# Solutions with single radial interface of the generalized Cahn-Hilliard flow

Chao Liu  School of Mathematics and Information Science, Guangzhou University,
Guangzhou 510006, People’s Republic of China<EMAIL_ADDRESS>and  Jun
Yang  School of Mathematics and Information Science, Guangzhou University,
Guangzhou 510006, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

We consider the generalized parabolic Cahn-Hilliard equation

$u_{t}=-\Delta\left[\Delta
u-W^{\prime}(u)\right]+W^{\prime\prime}(u)\left[\Delta
u-W^{\prime}(u)\right]\qquad\forall\,(t,x)\in\widetilde{{\mathbb{R}}}\times{\mathbb{R}}^{n},$

where $n=2$ or $n\geq 4$, $W(\cdot)$ is the typical double-well potential
function and $\widetilde{\mathbb{R}}$ is given by

$\widetilde{\mathbb{R}}=\left\\{\begin{array}[]{rl}(0,\infty),&\quad\mbox{if
}n=2,\\\ (-\infty,0),&\quad\mbox{if }n\geq 4.\end{array}\right.$

We construct a radial solution $u(t,x)$ possessing an interface. At main order
this solution consists of a traveling copy of the steady state $\omega(|x|)$,
which satisfies $\omega^{\prime\prime}(y)-W^{\prime}(\omega(y))=0$. Its
interface is resemble at main order copy of the sphere of the following form

$|x|=\sqrt[4]{-2(n-3)(n-1)^{2}t},\qquad\forall\,(t,x)\in\widetilde{{\mathbb{R}}}\times{\mathbb{R}}^{n},$

which is a solution to the Willmore flow in Differential Geometry. When $n=1$
or $3$, the result consists trivial solutions.

###### Key words and phrases:

Nonlinear parabolic equation; Cahn-Hilliard equation; Interfaces; Radial
solutions,

Mathematics Subject Classification: 35K30; 35K58

The work is supported by NSFC(No. 12171109) and Basic Research Project of
Guangzhou(No. 202201020094).

###### Contents

  1. 1 Introduction
    1. 1.1 Backgrounds
    2. 1.2 Main results
  2. 2 The setting-up: ansatz, the nonlinear projected problem for perturbation term
    1. 2.1 Some estimates of the basic layer
    2. 2.2 The setting-up of the problem
    3. 2.3 Estimates of the error terms
  3. 3 The Linear Problem
    1. 3.1 A few results of linear parabolic equations with a biharmonic operator
    2. 3.2 A priori estimate of a linear problem involving the operator $L$
      1. 3.2.1 Proof of Lemma 3.5
      2. 3.2.2 Proof of Assertion
    3. 3.3 The projected linear problem involving the operator $L$
    4. 3.4 The solvability of a linear projected problem
  4. 4 Solving the nonlinear projected problem
  5. 5 The reduction procedure: choosing the parameter $h$
    1. 5.1 Deriving the reduced equation involving $h(t)$
      1. 5.1.1
      2. 5.1.2
    2. 5.2 Solving the reduced equation involving $h$

##  1\. Introduction

We are interested in the construction of solutions of the following
generalized parabolic Cahn-Hilliard equation

$u_{t}=-\Delta\left[\Delta
u-W^{\prime}(u)\right]+W^{\prime\prime}(u)\left[\Delta
u-W^{\prime}(u)\right],\qquad\forall\,(t,x)\in\widetilde{\mathbb{R}}\times\mathbb{R}^{n},$
(1.1)

where $n=2$ or $n\geq 4$, $\widetilde{\mathbb{R}}$ is given by

$\widetilde{\mathbb{R}}=\left\\{\begin{array}[]{rl}(0,\infty),&\quad\mbox{if
}n=2,\\\\[5.69054pt] (-\infty,0),&\quad\mbox{if }n\geq 4,\end{array}\right.$

and the functions $W^{\prime}(s)$ and $W^{\prime\prime}(s)$ denote the
derivatives of first and second orders of $W$ respectively. The potential
$W(s)$ is a smooth function, which satisfies the following assumptions

