\rho)$ as
follows:

(3.1) $\Delta w=w\ \ \ \mbox{in}\ \Omega(t),\ t>0,$ (3.2) $-\Delta q=\mu w\ \
\ \mbox{in}\ \partial\Omega(t),\ t>0,$ (3.3) $\frac{\partial\rho}{\partial
t}=-\frac{1}{\varepsilon}\Big{(}\frac{\partial
p_{\ast}}{\partial\vec{n}}+\varepsilon\frac{\partial
q}{\partial\vec{n}}\Big{)}\sqrt{1+\frac{|\varepsilon\nabla_{\omega}\rho|^{2}}{(R_{\ast}(t)+\varepsilon\rho)^{2}}}-\frac{1}{\varepsilon}\frac{dR_{\ast}(t)}{dt}\
\ \mbox{on}\ \partial\Omega(t),\ t>0,$ (3.4)
$w=-\frac{1}{\varepsilon}\Big{[}\sigma_{\ast}(R_{\ast}(t)+\varepsilon\rho,t)-\sigma_{\ast}(R_{\ast}(t),t)\Big{]}\
\ \ \mbox{on}\ \partial\Omega(t),\ t>0,$ (3.5)
$q=-\frac{1}{\varepsilon}\big{[}p_{\ast}(R_{\ast}(t)+\varepsilon\rho,t)-\kappa\big{]}\
\ \ \mbox{on}\ \partial\Omega(t),\ t>0.$

By the Taylor expansion, one of the right-hand side terms of (3.3) can be
written in the following way,

(3.6)
$\begin{split}\sqrt{1+\frac{|\varepsilon\nabla_{\omega}\rho|^{2}}{(R_{\ast}(t)+\varepsilon\rho)^{2}}}\frac{\partial
p_{\ast}}{\partial\vec{n}}\Big{|}_{r=R_{\ast}(t)+\varepsilon\rho}&=\sqrt{1+\frac{|\varepsilon\nabla_{\omega}\rho|^{2}}{(R_{\ast}(t)+\varepsilon\rho)^{2}}}\nabla
p_{\ast}|_{r=R_{\ast}(t)+\varepsilon\rho}\cdot\vec{n}\\\ &=\frac{\partial
p_{\ast}(R_{\ast}(t)+\varepsilon\rho,t)}{\partial r}\\\ &=\frac{\partial
p_{\ast}(R_{\ast}(t),t)}{\partial
r}+\frac{\partial^{2}p_{\ast}(R_{\ast}(t),t)}{\partial
r^{2}}\varepsilon\rho+\varepsilon^{2}P_{\varepsilon}.\end{split}$

In addition,

(3.7)
$\begin{split}\sqrt{1+\frac{|\varepsilon\nabla_{\omega}\rho|^{2}}{(R_{\ast}(t)+\varepsilon\rho)^{2}}}\frac{\partial
q}{\partial\vec{n}}\Big{|}_{r=R_{\ast}(t)+\varepsilon\rho}&=\sqrt{1+\frac{|\varepsilon\nabla_{\omega}\rho|^{2}}{(R_{\ast}(t)+\varepsilon\rho)^{2}}}\nabla
q|_{r=R_{\ast}(t)+\varepsilon\rho}\cdot\vec{n}\\\ &=\frac{\partial q}{\partial
r}\Big{|}_{r=R_{\ast}(t)+\varepsilon\rho}-\frac{\varepsilon}{(R_{\ast}(t)+\varepsilon\rho)^{2}}\frac{\partial\rho}{\partial\theta}\frac{\partial
q}{\partial\theta}\Big{|}_{r=R_{\ast}(t)+\varepsilon\rho}\\\
&\quad-\frac{\varepsilon}{(R_{\ast}(t)+\varepsilon\rho)^{2}\sin^{2}\theta}\frac{\partial\rho}{\partial\varphi}\frac{\partial
q}{\partial\varphi}\Big{|}_{r=R_{\ast}(t)+\varepsilon\rho}.\end{split}$

Again by the Taylor expansion, the right-hand side terms of (3.4) and (3.5)
can be expanded into the following format, respectively,

(3.8)
$\sigma_{\ast}(R_{\ast}(t)+\varepsilon\rho,t)-\sigma_{\ast}(R_{\ast}(t),t)=\frac{\partial\sigma_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\varepsilon\rho+\varepsilon^{2}S_{\varepsilon},$

and

(3.9)
$\begin{split}p_{\ast}(R_{\ast}(t)+\varepsilon\rho,t)-\kappa=&p_{\ast}(R_{\ast}(t),t)+\frac{\partial
p_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\varepsilon\rho+\varepsilon^{2}K_{1\varepsilon}\\\
&-\frac{1}{R_{\ast}(t)}+\frac{\varepsilon}{R^{2}_{\ast}(t)}\Big{(}\rho+\frac{1}{2}\Delta_{\omega}\rho\Big{)}+\varepsilon^{2}K_{2\varepsilon}\\\
=&\varepsilon\Big{[}\frac{1}{R_{\ast}^{2}(t)}\Big{(}\rho+\frac{1}{2}\Delta_{\omega}\rho\Big{)}+\frac{\partial
p_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\rho\Big{]}+\varepsilon^{2}K_{\varepsilon},\end{split}$

where we have used the fact, see [17, Theorem 8.1],

$\kappa=\frac{1}{R}-\frac{\varepsilon}{R^{2}}\Big{(}\rho+\frac{1}{2}\Delta_{\omega}\rho\Big{)}+\varepsilon^{2}K$

with

$\Delta_{\omega}\rho=\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\rho}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}\rho}{\partial\varphi^{2}}.$

The Hanzawa transformation is defined by

$r=r^{\prime}+\chi(R-r^{\prime})\varepsilon\rho(\theta,\varphi,t),\ \ \
t=t^{\prime},\ \ \ \ \theta=\theta^{\prime},\ \ \ \varphi=\varphi^{\prime},$

here

$\chi(z)\in C^{\infty},\ \ \ \ \chi(z)=\left\\{\begin{aligned} &0,\ \
\mbox{if}\ |z|\geq\frac{3}{4}\delta_{0}\\\ &1,\ \ \mbox{if}\
|z|<\frac{1}{4}\delta_{0}\end{aligned}\right.,\ \ \ \
\left|\frac{d^{k}\chi}{dz^{k}}\right|\leq\frac{C}{\delta_{0}^{k}},\ \ \ \
(\delta_{0}\ \mbox{positive\ and\ small}).$

Under the Hanzawa transformation and (3.6)-(3.9), the system (3.1)-(3.5) is
transformed into

(3.10) $-\Delta^{\prime}w^{\prime}+w^{\prime}=\varepsilon
A_{\varepsilon}w^{\prime}\ \ \ \mbox{in}\ B_{R_{\ast}(t)},\ t>0,$ (3.11)
$-\Delta^{\prime}q^{\prime}-\mu w^{\prime}=\varepsilon
A_{\varepsilon}q^{\prime}\ \ \ \mbox{in}\ B_{R_{\ast}(t)},\ t>0,$ (3.12)
$\frac{\partial\rho^{\prime}}{\partial
t^{\prime}}=-\frac{\partial^{2}p_{\ast}}{\partial r^{\prime
2}}\rho^{\prime}-\frac{\partial q^{\prime}}{\partial r^{\prime}}+\varepsilon
B^{1}_{\varepsilon}\ \ \ \mbox{on}\ \partial B_{R_{\ast}(t)},\ t>0,$ (3.13)
$w^{\prime}=-\frac{\partial\sigma_{\ast}}{\partial
r^{\prime}}\rho^{\prime}+\varepsilon B^{2}_{\varepsilon}\ \ \ \mbox{on}\
\partial B_{R_{\ast}(t)},\ t>0,$ (3.14)
$q^{\prime}=-\frac{1}{R_{\ast}^{2}(t)}\Big{(}\rho^{\prime}+\frac{1}{2}\Delta_{\omega}\rho^{\prime}\Big{)}-\frac{\partial
p_{\ast}}{\partial r^{\prime}}\rho^{\prime}+\varepsilon B^{3}_{\varepsilon}\ \
\ \mbox{on}\ \partial B_{R_{\ast}(t)},\ t>0,$

where $B_{R_{\ast}(t)}$ is the ball with radius $R_{\ast}(t)$,
$A_{\varepsilon}$ given in [14] is a second order differential operator in
$(r^{\prime},\theta^{\prime},\varphi^{\prime})$, and

$\begin{split}B_{\varepsilon}^{1}=&-\frac{1}{\varepsilon^{2}}\frac{\partial
p_{\ast}(R_{\ast}(t^{\prime})+\varepsilon\rho^{\prime},t^{\prime})}{\partial
r^{\prime}}+\frac{1}{(R_{\ast}(t^{\prime})+\varepsilon\rho^{\prime})^{2}}\frac{\partial\rho^{\prime}}{\partial\theta^{\prime}}\frac{\partial
q^{\prime}}{\partial\theta^{\prime}}\\\
&+\frac{1}{(R_{\ast}(t^{\prime})+\varepsilon\rho^{\prime})^{2}\sin^{2}\theta^{\prime}}\frac{\partial\rho^{\prime}}{\partial\varphi^{\prime}}\frac{\partial
q^{\prime}}{\partial\varphi^{\prime}}-\frac{1}{\varepsilon^{2}}\frac{dR_{\ast}(t^{\prime})}{dt^{\prime}}+\frac{1}{\varepsilon}\frac{\partial^{2}p_{\ast}(R_{\ast}(t^{\prime}),t^{\prime})}{\partial
r^{\prime 2}}\rho^{\prime},\end{split}$
$\begin{split}B^{2}_{\varepsilon}=&-\frac{1}{\varepsilon^{2}}\big{[}\sigma_{\ast}(R_{\ast}(t^{\prime})+\varepsilon\rho^{\prime},t^{\prime})-\sigma_{\ast}(R_{\ast}(t^{\prime}),t^{\prime})\big{]}+\frac{1}{\varepsilon}\frac{\partial\sigma_{\ast}(R_{\ast}(t^{\prime}),t^{\prime})}{\partial
r^{\prime}}\rho^{\prime},\end{split}$
$B_{\varepsilon}^{3}=-\frac{1}{\varepsilon^{2}}\big{[}p_{\ast}(R_{\ast}(t^{\prime})+\varepsilon\rho^{\prime},t^{\prime})-\kappa\big{]}+\frac{1}{\varepsilon
R_{\ast}^{2}(t^{\prime})}\Big{(}\rho^{\prime}+\frac{1}{2}\Delta_{\omega}\rho^{\prime}\Big{)}+\frac{1}{\varepsilon}\frac{\partial
p_{\ast}(R_{\ast}(t^{\prime}),t^{\prime})}{\partial r^{\prime}}\rho^{\prime}.$

As pointed out in [14] that all terms of $A_{\varepsilon}$ do not involve any
singularity, then it follows from Theorem 2.1 that, for $T>1$,

$A_{\varepsilon}w^{\prime}\in
C^{2\alpha/3,\alpha/3}(B_{R_{\ast}(t)}\times[0,T]),$
$A_{\varepsilon}q^{\prime}\in
C^{\alpha,\alpha/3}(B_{R_{\ast}(t)}\times[0,T]).$

Notice that the term $\frac{1}{\varepsilon}$ appearing in $A_{\varepsilon}$ is
cancelled out by the coefficient that accompanies it, so that both
$C^{2\alpha/3,\alpha/3}$ norm of $A_{\varepsilon}w^{\prime}$ and
$C^{\alpha,\alpha/3}$ norm of $A_{\varepsilon}q^{\prime}$ are uniformly
bounded in $\varepsilon$.

Moreover, although $\sin^{2}\theta^{\prime}$ appears in the denominator in the
last term of $B_{\varepsilon}^{1}$, this incurs no singularities. Indeed, one
can simply choose a different coordinate system to deal with this problem,
i.e., $B_{\varepsilon}^{1}$ is a function defined on the unit sphere
$\Sigma=\\{x:|x|=1\\}$ (rather than in the variable $(\theta,\varphi)$). Then
(1.7) and Theorem 2.1 yield $B_{\varepsilon}^{1}\in
C^{1+\alpha,\alpha/3}(\partial B_{R_{\ast}(t)}\times[0,T])$. It follows from
(3.6) and (2.8) that $C^{1+\alpha,\alpha/3}$ norm of $B_{\varepsilon}^{1}$ is
uniformly bounded in $\varepsilon$. By (1.6) and Theorem 2.1, we find that
$B_{\varepsilon}^{2}\in C^{1+2\alpha/3,1+\alpha/3}(\partial
B_{R_{\ast}(t)}\times[0,T])$, and (3.8) implies that
$C^{1+2\alpha/3,1+\alpha/3}$ norm of $B_{\varepsilon}^{2}$ is uniformly
bounded in $\varepsilon$. Similarly, by (3.9), we immediately derive that
$B_{\varepsilon}^{3}\in C^{2+\alpha,\alpha/3}(\partial
B_{R_{\ast}(t)}\times[0,T])$, and its $C^{2+\alpha,\alpha/3}$ norm is
uniformly bounded in $\varepsilon$.

For notational convenience, we shall denote functions
$w^{\prime}(r^{\prime},\theta^{\prime},\varphi^{\prime},t^{\prime})$,
$q^{\prime}(r^{\prime},\theta^{\prime},\varphi^{\prime},t^{\prime})$ and
$\rho^{\prime}(\theta^{\prime},\varphi^{\prime},t^{\prime})$ again by
$w(r,\theta,\varphi,t)$, $q(r,\theta,\varphi,t)$ and $\rho(\theta,\varphi,t)$,
respectively, in the rest of this paper.

##  4\. The Inhomogeneous Linear System

In this section, we rewrite (3.10)-(3.14) as a inhomogeneous linear system
(4.1)-(4.5), and use spherical harmonic expansions to solve this inhomogeneous
system. Furthermore, we establish decay estimates for each mono-mode system
obtained from the spherical harmonic expansion.

Specifically, we consider the system (3.10)-(3.14) where the $\varepsilon$
terms of any order are replaced by given functions,

(4.1) $-\Delta w+w=\varepsilon f^{1}(r,\theta,\varphi,t)\ \ \ \mbox{in}\
B_{R_{\ast}(t)},\ t>0,$ (4.2) $-\Delta q=\mu w+\varepsilon
f^{2}(r,\theta,\varphi,t)\ \ \ \mbox{in}\ B_{R_{\ast}(t)},\ t>0,$ (4.3)
$\begin{split}\frac{\partial\rho}{\partial
t}=&-\frac{\partial^{2}p_{\ast}}{\partial
r^{2}}\Big{|}_{r=R_{\ast}(t)}\rho-\frac{\partial q}{\partial
r}\Big{|}_{r=R_{\ast}(t)}+\varepsilon b^{1}(\theta,\varphi,t)\ \ \ \
\mbox{on}\ \partial B_{R_{\ast}(t)},\ t>0,\end{split}$ (4.4)
$w=-\frac{\partial\sigma_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\rho+\varepsilon b^{2}(\theta,\varphi,t)\ \ \ \
\mbox{on}\ \partial B_{R_{\ast}(t)},\ t>0,$ (4.5)
$\begin{split}q=&-\frac{1}{R_{\ast}^{2}(t)}\Big{(}\rho+\frac{1}{2}\Delta_{\omega}\rho\Big{)}-\frac{\partial
p_{\ast}}{\partial r}\Big{|}_{r=R_{\ast}(t)}\rho+\varepsilon
b^{3}(\theta,\varphi,t)\ \ \ \mbox{on}\ \partial B_{R_{\ast}(t)},\
t>0,\end{split}$

and we add initial conditions (cf. (1.9))

(4.6) $w|_{t=0}=w_{0}(r,\theta,\varphi),\ \ \ \quad\ \ \
\rho|_{t=0}=\rho_{0}(\theta,\varphi).$

Assume that the functions $f^{i}$, $b^{j}$ satisfy the following properties

(4.7)
$\sqrt{|\varepsilon|}\left(\int_{0}^{\infty}e^{2\delta_{1}t}\|f^{1}(\cdot,t)\|^{2}_{L^{2}(B_{R_{\ast}(t)})}dt\right)^{1/2}\leq
1,$ (4.8)
$\sqrt{|\varepsilon|}\left(\int_{0}^{\infty}e^{2\delta_{1}t}\|f^{2}(\cdot,t)\|^{2}_{L^{2}(B_{R_{\ast}(t)})}dt\right)^{1/2}\leq
1,$ (4.9)
$\sqrt{|\varepsilon|}\left(\int_{0}^{\infty}e^{2\delta_{1}t}\|b^{1}(\cdot,t)\|^{2}_{H^{1/2}(\partial
B_{R_{\ast}(t)})}dt\right)^{1/2}\leq 1,$ (4.10)
$\sqrt{|\varepsilon|}\left(\int_{0}^{\infty}e^{2\delta_{1}t}\|b^{2}(\cdot,t)\|^{2}_{H^{1}(\partial
B_{R_{\ast}(t)})}dt\right)^{1/2}\leq 1,$ (4.11)
$\sqrt{|\varepsilon|}\left(\int_{0}^{\infty}e^{2\delta_{1}t}\|b^{3}(\cdot,t)\|^{2}_{H^{3/2}(\partial
B_{R_{\ast}(t)})}dt\right)^{1/2}\leq 1,$

where $\delta_{1}>0$ is sufficiently small, and for some $\alpha\in(0,1)$,

(4.12)
$\sqrt{|\varepsilon|}\|f^{1}\|_{C^{2\alpha/3,\alpha/3}(B_{R_{\ast}(t)}\times[0,\infty))}\leq
1,$ (4.13)
$\sqrt{|\varepsilon|}\|f^{2}\|_{C^{\alpha,\alpha/3}(B_{R_{\ast}(t)}\times[0,\infty))}\leq
1,$ (4.14) $\sqrt{|\varepsilon|}\|b^{1}\|_{C^{1+\alpha,\alpha/3}(\partial
B_{R_{\ast}(t)}\times[0,\infty))}\leq 1,$ (4.15)
$\sqrt{|\varepsilon|}\|b^{2}\|_{C^{2+2\alpha/3,1+\alpha/3}(\partial
B_{R_{\ast}(t)}\times[0,\infty))}\leq 1,$ (4.16)
$\sqrt{|\varepsilon|}\|b^{3}\|_{C^{2+\alpha,\alpha/3}(\partial
B_{R_{\ast}(t)}\times[0,\infty))}\leq 1.$

For simplicity, we also assume that

(4.17) $\rho_{0}\in C^{4+\alpha}(\overline{B}_{R_{\ast}(0)}),\ \ \ \ \
w_{0}\in C^{2+2\alpha/3}(\overline{B}_{R_{\ast}(0)}),$

and the compatibility condition of order 2 for $w$ is satisfied.

######  Remark 4.1.

For given functions $f^{i}$, $b^{j}$ in some subset $X_{1}$ of a Banach space
$X$, we solve the inhomogeneous linear system (4.1)-(4.5), and derive the
estimate of $(w,q,\rho)$. Then we define the new functions
$\widetilde{f}^{i}$, $\widetilde{b}^{j}$ by

(4.18) $\widetilde{f}^{1}=A_{\varepsilon}w,\ \ \ \
\widetilde{f}^{2}=A_{\varepsilon}q,\ \ \ \
\widetilde{b}^{1}=B_{\varepsilon}^{1},\ \ \
\widetilde{b}^{2}=B_{\varepsilon}^{2},\ \ \
\widetilde{b}^{3}=B_{\varepsilon}^{3}.$

We shall show that the mapping
$S:(f^{i},b^{j})\rightarrow(\widetilde{f}^{i},\widetilde{b}^{j})$ admits a
fixed point which is carried out in Section 5, leading to the asymptotic
stability.