$\left\\{\begin{array}[]{l}W(s)>W(-1)=W(1)\qquad\text{in}\
(-1,1),\\\\[2.84526pt] W(s)=W(-s),\qquad\text{for all}\
s\in\mathbb{R},\\\\[2.84526pt] W^{\prime}(-1)=W^{\prime}(1)=0,\\\\[2.84526pt]
W^{\prime\prime}(-1)=W^{\prime\prime}(1)>0.\end{array}\right.$ (1.2)

The potential $W(u)$ has two non-degenerate local minimum points $u=+1$ and
$u=-1$, which are stable equilibria of (1.1). In particular, the function
$W(s)=\frac{1}{4}(1-s^{2})^{2}$ obviously satisfies the above conditions in
(1.2). Up to a scaling, the function $W(s)=\cos(s)$ also satisfies (1.2).

###  1.1. Backgrounds

Generally speaking, the Cahn-Hilliard equation means the following equation

$\left\\{\begin{array}[]{l}u_{t}=-\Delta\left[\Delta
u-W^{\prime}(u)\right],\qquad\forall\,(t,x)\in(-\infty,+\infty)\times\Omega,\\\\[5.69054pt]
u(0,x)=u_{0}(x),\hskip 56.9055pt\forall\,x\in\Omega,\end{array}\right.$ (1.3)

where $\Omega$ denotes a smooth bounded domain in $\mathbb{R}^{n}$ or the
whole space $\mathbb{R}^{n}$($n\geq 1$), which describes phase separation
processes of binary alloys in [7]. Various kinds of problems of this class
equation have been extensively studied in recent thirty years. By applying a
priori estimate and continuity argument, the existence and asymptotic
behaviors of global smooth solutions of Cauchy problem (1.3) have been proved
in [30] when the initial value $u_{0}$ is close to stable equilibria $\bar{u}$
($W(\bar{u})=0$) in the $L^{\infty}\cap L^{1}(\mathbb{R}^{n})$ space. Using
Fourier transform and estimates on the kernel of a linear parabolic operator,
a uniform $L^{\infty}$ bound estimate for solutions of perturbed Cauchy
problem (1.3) with additional nonlinear terms, was established by Caffarelli
and Muler in [12]. In a bounded domain, the global well-posedness and long-
time behavior of solutions to problem (1.3) with several types of dynamic
boundary conditions were studied in [10, 11, 25, 31, 36].

Based on the works of De Giorgi in [14, 15], numerous authors [3, 35, 38]
studied the diffuse approximation of the Willmore functional:

$\mathcal{W}(S,\Omega)=\frac{1}{2}\int_{\partial S\cap\Omega}\left|H_{\partial
S}(x)\right|^{2}{\mathrm{d}}\mu^{n-1},$ (1.4)

where $\Omega$ is a given open set in $\mathbb{R}^{n}$, the set
$S\subset\mathbb{R}^{n}$ with smooth boundary $\partial S\subset\Omega$, and
$H_{\partial S}(x)$ is the mean curvature of surface $\partial S$ at point
$x\in\partial S$. The approximating functional is defined by

$\mathbb{W}_{\varepsilon}(u)=\left\\{\begin{array}[]{l}\frac{1}{2\varepsilon}\int_{\Omega}\Big{(}\varepsilon\Delta
u-\frac{W^{\prime}(u)}{\varepsilon}\Big{)}^{2}{\mathrm{d}}x,\qquad\text{if}\
u\in L^{1}(\Omega)\cap W^{2,2}(\Omega),\\\\[5.69054pt] +\infty,\hskip
110.96556pt\text{otherwise in}\ L^{1}(\Omega),\end{array}\right.$ (1.5)