######  Remark 4.2.

The initial data are supposed to satisfy the conditions of Theorem 2.1 rather
than (4.17). However, in the process of showing that the mapping $S$ admits a
fixed point, we shall need Hölder estimates of $D_{x}^{2}\sigma$ which
requires the consistency condition of order 2 for $\sigma$ at
$\partial\Omega(0)$. This is quite restrictive. As stated in [14, Remark 3.2],
we can avoid it by taking the initial time at $t=\frac{T}{2}$ instead of at
$t=0$. To simplify notation, we shall denote $t=\frac{T}{2}$ and the initial
data $(w|_{t=T/2},\rho|_{t=T/2})$ by $t=0$ and $(w_{0},\rho_{0})$,
respectively. Then the consistency condition of order 2 is satisfied at $t=0$,
with $\rho_{0}\in C^{4+\alpha}(\overline{B}_{R_{\ast}(0)})$ and $w_{0}\in
C^{2+2\alpha/3}(\overline{B}_{R_{\ast}(0)})$ (cf. (4.17)).

Now we proceed to use spherical harmonic expansions to solve the inhomogeneous
linear system (4.1)-(4.5). We first formally expand all the functions $f^{i}$,
$b^{j}$ in terms of spherical harmonics

$f^{i}(r,\theta,\varphi,t)=\sum^{\infty}_{n=0}\sum^{n}_{m=-n}f^{i}_{n,m}(r,t)Y_{n,m}(\theta,\varphi),\
\ \ \ i=1,2,$
$b^{j}(\theta,\varphi,t)=\sum^{\infty}_{n=0}\sum^{n}_{m=-n}b^{j}_{n,m}(t)Y_{n,m}(\theta,\varphi),\
\ \ \ j=1,2,3.$

Recall [14] that (4.7)-(4.11) imply

(4.19)
$|\varepsilon|\int_{0}^{\infty}e^{2\delta_{1}t}\|f_{n,m}^{1}(\cdot,t)\|^{2}_{L^{2}(B_{R_{\ast}(t)})}dt=F_{n,m}^{1},\
\ \ \ \ \sum_{n,m}F_{n,m}^{1}\leq 1,$ (4.20)
$|\varepsilon|\int_{0}^{\infty}e^{2\delta_{1}t}\|f_{n,m}^{2}(\cdot,t)\|^{2}_{L^{2}(B_{R_{\ast}(t)})}dt=F_{n,m}^{2},\
\ \ \ \ \sum_{n,m}F_{n,m}^{2}\leq 1,$ (4.21)
$|\varepsilon|(n+1)\int_{0}^{\infty}e^{2\delta_{1}t}|b_{n,m}^{1}(t)|^{2}dt=B_{n,m}^{1},\
\ \ \ \ \sum_{n,m}B_{n,m}^{1}\leq C,$ (4.22)
$|\varepsilon|(n+1)^{2}\int_{0}^{\infty}e^{2\delta_{1}t}|b_{n,m}^{2}(t)|^{2}dt=B_{n,m}^{2},\
\ \ \ \ \sum_{n,m}B_{n,m}^{2}\leq C,$ (4.23)
$|\varepsilon|(n+1)^{3}\int_{0}^{\infty}e^{2\delta_{1}t}|b_{n,m}^{3}(t)|^{2}dt=B_{n,m}^{3},\
\ \ \ \ \sum_{n,m}B_{n,m}^{3}\leq C.$

We then look for a solution of the following form:

$\displaystyle w(r,\theta,\varphi,t)$ $\displaystyle=$
$\displaystyle\sum^{\infty}_{n=0}\sum^{n}_{m=-n}w_{n,m}(r,t)Y_{n,m}(\theta,\varphi),$
$\displaystyle q(r,\theta,\varphi,t)$ $\displaystyle=$
$\displaystyle\sum^{\infty}_{n=0}\sum^{n}_{m=-n}q_{n,m}(r,t)Y_{n,m}(\theta,\varphi),$
$\displaystyle\rho(\theta,\varphi,t)$ $\displaystyle=$
$\displaystyle\sum^{\infty}_{n=0}\sum^{n}_{m=-n}\rho_{n,m}(t)Y_{n,m}(\theta,\varphi).$

By the relation

$\Delta_{\omega}Y_{n,m}(\theta,\varphi)+n(n+1)Y_{n,m}(\theta,\varphi)=0\ \ \
\mbox{and}\ \ \ \Delta=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\Big{(}r^{2}\frac{\partial}{\partial
r}\Big{)}+\frac{1}{r^{2}}\Delta_{\omega},$

we obtain that $w_{n,m}(r,t)$, $q_{n,m}(r,t)$ and $\rho_{n,m}(t)$ satisfy

(4.24) $-\Delta
w_{n,m}(r,t)+\left(\frac{n(n+1)}{r^{2}}+1\right)w_{n,m}(r,t)=\varepsilon
f^{1}_{n,m}(r,t)\ \ \ \mbox{in}\ B_{R_{\ast}(t)},\ t>0,$ (4.25) $-\Delta
q_{n,m}(r,t)+\frac{n(n+1)}{r^{2}}q_{n,m}(r,t)=\mu w_{n,m}(r,t)+\varepsilon
f^{2}_{n,m}(r,t)\ \ \ \mbox{in}\ B_{R_{\ast}(t)},\ t>0,$ (4.26)
$\begin{split}\frac{d\rho_{n,m}(t)}{dt}=&-\frac{\partial^{2}p_{\ast}}{\partial
r^{2}}\Big{|}_{r=R_{\ast}(t)}\rho_{n,m}(t)-\frac{\partial
q_{n,m}(r,t)}{\partial r}\Big{|}_{r=R_{\ast}(t)}+\varepsilon b^{1}_{n,m}(t),\
\ \ t>0,\end{split}$ (4.27)
$w_{n,m}(R_{\ast}(t),t)=-\frac{\partial\sigma_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\rho_{n,m}(t)+\varepsilon b^{2}_{n,m}(t),\ \ \ \ \
t>0,$ (4.28)
$\begin{split}q_{n,m}(R_{\ast}(t),t)=&-\frac{1}{R_{\ast}^{2}(t)}\left(1-\frac{n(n+1)}{2}\right)\rho_{n,m}(t)\\\
&-\frac{\partial p_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\rho_{n,m}(t)+\varepsilon b^{3}_{n,m}(t),\ \ \
t>0,\end{split}$ (4.29) $\rho_{n,m}|_{t=0}=\rho_{0,n,m},\ \ \ \ \
w_{n,m}|_{t=0}=w_{0,n,m}(r)\ \ \ \ \mbox{in}\ \ B_{R_{\ast}(0)}.$

As in [23], we can solve (4.24) and (4.27) in the form

(4.30)
$\begin{split}w_{n,m}(r,t)=&\Big{[}-\frac{\partial\sigma_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\rho_{n,m}(t)+\varepsilon
b^{2}_{n,m}(t)\Big{]}\frac{R_{\ast}^{1/2}(t)}{I_{n+1/2}(R_{\ast}(t))}\frac{I_{n+1/2}(r)}{r^{1/2}}\\\
&+\varepsilon\xi_{1,n,m}(r,t),\end{split}$

where $\xi_{1,n,m}(r,t)$ is the solution of

(4.31)
$\begin{split}-\Delta\xi_{1,n,m}(r,t)+\left(\frac{n(n+1)}{r^{2}}+1\right)\xi_{1,n,m}(r,t)&=f^{1}_{n,m}(r,t)\
\ \ \ \mbox{in}\ B_{R_{\ast}(t)},\\\
\xi_{1,n,m}(R_{\ast}(t),t)&=0.\end{split}$

Let

(4.32) $\psi(r,t)=q_{n,m}(r,t)+\mu w_{n,m}(r,t),$

then $\psi(r,t)$ satisfies

(4.33) $-\Delta\psi(r,t)+\frac{n(n+1)}{r^{2}}\psi(r,t)=\varepsilon\mu
f^{1}_{n,m}(r,t)+\varepsilon f^{2}_{n,m}(r,t)\ \ \ \ \mbox{in}\
B_{R_{\ast}(t)},$

and by (4.27) and (4.28), $\psi(r,t)$ satisfies the following boundary
condition

(4.34)
$\begin{split}\psi(R_{\ast}(t),t)=&\frac{1}{R_{\ast}^{2}(t)}\left(\frac{n(n+1)}{2}-1\right)\rho_{n,m}(t)-\frac{\partial
p_{\ast}}{\partial r}\Big{|}_{r=R_{\ast}(t)}\rho_{n,m}(t)\\\
&-\mu\frac{\partial\sigma_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\rho_{n,m}(t)+\varepsilon
b^{3}_{n,m}(t)+\mu\varepsilon b^{2}_{n,m}(t).\end{split}$

The solution of the problem (4.33)-(4.34) is given by

(4.35)
$\begin{split}\psi(r,t)=&\frac{r^{n}}{R^{n}_{\ast}(t)}\Big{\\{}\frac{1}{R_{\ast}^{2}(t)}\left(\frac{n(n+1)}{2}-1\right)\rho_{n,m}(t)-\frac{\partial
p_{\ast}}{\partial r}\Big{|}_{r=R_{\ast}(t)}\rho_{n,m}(t)\\\
&-\mu\frac{\partial\sigma_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\rho_{n,m}(t)+\varepsilon
b^{3}_{n,m}(t)+\mu\varepsilon
b^{2}_{n,m}(t)\Big{\\}}+\varepsilon\psi_{1,n,m}(r,t),\end{split}$

where $\psi_{1,n,m}(r,t)$ is the solution of

(4.36)
$\begin{split}-\Delta\psi_{1,n,m}(r,t)+\frac{n(n+1)}{r^{2}}\psi_{1,n,m}(r,t)&=\mu
f^{1}_{n,m}(r,t)+f^{2}_{n,m}(r,t)\ \ \ \ \mbox{in}\ B_{R_{\ast}(t)},\\\
\psi_{1,n,m}(R_{\ast}(t),t)&=0.\end{split}$

It follows from (4.32), (4.35), (4.30) and (2.5) that

$\begin{split}\frac{\partial q_{n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}=&\frac{\partial\psi}{\partial
r}\Big{|}_{r=R_{\ast}(t)}-\mu\frac{\partial w_{n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\\\
=&\frac{n}{R_{\ast}(t)}\Big{\\{}\frac{1}{R_{\ast}^{2}(t)}\left(\frac{n(n+1)}{2}-1\right)\rho_{n,m}(t)-\frac{\partial
p_{\ast}}{\partial r}\Big{|}_{r=R_{\ast}(t)}\rho_{n,m}(t)\\\
&-\mu\frac{\partial\sigma_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\rho_{n,m}(t)+\varepsilon
b^{3}_{n,m}(t)+\mu\varepsilon
b^{2}_{n,m}(t)\Big{\\}}+\varepsilon\frac{\partial\psi_{1,n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\\\
&-\mu\Big{[}-\frac{\partial\sigma_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\rho_{n,m}(t)+\varepsilon
b^{2}_{n,m}(t)\Big{]}\left\\{\frac{I_{n+3/2}(R_{\ast}(t))}{I_{n+1/2}(R_{\ast}(t))}+\frac{n}{R_{\ast}(t)}\right\\}\\\
&-\varepsilon\mu\frac{\partial\xi_{1,n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}.\end{split}$

After a direct computation, by (2.1), (2.7) and (2.8), we obtain

$\begin{split}\frac{\partial q_{n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}=&\frac{n}{R_{\ast}(t)}\Big{\\{}\frac{1}{R_{\ast}^{2}(t)}\left(\frac{n(n+1)}{2}-1\right)\rho_{n,m}(t)-\frac{\partial
p_{\ast}}{\partial r}\Big{|}_{r=R_{\ast}(t)}\rho_{n,m}(t)\Big{\\}}\\\
&+\mu\frac{\partial\sigma_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}R_{\ast}(t)P_{n}(R_{\ast}(t))\rho_{n,m}(t)+\frac{n}{R_{\ast}(t)}\varepsilon
b^{3}_{n,m}(t)\\\ &-\varepsilon\mu
R_{\ast}(t)P_{n}(R_{\ast}(t))b^{2}_{n,m}(t)-\varepsilon\mu\frac{\partial\xi_{1,n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}+\varepsilon\frac{\partial\psi_{1,n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\\\
=&\Big{\\{}\frac{n}{R_{\ast}^{3}(t)}\left(\frac{n(n+1)}{2}-1\right)+\frac{n}{R_{\ast}(t)}\frac{dR_{\ast}(t)}{dt}\\\
&+\mu\phi(t)R^{2}_{\ast}(t)P_{0}(R_{\ast}(t))P_{n}(R_{\ast}(t))\Big{\\}}\rho_{n,m}(t)+\varepsilon\frac{n}{R_{\ast}(t)}b^{3}_{n,m}(t)\\\
&-\varepsilon\mu
R_{\ast}(t)P_{n}(R_{\ast}(t))b^{2}_{n,m}(t)-\varepsilon\mu\frac{\partial\xi_{1,n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}+\varepsilon\frac{\partial\psi_{1,n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}.\end{split}$

Inserting the above expression into (4.26) and using the relation (2.9), we
derive

$\begin{split}\frac{d\rho_{n,m}(t)}{dt}=&\Big{\\{}-\frac{n-1}{R_{\ast}(t)}\frac{dR_{\ast}(t)}{dt}-\frac{n}{R_{\ast}^{3}(t)}\Big{(}\frac{n(n+1)}{2}-1\Big{)}\\\
&+\mu\phi(t)R^{2}_{\ast}(t)P_{0}(R_{\ast}(t))\big{[}P_{1}(R_{\ast}(t))-P_{n}(R_{\ast}(t))\big{]}\Big{\\}}\rho_{n,m}(t)\\\
&+\varepsilon\mu\frac{\partial\xi_{1,n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}-\varepsilon\frac{\partial\psi_{1,n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\\\ &+\varepsilon b^{1}_{n,m}(t)+\varepsilon\mu
R_{\ast}(t)P_{n}(R_{\ast}(t))b^{2}_{n,m}(t)-\varepsilon\frac{n}{R_{\ast}(t)}b^{3}_{n,m}(t).\end{split}$

Then we solve this ODE in the following form

(4.37)
$\begin{split}\rho_{n,m}(t)=\rho_{0,n,m}e^{\int_{0}^{t}H_{n}(\tau)d\tau}+\varepsilon\int_{0}^{t}Q_{n,m}(s)e^{\int_{s}^{t}H_{n}(\tau)d\tau}ds,\end{split}$

where

(4.38)
$\begin{split}H_{n}(\tau)=&-\frac{n-1}{R_{\ast}(\tau)}\frac{dR_{\ast}(\tau)}{d\tau}-\frac{n}{R_{\ast}^{3}(\tau)}\Big{(}\frac{n(n+1)}{2}-1\Big{)}\\\
&+\mu\phi(\tau)R^{2}_{\ast}(\tau)P_{0}(R_{\ast}(\tau))\big{[}P_{1}(R_{\ast}(\tau))-P_{n}(R_{\ast}(\tau))\big{]}\\\
=&\Big{\\{}R_{\ast}(\tau)\big{[}P_{1}(R_{\ast}(\tau))-P_{n}(R_{\ast}(\tau))\big{]}-\frac{n-1}{R_{\ast}(\tau)}\Big{\\}}\frac{dR_{\ast}(\tau)}{d\tau}\\\
&-\frac{n}{R_{\ast}^{3}(\tau)}\Big{(}\frac{n(n+1)}{2}-1\Big{)}+\frac{\mu\widetilde{\sigma}}{3}R_{\ast}^{2}(\tau)\big{[}P_{1}(R_{\ast}(\tau))-P_{n}(R_{\ast}(\tau))\big{]},\end{split}$

here we have used the fact, by (1.8),

$\mu\phi(\tau)P_{0}(R_{\ast}(\tau))=\frac{1}{R_{\ast}(\tau)}\frac{dR_{\ast}(\tau)}{d\tau}+\frac{\mu\widetilde{\sigma}}{3},$

and

(4.39) $\begin{split}Q_{n,m}(s)=&\mu\frac{\partial\xi_{1,n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(s)}-\frac{\partial\psi_{1,n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(s)}\\\ &+b_{n,m}^{1}(s)+\mu
R_{\ast}(s)P_{n}(R_{\ast}(s))b_{n,m}^{2}(s)-\frac{n}{R_{\ast}(s)}b_{n,m}^{3}(s).\end{split}$

Note that $\rho_{M,n,m}\triangleq\rho_{0,n,m}e^{\int_{0}^{t}H_{n}(\tau)d\tau}$
is the function $\rho_{n,m}$ corresponding to the case $f_{n,m}^{1}\equiv
f_{n,m}^{2}\equiv b_{n,m}^{1}\equiv b_{n,m}^{2}\equiv b_{n,m}^{3}\equiv 0$ of
the system (4.24)-(4.28). The corresponding $w_{n,m}$ and $q_{n,m}$ are
denoted by $w_{M,n,m}$ and $q_{M,n,m}$. Let
$w_{M}=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}w_{M,n,m}Y_{n,m}$,
$q_{M}=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}q_{M,n,m}Y_{n,m}$ and
$\rho_{M}=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\rho_{M,n,m}Y_{n,m}$.

We shall estimate $\rho_{n,m}(t)$ by dividing into three cases: $n=0$, $n=1$
and $n\geq 2$.

###  4.1. Case 1: $n=0$

It follows from (4.37) and (4.39) that

(4.40)
$\begin{split}\rho_{0,0}(t)=&\rho_{0,0,0}e^{\int_{0}^{t}H_{0}(\tau)d\tau}+\varepsilon\int_{0}^{t}Q_{0,0}(s)e^{\int_{s}^{t}H_{0}(\tau)d\tau}ds\\\
=&\rho_{0,0,0}e^{\int_{0}^{t}H_{0}(\tau)d\tau}+\varepsilon\int_{0}^{t}\Big{[}\mu\frac{\partial\xi_{1,0,0}}{\partial
r}\Big{|}_{r=R_{\ast}(s)}-\frac{\partial\psi_{1,0,0}}{\partial
r}\Big{|}_{r=R_{\ast}(s)}\Big{]}e^{\int_{s}^{t}H_{0}(\tau)d\tau}ds\\\
&+\varepsilon\int_{0}^{t}b_{0,0}^{1}(s)e^{\int_{s}^{t}H_{0}(\tau)d\tau}ds+\varepsilon\int_{0}^{t}\mu
R_{\ast}(s)P_{0}(R_{\ast}(s))b_{0,0}^{2}(s)e^{\int_{s}^{t}H_{0}(\tau)d\tau}ds\\\
\equiv&\rho_{M,0,0}+\varepsilon L_{1}+\varepsilon L_{2}+\varepsilon
L_{3},\end{split}$

where, by (4.38),

(4.41)
$\begin{split}H_{0}(\tau)=&\Big{\\{}R_{\ast}(\tau)\big{[}P_{1}(R_{\ast}(\tau))-P_{0}(R_{\ast}(\tau))\big{]}+\frac{1}{R_{\ast}(\tau)}\Big{\\}}\frac{dR_{\ast}(\tau)}{d\tau}\\\
&+\frac{\mu\widetilde{\sigma}}{3}R_{\ast}^{2}(\tau)\big{[}P_{1}(R_{\ast}(\tau))-P_{0}(R_{\ast}(\tau))\big{]}.\end{split}$

We now proceed to estimate the last three terms on the right-hand of (4.40),
respectively. Before doing that, we establish the following lemma.