where the function $W$ satisfies (1.2). The essential and more challenging
work is to rigourously prove that the approximating functional
$\mathbb{W}_{\varepsilon}(u)$ $\Gamma$-converges to the Willmore functional
$\mathcal{W}(S,\Omega)$ as $\varepsilon$ goes to $0$. Bellettini and Paolini
in [2] proved the $\Gamma$-lim sup inequality for smooth Willmore
hypersurfaces. However, the $\Gamma$-lim inf inequality is more hard to prove.
Up to now, it has been proved in $\mathbb{R}^{n}$ with $n=2,3$ in [35] or
$n=2$ in [34]. This problem is still open in $\mathbb{R}^{n}$ with $n\geq 4$.
The relation of the critical points of (1.4) and (1.5) was exemplified by M.
Rizzi [37] and also A. Malchiodi, R. Mandel, M. Rizzi [33]. On the other hand,
for the parabolic Cahn-Hilliard equation (1.3), the gamma convergence results
have obtained by Le in [32] under suitable conditions. And see [5] for the
case of (1.3) on the one-dimensional torus.

The $L^{2}$ gradient flow of the approximating energy
$\mathbb{W}_{\varepsilon}(u)$ is equivalent to the evolution equation:

$\partial_{t}u_{\varepsilon}=-\Delta\Big{[}\Delta
u_{\varepsilon}-\frac{1}{\varepsilon^{2}}W^{\prime}(u_{\varepsilon})\Big{]}+\frac{1}{\varepsilon^{2}}W^{\prime\prime}(u_{\varepsilon})\Big{[}\Delta
u_{\varepsilon}-\frac{1}{\varepsilon^{2}}W^{\prime}(u_{\varepsilon})\Big{]}\qquad\text{in}\
(-\infty,+\infty)\times\Omega,$ (1.6)

which was introduced in [20] to describe the deformation of a vesicle membrane
under the elastic bending energy, with prescribed bulk volume and surface
area. The well-posedness of the phase field model (1.6) with fixed
$\varepsilon$ has been proved in [8] providing a volume constraint for the
average of $u$, or in [9] with both volume and area constraints. By applying
formal method of matched asymptotic expansions, Loreti and March in [29] (or
Wang in [39]) showed that if $\Gamma(t)\subset\mathbb{R}^{n}$ with $n=2$ or
$3$, is a family of compact closed smooth interfaces and evolves by Willmore
flow, it can be approximated by nodal set of the solution $u_{\varepsilon}$ to
the phase field (1.6) when $\varepsilon$ goes to $0$. The Willmore flow
equation is given by

$V(t)=\Delta_{\Sigma(t)}H-\frac{1}{2}H^{3}+H\|A\|^{2},$ (1.7)

which is the $L^{2}$ gradient flow for (1.4) with $\partial S(t)=\Sigma(t)$,
where $V(t)$ denotes the outer normal velocity at $x\in\Sigma(t)$,
$\Delta_{\Sigma(t)}$ is the Laplace-Beltrami operator on surface $\Sigma(t)$,
$H$ and $A$ are the mean curvature and the second fundamental form of
$\Sigma(t)$ respectively, and $\|A\|^{2}$ is the sum of squared coefficients
of $A$. Fei and Liu [22] proved that for given a solution $\Gamma_{0}(t)$ of
(1.7) in $\mathbb{R}^{n}(n=2,3)$, there exists a solution $u_{\varepsilon}$ of
equation (1.6) with Neumann boundary condition such that its level set
convergence to $\Gamma_{0}(t)$ as the parameter $\varepsilon$ goes to zero.
Moreover, a variety of problems for the Willmore flow (1.7) has been
investigated by Kuwert and Schätzle in several papers (for example see [26,
27, 28]). However, the study of connections between equation (1.6) and the
Willmore flow (1.7) is a challenging work when $n\geq 4$. The present paper is
one of the first attempts in this direction.