######  Lemma 4.1.

For $\mu>0$, there exists a positive number $\delta$, depending on $\mu$ and
$R_{\ast}$, such that the following is true:

(4.42) $e^{\int_{s}^{t}H_{0}(\tau)d\tau}\leq Ce^{-\delta(t-s)}.$

Proof. For any $t>0$ and $0<s<t$, there exist $m\in\mathbb{N}$ and $z\in[0,T)$
such that $t-s=mT+z$, then

$\begin{split}&e^{\int_{s}^{t}\left\\{R_{\ast}(\tau)\big{[}P_{1}(R_{\ast}(\tau))-P_{0}(R_{\ast}(\tau))\big{]}+\frac{1}{R_{\ast}(\tau)}\right\\}\frac{dR_{\ast}(\tau)}{d\tau}d\tau}\\\
&\quad=e^{\int_{s}^{s+z}\left\\{R_{\ast}(\tau)\big{[}P_{1}(R_{\ast}(\tau))-P_{0}(R_{\ast}(\tau))\big{]}+\frac{1}{R_{\ast}(\tau)}\right\\}dR_{\ast}(\tau)}\leq
C.\end{split}$

Furthermore, by the boundedness of $R_{\ast}(t)$ and (2.4), we obtain

$\begin{split}e^{\int_{s}^{t}H_{0}(\tau)d\tau}&\leq
Ce^{\int_{s}^{t}\frac{\mu\widetilde{\sigma}}{3}R_{\ast}^{2}(\tau)\big{[}P_{1}(R_{\ast}(\tau))-P_{0}(R_{\ast}(\tau))\big{]}d\tau}\\\
&\leq Ce^{-\delta(t-s)}\end{split}$

for some $\delta>0$. Hence, the proof is complete. $\Box$

######  Lemma 4.2.

If

$\int_{0}^{\infty}e^{2\delta_{1}t}|b(t)|^{2}dt\leq A,\ \ \ \ \
0<\delta_{1}<\delta,$

then

(4.43)
$\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\int_{0}^{t}b(s)e^{\int_{s}^{t}H_{0}(\tau)d\tau}ds\Big{|}^{2}dt\leq
CA.$

Proof. Lemma 4.1 yields

$\begin{split}J_{0}&\equiv\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\int_{0}^{t}b(s)e^{\int_{s}^{t}H_{0}(\tau)d\tau}ds\Big{|}^{2}dt\\\
&\leq
C\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\int_{0}^{t}|b(s)|e^{-\delta(t-s)}ds\Big{|}^{2}dt\\\
&=C\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\int_{0}^{t}|b(t-s)|e^{-\delta
s}ds\Big{|}^{2}dt\\\ &\leq
C\int_{0}^{\infty}\left\\{\int_{0}^{t}|b(t-s)|^{2}e^{2\delta_{1}(t-s)}e^{-\delta
s+\delta_{1}s}ds\int_{0}^{t}e^{-\delta s+\delta_{1}s}ds\right\\}dt\\\ &\leq
C\int_{0}^{\infty}\int_{0}^{t}|b(t-s)|^{2}e^{2\delta_{1}(t-s)}e^{-(\delta-\delta_{1})s}dsdt.\end{split}$

Changing the order of the integration, we obtain

$\begin{split}J_{0}&\leq
C\int_{0}^{\infty}\int_{s}^{\infty}|b(t-s)|^{2}e^{2\delta_{1}(t-s)}e^{-(\delta-\delta_{1})s}dtds\\\
&=C\int_{0}^{\infty}\int_{0}^{\infty}|b(t)|^{2}e^{2\delta_{1}t}e^{-(\delta-\delta_{1})s}dtds\\\
&\leq CA\int_{0}^{\infty}e^{-(\delta-\delta_{1})s}ds\\\ &\leq CA,\end{split}$

which completes our proof. $\Box$

######  Lemma 4.3.

For $0<\delta_{1}<\delta$, the following estimate holds:

(4.44) $\int_{0}^{\infty}e^{2\delta_{1}t}\big{|}L_{1}\big{|}^{2}dt\leq
C|\varepsilon|^{-1}\big{(}F_{0,0}^{1}+F_{0,0}^{2}\big{)}.$

Proof. Recall [13, Lemmas 3.2 and 3.3] that

(4.45) $\left|\frac{\partial\xi_{1,n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\right|^{2}\leq\frac{C}{n+1}\|f_{n,m}^{1}(\cdot,t)\|^{2}_{L^{2}(B_{R_{\ast}(t)})},$
(4.46) $\left|\frac{\partial\psi_{1,n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\right|^{2}\leq\frac{C}{n+1}\big{(}\|f_{n,m}^{1}(\cdot,t)\|^{2}_{L^{2}(B_{R_{\ast}(t)})}+\|f_{n,m}^{2}(\cdot,t)\|^{2}_{L^{2}(B_{R_{\ast}(t)})}\big{)},$

so that, by (4.19) and (4.20), we obtain

$\begin{split}&\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\mu\frac{\partial\xi_{1,0,0}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}-\frac{\partial\psi_{1,0,0}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\Big{|}^{2}dt\\\ &\quad\leq
C\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\frac{\partial\xi_{1,0,0}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\Big{|}^{2}dt+C\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\frac{\partial\psi_{1,0,0}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\Big{|}^{2}dt\\\ &\quad\leq
C\int_{0}^{\infty}e^{2\delta_{1}t}\|f_{0,0}^{1}\|^{2}_{L^{2}(B_{R_{\ast}(t)})}dt+C\int_{0}^{\infty}e^{2\delta_{1}t}\big{(}\|f_{0,0}^{1}\|^{2}_{L^{2}(B_{R_{\ast}(t)})}+\|f_{0,0}^{2}\|^{2}_{L^{2}(B_{R_{\ast}(t)})}\big{)}dt\\\
&\quad\leq
C|\varepsilon|^{-1}\big{(}F_{0,0}^{1}+F_{0,0}^{2}\big{)}.\end{split}$

Hence, it follows from Lemma 4.2 that (4.44) holds. $\Box$

######  Lemma 4.4.

The following estimates hold:

(4.47) $\int_{0}^{\infty}e^{2\delta_{1}t}\big{|}L_{2}\big{|}^{2}dt\leq
C|\varepsilon|^{-1}B_{0,0}^{1},$ (4.48)
$\int_{0}^{\infty}e^{2\delta_{1}t}\big{|}L_{3}\big{|}^{2}dt\leq
C|\varepsilon|^{-1}B_{0,0}^{2}.$

Proof. (4.21) and Lemma 4.2 yield (4.47). Moreover, by (2.6), (4.22) and Lemma
4.2, we get (4.48). Therefore, the proof is complete. $\Box$

Combining Lemmas 4.3 and 4.4, we establish an estimate for $\rho_{0,0}(t)$ as
follows:

(4.49)
$\int_{0}^{\infty}e^{2\delta_{1}t}\big{|}\rho_{0,0}(t)-\rho_{M,0,0}(t)\big{|}^{2}dt\leq
C|\varepsilon|\big{(}F_{0,0}^{1}+F_{0,0}^{2}+B_{0,0}^{1}+B_{0,0}^{2}\big{)}.$

###  4.2. Case 2: $n=1$

It can be easily seen from (4.38) that

$\begin{split}H_{1}(\tau)=0.\end{split}$

Therefore, (4.37) implies that

(4.50) $\rho_{1,m}(t)=\rho_{0,1,m}+\varepsilon\int_{0}^{t}Q_{1,m}(s)ds,$

where

(4.51) $\begin{split}Q_{1,m}(s)=&\mu\frac{\partial\xi_{1,1,m}}{\partial
r}\Big{|}_{r=R_{\ast}(s)}-\frac{\partial\psi_{1,1,m}}{\partial
r}\Big{|}_{r=R_{\ast}(s)}\\\ &+b_{1,m}^{1}(s)+\mu
R_{\ast}(s)P_{1}(R_{\ast}(s))b_{1,m}^{2}(s)-\frac{1}{R_{\ast}(s)}b_{1,m}^{3}(s).\end{split}$

As pointed out in the introduction, after the initial values perturbed, say
(1.9), the domain of the system (1.1)-(1.5) (or (4.1)-(4.5)) undergoes a
translation owing to the perturbations of mode 1, that is to say, the
perturbations of mode 1 terms result in the decay estimates in time $t$, as
well as the translation of the origin. In order to establish the asymptotic
stability, we need to take care of the center of the limiting sphere. We shall
employ on an iterative procedure in which at each iteration we shift the
domain to a “nearly” optimal location with a new center $\varepsilon
a=\varepsilon(a_{1},a_{2},a_{3})$. Note that a translation of the origin dose
not change the equations (1.1)-(1.5) (or (4.1)-(4.5)), but change the initial
data. As stated in [14, Section 6], in the new coordinate system, the initial
data are changed as follows, $m=-1,0,1$,

(4.52) $\begin{split}w_{0,1,m}\ \mbox{is\ replaced\ by}\
w_{0,1,m}+b_{m+2}\frac{\partial\sigma_{\ast}(r,0)}{\partial r}+\varepsilon
A_{m},\\\ \mbox{and}\ \rho_{0,1,m}\ \mbox{is\ replaced\ by}\
\rho_{0,1,m}-b_{m+2}+\varepsilon B_{m},\end{split}$

where $A_{m}$ and $B_{m}$ are bounded functions of $(a,\varepsilon)$, and
$b_{m+2}$ satisfies

$b_{1}-b_{3}=a_{1}\sqrt{\frac{8\pi}{3}},\ \ \ \
i(b_{1}+b_{3})=-a_{2}\sqrt{\frac{8\pi}{3}},\ \ \ \
b_{2}=a_{3}\sqrt{\frac{4\pi}{3}}.$

Then, in the new coordinate system, the expression $\rho_{1,m}(t)$ is given by

(4.53) $\rho_{1,m}(t)=\rho_{0,1,m}-b_{m+2}+\varepsilon
B_{m}+\varepsilon\int_{0}^{t}Q_{1,m}(s)ds.$

We note that as we go to the system (4.1)-(4.5), we shall always use the
translated initial data. As in [14, Page 632], we take $f^{i}$, $b^{j}$ in
(4.1)-(4.5) to be functions not only of $(r,\theta,\varphi,t)$, but also of
the center $a$, which aims to accommodate the consistency condition of order
2. The function $Q_{1,m}(s)$ depends on $a$ implicitly through the dependence
of $\xi_{1,1,m}$, $\psi_{1,1,m}$ and $b^{j}_{1,m}$ on $a$.

In order to find the center of the limiting sphere and establish the decay
estimates in time $t$, we rewrite $\rho_{1,m}(t)$ as

(4.54) $\begin{split}\rho_{1,m}(t)&=\rho_{0,1,m}-b_{m+2}+\varepsilon
B_{m}+\varepsilon\int_{0}^{\infty}Q_{1,m}(s)ds-\varepsilon\int_{t}^{\infty}Q_{1,m}(s)ds\\\
&\triangleq F_{m}(a)-\varepsilon\int_{t}^{\infty}Q_{1,m}(s)ds.\end{split}$

Let $F(a)=(F_{-1}(a),F_{0}(a),F_{1}(a))$. We have the following critical
theorem.

######  Theorem 4.1.

For $|\varepsilon|$ small, there exists a new center $\varepsilon
a^{\ast}(\varepsilon)$ such that

$F(a^{\ast}(\varepsilon))=0.$

Proof. By (4.54), we have

$\begin{split}F(a)=E(a)+\varepsilon G(a).\end{split}$

Clearly, there exists $a_{0}$ such that $E(a_{0})=0$ and $E^{\prime}(a_{0})$
is invertible. Moreover, $E^{\prime}(a)$ and $G^{\prime}(a)$ are continuous
for $a\in\overline{B}_{1}(a_{0})$. Hence, by applying Theorem 2.2, we complete
the proof. $\Box$

Now we proceed to estimate the last term of the right-hand side of (4.54).
Recall [14, Lemma 5.4] that

$\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\int_{t}^{\infty}b(\tau)d\tau\Big{|}^{2}dt\leq\frac{1}{\delta_{1}^{2}}\int_{0}^{\infty}e^{2\delta_{1}t}|b(t)|^{2}dt,$

so that, by (2.6), (4.45), (4.46) and (4.19)-(4.23), we have the following
estimate:

$\begin{split}&\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\int_{t}^{\infty}Q_{1,m}(s)ds\Big{|}^{2}dt\\\
&\leq C\int_{0}^{\infty}e^{2\delta_{1}t}\big{|}Q_{1,m}(t)\big{|}^{2}dt\\\
&\leq
C\int_{0}^{\infty}e^{2\delta_{1}t}\Big{(}\Big{|}\frac{\partial\xi_{1,1,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\Big{|}^{2}+\Big{|}\frac{\partial\psi_{1,1,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\Big{|}^{2}+|b_{1,m}^{1}|^{2}+|b_{1,m}^{2}|^{2}+|b_{1,m}^{3}|^{2}\Big{)}dt\\\
&\leq
C\int_{0}^{\infty}e^{2\delta_{1}t}\Big{(}\|f_{1,m}^{1}\|^{2}_{L^{2}}+\|f_{1,m}^{2}\|^{2}_{L^{2}}+|b_{1,m}^{1}|^{2}+|b_{1,m}^{2}|^{2}+|b_{1,m}^{3}|^{2}\Big{)}dt\\\
&\leq
C|\varepsilon|^{-1}\big{(}F_{1,m}^{1}+F_{1,m}^{2}+B_{1,m}^{1}+B_{1,m}^{2}+B_{1,m}^{3}\big{)}.\end{split}$

It follows that

(4.55)
$\begin{split}\int_{0}^{\infty}e^{2\delta_{1}t}|\rho_{1,m}(t)-F_{m}(a^{\ast}(\varepsilon))|^{2}&=|\varepsilon|^{2}\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\int_{t}^{\infty}Q_{1,m}(s)ds\Big{|}^{2}dt\\\
&\leq
C|\varepsilon|\big{(}F_{1,m}^{1}+F_{1,m}^{2}+B_{1,m}^{1}+B_{1,m}^{2}+B_{1,m}^{3}\big{)}.\end{split}$

###  4.3. Case 3: $n\geq 2$

In this subsection, we shall deal with the case $n\geq 2$. We first establish
the following lemma:

######  Lemma 4.5.

Let $\mu<\mu_{\ast}$, where $\mu_{\ast}$ is defined by (1.11). For $n\geq 2$,
there exists a small positive number $\delta$, independent of $n$, such that

(4.56) $e^{\int_{s}^{t}H_{n}(\tau)d\tau}\leq e^{-\delta(n^{3}+1)(t-s)}.$

Proof. Since $0<\mu<\mu_{\ast}$ and $R_{\ast}(t)$ is periodic and bounded, it
follows from (4.38) and (2.6) that there exists $\gamma>0$ ($\gamma$ is
independent of $n$) such that

$\begin{split}H_{n}(\tau)\leq-\gamma(n^{3}+1)\end{split}$

for $n$ sufficiently large, say $n>n_{0}$, which yields (4.56).

Recall [22, Lemma 4.4] that

$\mu_{\ast}\leq\frac{\int_{0}^{T}\frac{j}{R_{\ast}^{3}(\tau)}\Big{(}\frac{j(j+1)}{2}-1\Big{)}d\tau}{\int_{0}^{T}\frac{\widetilde{\sigma}}{3}R_{\ast}^{2}(\tau)\big{[}P_{1}(R_{\ast}(\tau))-P_{j}(R_{\ast}(\tau))\big{]}d\tau},\
\ \ \ \ j\geq 2.$

Then for each fixed $j\in[2,n_{0}]$ and $0<\mu<\mu_{\ast}$, we have

(4.57)
$\begin{split}\int_{s}^{s+T}&\frac{j}{R_{\ast}^{3}(\tau)}\Big{(}\frac{j(j+1)}{2}-1\Big{)}d\tau-\mu\int_{s}^{s+T}\frac{\widetilde{\sigma}}{3}R_{\ast}^{2}(\tau)\big{[}P_{1}(R_{\ast}(\tau))-P_{j}(R_{\ast}(\tau))\big{]}d\tau>\gamma_{j},\end{split}$

where $\gamma_{j}>0$ is small enough, and dependent of $(R_{\ast},T,\mu)$ and
$j$. For $t>s$, there exists $m\in\mathbb{N}$ and $\nu\in[0,T)$ such that
$t-s=mT+\nu$. Furthermore, by (4.38) and (4.57), we obtain

$\begin{split}e^{\int_{s}^{t}H_{n}(\tau)d\tau}=&e^{\int_{s}^{s+\nu}H_{n}(\tau)d\tau+m\int_{s}^{s+T}H_{n}(\tau)d\tau}\\\
=&\mbox{exp}\Big{\\{}\int_{s}^{s+\nu}\Big{[}R_{\ast}(\tau)\big{[}P_{1}(R_{\ast}(\tau))-P_{j}(R_{\ast}(\tau))\big{]}-\frac{j-1}{R_{\ast}(\tau)}\Big{]}\frac{dR_{\ast}(\tau)}{d\tau}d\tau\\\
&-\int_{s}^{s+\nu}\frac{j}{R_{\ast}^{3}(\tau)}\Big{(}\frac{j(j+1)}{2}-1\Big{)}d\tau\\\
&+\mu\int_{s}^{s+\nu}\frac{\widetilde{\sigma}}{3}R_{\ast}^{2}(\tau)\big{[}P_{1}(R_{\ast}(\tau))-P_{j}(R_{\ast}(\tau))\big{]}d\tau\\\
&-m\int_{s}^{s+T}\frac{j}{R_{\ast}^{3}(\tau)}\Big{(}\frac{j(j+1)}{2}-1\Big{)}d\tau\\\
&+m\mu\int_{s}^{s+T}\frac{\widetilde{\sigma}}{3}R_{\ast}^{2}(\tau)\big{[}P_{1}(R_{\ast}(\tau))-P_{j}(R_{\ast}(\tau))\big{]}d\tau\Big{\\}}\\\
\leq&e^{-m\widetilde{\delta}_{j}}=e^{-\frac{t-s-\nu}{T}\widetilde{\delta}_{j}}\leq
e^{-\delta_{j}(t-s)}\end{split}$

for some $\widetilde{\delta}_{j}>0$ and $\delta_{j}>0$ (both
$\widetilde{\delta}_{j}$ and $\delta_{j}>0$ depend on $(R_{\ast},T,\mu)$ and
$j$).

Let
$\delta=\min\\{\gamma,\frac{\delta_{2}}{2^{3}+1},\frac{\delta_{3}}{3^{3}+1},\cdots,\frac{\delta_{n_{0}}}{n_{0}^{3}+1}\\}$,
then for $n\geq 2$, we get (4.56), and this completes the proof. $\Box$

######  Lemma 4.6.

If

$\int_{0}^{\infty}e^{2\delta_{1}t}|b(t)|^{2}dt\leq A,\ \ \ \ \
0<\delta_{1}<\delta,$

then, for $n\geq 2$,

(4.58)
$\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\int_{0}^{t}b(s)e^{\int_{s}^{t}H_{n}(\tau)d\tau}ds\Big{|}^{2}dt\leq
CA(n+1)^{-6},$

where $C$ is a constant independent of $n$.

Proof. It follows from Lemma 4.5 that

$\begin{split}J_{1}&\equiv\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\int_{0}^{t}b(s)e^{\int_{s}^{t}H_{n}(\tau)d\tau}ds\Big{|}^{2}dt\\\
&\leq\int_{0}^{\infty}e^{2\delta_{1}t}\Big{(}\int_{0}^{t}|b(s)|e^{-\delta(n^{3}+1)(t-s)}ds\Big{)}^{2}dt\\\
&=\int_{0}^{\infty}e^{2\delta_{1}t}\Big{(}\int_{0}^{t}|b(t-\tau)|e^{-\delta(n^{3}+1)\tau}d\tau\Big{)}^{2}dt\\\
&\leq\int_{0}^{\infty}\left\\{\int_{0}^{t}|b(t-\tau)|^{2}e^{2\delta_{1}(t-\tau)}e^{-\delta(n^{3}+1)\tau+\delta_{1}\tau}d\tau\int_{0}^{t}e^{-\delta(n^{3}+1)\tau+\delta_{1}\tau}d\tau\right\\}dt\\\
&\leq
C(n+1)^{-3}\int_{0}^{\infty}\int_{0}^{t}|b(t-\tau)|^{2}e^{2\delta_{1}(t-\tau)}e^{-\delta(n^{3}+1)\tau+\delta_{1}\tau}d\tau
dt.\end{split}$

Changing the order of the integration, we obtain

$\begin{split}J_{1}&\leq
C(n+1)^{-3}\int_{0}^{\infty}\int_{\tau}^{\infty}|b(t-\tau)|^{2}e^{2\delta_{1}(t-\tau)}e^{-\delta(n^{3}+1)\tau+\delta_{1}\tau}dtd\tau\\\
&=C(n+1)^{-3}\int_{0}^{\infty}\int_{0}^{\infty}|b(t)|^{2}e^{2\delta_{1}t}e^{-\delta(n^{3}+1)\tau+\delta_{1}\tau}dtd\tau\\\
&\leq
CA(n+1)^{-3}\int_{0}^{\infty}e^{-\delta(n^{3}+1)\tau+\delta_{1}\tau}d\tau\\\
&\leq CA(n+1)^{-6},\end{split}$

which completes our proof. $\Box$

######  Lemma 4.7.

For all $n\geq 2$ and $|m|\leq n$,

$\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\rho_{n,m}-\rho_{M,n,m}\Big{|}^{2}dt\leq
C|\varepsilon|(n+1)^{-7}\big{(}F_{n,m}^{1}+F_{n,m}^{2}+B_{n,m}^{1}+B_{n,m}^{2}+B_{n,m}^{3}\big{)},$

where $C$ is a constant independent of $n$.

Proof. By (4.39), (2.6), (4.45), (4.46) and (4.19)-(4.23), we get

$\begin{split}&\int_{0}^{\infty}e^{2\delta_{1}t}|Q_{n,m}(t)|^{2}dt\\\ &\leq
C\int_{0}^{\infty}e^{2\delta_{1}t}\Big{(}\Big{|}\frac{\partial\xi_{1,n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\Big{|}^{2}+\Big{|}\frac{\partial\psi_{1,n,m}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}\Big{|}^{2}+|b_{n,m}^{1}|^{2}+|b_{n,m}^{2}|^{2}+n^{2}|b_{n,m}^{3}|^{2}\Big{)}dt\\\
&\leq
C\int_{0}^{\infty}e^{2\delta_{1}t}\Big{(}\frac{1}{n+1}\|f_{n,m}^{1}\|^{2}_{L^{2}}+\frac{1}{n+1}\|f_{n,m}^{2}\|^{2}_{L^{2}}+|b_{n,m}^{1}|^{2}+|b_{n,m}^{2}|^{2}+n^{2}|b_{n,m}^{3}|^{2}\Big{)}dt\\\
&\leq
C|\varepsilon|^{-1}(n+1)^{-1}\big{(}F_{n,m}^{1}+F_{n,m}^{2}+B_{n,m}^{1}+B_{n,m}^{2}+B_{n,m}^{3}\big{)}.\end{split}$

Furthermore, it follows from Lemma 4.6 that

$\begin{split}\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\rho_{n,m}-\rho_{M,n,m}\Big{|}^{2}dt=&|\varepsilon|^{2}\int_{0}^{\infty}e^{2\delta_{1}t}\Big{|}\int_{0}^{t}Q_{n,m}(s)e^{\int_{s}^{t}H_{n}(\tau)d\tau}ds\Big{|}^{2}dt\\\
\leq&C|\varepsilon|(n+1)^{-7}\big{(}F_{n,m}^{1}+F_{n,m}^{2}+B_{n,m}^{1}+B_{n,m}^{2}+B_{n,m}^{3}\big{)}.\end{split}$

Hence, the proof is complete. $\Box$

Combining (4.49), (4.55) and Lemma 4.7, we obtain, in the new coordinate
system,

(4.59)
$\int_{0}^{\infty}e^{2\delta_{1}t}\|\rho-\rho_{M}\|^{2}_{H^{7/2}(\partial
B_{R_{\ast}(t)})}dt\leq C|\varepsilon|.$

######  Remark 4.3.

In fact, $(w_{M},q_{M},\rho_{M})$ is the solution of the corresponding linear
system of the original problem (1.1)-(1.5), and its decay estimate in time $t$
has been derived in [25] for the two-space dimensional case and [22] for the
three-space dimensional case, respectively.

Moreover, we have the following estimates for $w$, $q$ and
$\frac{\partial\rho}{\partial t}$.

######  Lemma 4.8.

If $\delta_{1}$ is sufficiently small then

(4.60)
$\int_{0}^{\infty}e^{2\delta_{1}t}\|w-w_{M}\|^{2}_{H^{1}(B_{R_{\ast}(t)})}dt\leq
C|\varepsilon|,$ (4.61)
$\int_{0}^{\infty}e^{2\delta_{1}t}\|q-q_{M}\|^{2}_{H^{2}(B_{R_{\ast}(t)})}dt\leq
C|\varepsilon|,$ (4.62)
$\int_{0}^{\infty}e^{2\delta_{1}t}\Big{\|}\frac{\partial}{\partial
t}(\rho-\rho_{M})\Big{\|}^{2}_{H^{1/2}(\partial B_{R_{\ast}(t)})}dt\leq
C|\varepsilon|.$

Proof. Observe that $\widetilde{w}\equiv w_{n,m}-w_{M,n,m}$ satisfies (4.24)
and (4.27) with $\rho_{n,m}$ replaced by
$\widetilde{\rho}\equiv\rho_{n,m}-\rho_{M,n,m}$, i.e.,

$\begin{split}-\Delta\widetilde{w}+\left(\frac{n(n+1)}{r^{2}}+1\right)\widetilde{w}&=\varepsilon
f_{n,m}^{1}(r,t)\qquad\qquad\quad\ \ \ \ \ \mbox{in}\ B_{R_{\ast}(t)},\
t>0,\\\ \widetilde{w}(R_{\ast}(t),t)&=-\lambda\widetilde{\rho}(t)+\varepsilon
b_{n,m}^{2}(t),\ \qquad\ \ \ t>0,\end{split}$

where $\lambda=\frac{\partial\sigma_{\ast}}{\partial
r}\Big{|}_{r=R_{\ast}(t)}$.

Define $V=\widetilde{w}-\widetilde{w}|_{r=R_{\ast}(t)}$, we obtain

$\begin{split}-\Delta V+\left(\frac{n(n+1)}{r^{2}}+1\right)V&=\varepsilon
f_{n,m}^{1}+\left(\frac{n(n+1)}{r^{2}}+1\right)(\lambda\widetilde{\rho}-\varepsilon
b_{n,m}^{2})\ \ \ \mbox{in}\ B_{R_{\ast}(t)},\ t>0,\\\
V(R_{\ast}(t),t)&=0,\qquad\qquad\quad\ t>0.\end{split}$

Multiplying both sides with $V$ and integrating over $B_{R_{\ast}(t)}$, we get

$\begin{split}&\int_{B_{R_{\ast}(t)}}|\nabla
V|^{2}dx+\int_{B_{R_{\ast}(t)}}\left(\frac{n(n+1)}{r^{2}}+1\right)|V|^{2}dx\\\
&=\int_{B_{R_{\ast}(t)}}\varepsilon
f_{n,m}^{1}Vdx+\int_{B_{R_{\ast}(t)}}\left(\frac{n(n+1)}{r^{2}}+1\right)\lambda\widetilde{\rho}Vdx-\int_{B_{R_{\ast}(t)}}\left(\frac{n(n+1)}{r^{2}}+1\right)\varepsilon
b_{n,m}^{2}Vdx.\end{split}$

By Young’s inequality, we further have

$\begin{split}&\int_{B_{R_{\ast}(t)}}|\nabla
V|^{2}dx+\int_{B_{R_{\ast}(t)}}\left(\frac{n(n+1)}{r^{2}}+1\right)|V|^{2}dx\\\
&\leq
C|\varepsilon|^{2}\|f_{n,m}^{1}\|^{2}_{L^{2}(B_{R_{\ast}(t)})}+C(n^{2}+1)|\widetilde{\rho}(t)|^{2}+C(n^{2}+1)|\varepsilon|^{2}|b^{2}_{n,m}(t)|^{2}.\end{split}$

Then integrating with $e^{2\delta_{1}t}dt$ and using (4.7), (4.59), (4.10), we
obtain

$\begin{split}\int_{0}^{\infty}e^{2\delta_{1}t}\|w-{w}_{M}\|^{2}_{H^{1}(B_{R_{\ast}(t)})}dt\leq&C|\varepsilon|^{2}\int_{0}^{\infty}e^{2\delta_{1}t}\|f^{1}\|^{2}_{L^{2}(B_{R_{\ast}(t)})}dt\\\
&+C\int_{0}^{\infty}e^{2\delta_{1}t}\|\rho-\widetilde{\rho}\|^{2}_{H^{7/2}(\partial
B_{R_{\ast}(t)})}dt\\\
&+C|\varepsilon|^{2}\int_{0}^{\infty}e^{2\delta_{1}t}\|b^{2}\|^{2}_{H^{1}(\partial
B_{R_{\ast}(t)})}dt\\\ \leq&C|\varepsilon|,\end{split}$

which completes (4.60).

Using (4.60) and the elliptic estimates, we derive (4.61). It follows from
(4.3), (4.59), (4.61) and (4.9) that (4.62) holds. Therefore, the proof is
complete. $\Box$

##  5\. Stability for $\mu<\mu_{\ast}$

In this section, we shall show that the mapping $S$ defined in Remark 4.1
admits a fixed point, thereby establishing the global existence of the system
(1.1)-(1.5) and the asymptotic stability of the radially symmetric
$T$-periodic positive solution
$(\sigma_{\ast}(r,t),p_{\ast}(r,t),R_{\ast}(t))$. Since the proof is similar
to that of the corresponding proof in [14, Page 633-637], we only give an
outline of the proof and omit most of its details.

We first have the following Hölder estimates on $\rho$, $w$, $q$.

######  Lemma 5.1.

After performing a change of variables $x\rightarrow x+\varepsilon
a^{\ast}(\varepsilon)$ on the initial data through (4.52), then for all $t>0$,
there exists a unique solution $(w,q,\rho)$ of the problem (4.1)-(4.5)
satisfying

(5.1)
$\begin{split}\|w\|_{C^{2+2\alpha/3,1+\alpha/3}(B_{R_{\ast}(t)}\times[0,\infty))}\leq
C,\end{split}$ (5.2)
$\begin{split}\|q\|_{C^{2+\alpha,\alpha/3}(B_{R_{\ast}(t)}\times[0,\infty))}\leq
C,\end{split}$ (5.3)
$\begin{split}\|\rho,D_{x}\rho\|_{C^{3+\alpha,1+\alpha/3}(\partial
B_{R_{\ast}(t)}\times[0,\infty))}\leq C.\end{split}$

The proof of this lemma is similar to that of [14, Lemma 7.1], so we omit
here.

Introduce the space $X$ of functions $\Phi=(f^{1},f^{2},b^{1},b^{2},b^{3})$,
where the norm $\|\Phi\|$ defined by the maximum of the left-hand sides of
(4.7)-(4.16) with $\sqrt{|\varepsilon|}$ dropped, and set

$X_{1}=\\{\Phi\in X:\ \sqrt{|\varepsilon|}\|\Phi\|\leq 1\\},$

then we define a new function $\widetilde{\Phi}\equiv
S\Phi=(\widetilde{f}^{1},\widetilde{f}^{2},\widetilde{b}^{1},\widetilde{b}^{2},\widetilde{b}^{3})$
as follows (cf. (4.18)):

$\begin{split}\widetilde{f}^{1}=A_{\varepsilon}w,\ \ \ \
\widetilde{f}^{2}=A_{\varepsilon}q,\ \ \ \ \
\widetilde{b}^{1}=B_{\varepsilon}^{1},\ \ \ \ \
\widetilde{b}^{2}=B_{\varepsilon}^{2},\ \ \ \ \
\widetilde{b}^{3}=B_{\varepsilon}^{3},\end{split}$

where $w$, $q$, $\rho$ is the solution of (4.1)-(4.5). As in the proof of [14,
Lemma 7.2] which is based on (5.1)-(5.3) and (4.59)-(4.62), we obtain that $S$
maps $X_{1}$ into itself and $S$ is a contraction mapping. Hence, the mapping
$S$ admits a unique fixed point.

By the above argument, we can now state the main result of this paper.

######  Theorem 5.1.

Consider the problem (1.1)-(1.5) with the initial data (1.9) satisfying (2.10)
and let $\mu<\mu_{\ast}$. If $|\varepsilon|$ is sufficiently small, then there
exists a unique global solution of this problem. And there exists a new center
$\varepsilon a^{\ast}(\varepsilon)$, where $a^{\ast}(\varepsilon)$ is a
bounded function of $\varepsilon$, such that for some $\overline{t}>0$,
$\partial\Omega(t)$ behaves like

$\partial B_{R_{\ast}(t)}(\varepsilon
a^{\ast}(\varepsilon))=\big{\\{}x:|x-\varepsilon
a^{\ast}(\varepsilon)|=R_{\ast}(t)\big{\\}},\ \ \ \ \ t>\overline{t}.$

######  Remark 5.1.

The proof of Theorem 5.1 shows that after the translation of the origin
$0\rightarrow\varepsilon a^{\ast}(\varepsilon)$ and the Hanzawa
transformation, the global solution in the new variables $(r,\theta,\varphi)$
is of the form:

$\begin{split}\sigma(r,\theta,\varphi,t)&=\sigma_{\ast}(r,t)+\varepsilon
w(r,\theta,\varphi,t),\\\ p(r,\theta,\varphi,t)&=p_{\ast}(r,t)+\varepsilon
q(r,\theta,\varphi,t),\\\ \partial\Omega(t):\
r&=R_{\ast}(t)+\varepsilon\rho(\theta,\varphi,t),\end{split}$

where $w$, $q$, $\rho$ satisfy (5.1)-(5.3).

##  6\. Acknowledgments

This research is supported by China Postdoctoral Science Foundation (Grant No.
2020M683014). I would like to thank Professor S.B. Cui for his encouragement
and helpful suggestions. I also thank Professor B. Hu for his advices in the
preparation of this paper and constant encouragement.

## References

  * [1] B. Bazaliy and A. Friedman, Global existence and asymptotic stability for an elliptic-parabolic free boundary problem: an application to a model of tumor growth, Indiana Univ. Math. J. 52 (2003) 1265-1304.
  * [2] H.M. Byrne, The importance of intercellular adhesion in the development of carcinomas, IMA J. Math. Appl. Med. Biol. 14 (1997) 305-323.
  * [3] H.M. Byrne and M.A.J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci. 130 (1995) 151-181.
  * [4] H.M. Byrne and M.A.J. Chaplain, Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Math. Comput. Modelling, 24 (1996) 1-17.
  * [5] X.F. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth, SIAM J. Math. Anal. 35 (2003) 974-986.
  * [6] V. Cristini, J. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth, J. Math. Biol. 46 (2003) 191-224.
  * [7] S.B. Cui, Lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth, J. Differential Equations, 246 (2009) 1845-1882.
  * [8] S.B. Cui, Asymptotic stability of the stationary solution for a parabolic-hyperbolic free boundary problem modeling tumor growth, SIAM J. Math. Anal. 45 (2013) 2870-2893.
  * [9] S.B. Cui and J. Escher, Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth, Comm. Partial Differential Equations, 33 (2008) 636-655.
  * [10] S.B. Cui and J. Escher, Well-posedness and stability of a multi-dimensional tumor growth model, Arch. Ration. Mech. Anal. 191 (2009) 173-193.
  * [11] J. Escher and A.-V. Matioc, Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011) 573-596.
  * [12] D. Forger. Biological clocks, Rhythms, and Oscillations: The theory of biological timekeeping, MIT Press, Cambridge, MA, 2017.
  * [13] A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. Ration. Mech. Anal. 180 (2006) 293-330.
  * [14] A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equations, 227 (2006) 598-639.
  * [15] A. Friedman and K.-Y. Lam, Analysis of a free-boundary tumor model with angiogenesis, J. Differential Equations, 259 (2015) 7636-7661.
  * [16] A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol. 38 (1999) 262-284.
  * [17] A. Friedman and F. Reitich, Nonlinear stability of a quasi-static Stefan problem with surface tension: a continuation approach, Ann. Sc. Norm. Super Pisa CI. Sci. 30 (2001) 341-403.
  * [18] A. Friedman and F. Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth, Trans. Amer. Math. Soc. 353 (2001) 1587-1634.
  * [19] H.P. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theoret. Biol. 56 (1976) 229-242.
  * [20] W.R. Hao, J. Hauenstein, B. Hu, T. McCoy and A. Sommese, Computing steady-state solutions for a free boundary problem modeling tumor growth by Stokes equation, J. Comput. Appl. Math. 237 (2013) 326-334.
  * [21] W.R. Hao, J. Hauenstein, B. Hu and A. Sommese, A three-dimensional steady-state tumor system, Appl. Math. Comput. 218 (2011) 2661-2669.
  * [22] W.H. He and R.X. Xing, The existence and linear stability of periodic solution for a free boundary problem modeling tumor growth with a periodic supply of external nutrients, Nonlinear Anal. Real World Appl. 60 (2021) 103290, 16pp.
  * [23] Y.D. Huang, Z.C. Zhang and B. Hu, Bifurcation for a free-boundary tumor model with angiogenesis, Nonlinear Anal. Real World Appl. 35 (2017) 483-502.
  * [24] Y.D. Huang, Z.C. Zhang and B. Hu, Bifurcation from stability to instability for a free boundary tumor model with angiogenesis, Discrete Cont. Dyn. Syst. 39 (2019) 2473-2510.
  * [25] Y.D. Huang, Z.C. Zhang and B. Hu, Linear stability for a free-boundary tumor model with a periodic supply of external nutrients, Math. Methods Appl. Sci. 42 (2019) 1039-1054.
  * [26] Y.D. Huang, Z.C. Zhang and B. Hu, Asymptotic stability for a free boundary tumor model with angiogenesis, J. Differential Equations, 270 (2021) 961-993.
  * [27] F.J. Li and B.C. Liu, Bifurcation for a free boundary problem modeling the growth of tumors with a drug induced nonlinear proliferation rate, J. Differential Equations, 263 (2017) 7627-7646.
  * [28] J. Lowengrub, H. Frieboes, F. Jin, Y. Chuang, X. Li, P. Macklin, S. Wise and V. Cristini, Nonlinear modelling of cancer: bridging the gap between cells and tumours, Nonlinearity, 23 (2010) 1-91.
  * [29] H.J. Song, B. Hu and Z.J. Wang, Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021) 667-691.
  * [30] M. Tindall and C. Please, Modelling the cell cycle and cell movement in multicellular tumour spheroids, Bull. Math. Biol. 69 (2007) 1147-1165.
  * [31] Z.J. Wang, Bifurcation for a free boundary problem modeling tumor growth with inhibitors, Nonlinear Anal. Real World Appl. 19 (2014) 45-53.
  * [32] J.P. Ward and J.R. King, Mathematical modelling of avascular-tumour growth II: Modelling growth saturation, IMA J. Math. Appl. Med. Biol. 16 (1999) 171-211.
  * [33] J.D. Wu and S.B. Cui, Asymptotic stability of stationary solutions of a free boundary problem modeling the growth of tumors with fluid tissues, SIAM J. Math. Anal. 41 (2009) 391-414.
  * [34] J.D. Wu and S.B. Xu, Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020) 2453-2460.
  * [35] J.D. Wu and F.J. Zhou, Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with fluid-like tissue under the action of inhibitors, Trans. Amer. Math. Soc. 365 (2013) 4181-4207.
  * [36] J.D. Wu and F.J. Zhou, Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs-Thomson relation, J. Differential Equations, 262 (2017) 4907-4930.
  * [37] X.E. Zhao and B. Hu, The impact of time delay in a tumor model, Nonlinear Anal. Real World Appl. 51 (2020) 103015, 29pp.
  * [38] X.E. Zhao and B. Hu, Symmetry-breaking bifurcation for a free-boundary tumor model with time delay, J. Differential Equations 269 (2020), 1829–1862.
  * [39] J.Y. Zheng and R.X. Xing, Bifurcation for a free-boundary tumor model with extracellular matrix and matrix degrading enzymes, J. Differential Equations, 268 (2020) 3152-3170.
  * [40] Y.H. Zhuang and S.B. Cui, Analysis of a free boundary problem modeling the growth multicell spheroids with angiogenesis, J. Differential Equations, 265 (2018) 620-644.