###  1.2. Main results

In this paper, we want to find solutions of (1.1) whose values lie at all
times in $[-1,1]$, and approach either $+1$ or $-1$ in the most of the space
$\mathbb{R}^{n}$. This type of solution corresponds to a continuous
realization of a material, in which the two states ($u=-1$ and $u=+1$)
coexist. The main difficult point of the study of this type solution of (1.1),
is to derive qualitative information on the interface region(the walls
separating the two phases). It is easy to find that $u(t,x)$ is a solution of
(1.1) if and only if
$u_{\varepsilon}(t,x):=u(\varepsilon^{-4}t,\varepsilon^{-1}x)$ satisfies
equation (1.6). Basing on the results in [29] or [39] with $n=2$ and $3$, we
know that the nodal set of $u_{\varepsilon}(t,x)$ approximates to the solution
of Willmore flow (1.7). Let us consider the sphere $\Gamma_{n}(t)$ evolving by
the Willmore flow in (1.7). Then we have that the radius $\gamma_{n}(t)$ of
$\Gamma_{n}(t)$ satisfies the equation

$\gamma_{n}^{\prime}(t)=-\frac{1}{2}\Big{(}\frac{n-1}{\gamma_{n}(t)}\Big{)}^{3}+\frac{\big{(}n-1\big{)}^{2}}{\big{(}\gamma_{n}(t)\big{)}^{3}},$
(1.8)

which has a solution

$\gamma_{n}(t):=\sqrt[4]{-2(n-3)(n-1)^{2}t},$ (1.9)

where $t\leq 0$ when $n\geq 3$ and $t\geq 0$ when $n=2$. Due to the self-
similarity, there holds that the sphere $\left|x\right|=\gamma_{n}(t)$ is also
the transition layer (nodal set) for $u(t,x)$, which is a solution of (1.1).

Our aim is to construct solutions to equation (1.1) with one transition layer
closes to the sphere $\left|x\right|=\gamma_{n}(t)$. In which, $t\leq 0$ when
$n\geq 4$, these solutions are called ancient solutions. And $t\geq 0$ for
$n=2$, they are long time solutions. We shall mention that the ancient
radially symmetric solutions for the parabolic Allen-Cahn equation

$u_{t}=\Delta u+u-u^{3}\qquad\text{in}\ (-\infty,0]\times\mathbb{R}^{n},$
(1.10)

have been obtained by del Pino and Gkikas in [18] for $n=1$ and [19] with
$n\geq 2$.

Let us introduce a layer function by considering the following problem of
semilinear elliptic equation

$\omega^{\prime\prime}(y)-W^{\prime}\big{(}\omega(y)\big{)}=0,\quad\omega^{\prime}(y)>0,\quad
y\in\mathbb{R},\quad\omega(0)=0\quad\text{and}\quad\lim\limits_{y\rightarrow\pm\infty}\omega(y)=\pm
1,$ (1.11)

which has a unique smooth solution $\omega(y)$ obtained in [1], where $W$
satisfies the conditions in (1.2). In particular if we choose
$W^{\prime}(s)=s^{3}-s$, then (1.11) is the elliptic Allen-Cahn equation, and
its solution is written as

$\omega(y)=\tanh(\frac{y}{\sqrt{2}}).$

For general functions $W$ satisfying (1.2), the solution $\omega$ of (1.11)
has no explicit expression. In [1], the inverse function of $\omega$ is given
by

$\lambda(s):=\int^{s}_{0}\frac{1}{\sqrt{2(W(\tau)-W(1))}}\mathrm{d}\tau,\qquad
s\in(-1,1).$ (1.12)

Thanks to (1.2), the function $\lambda(s)$ is well-defined. Taking $\omega$ as
a basic layer (the name layer is motivated by the fact that $\omega$
approaches the limits $1$ and $-1$ at $\pm\infty$), we will construct a
solution of (1.1) with a ‘transition layer’ which is symmetric about the
sphere $\left|x\right|=\gamma_{n}(t)$ in (1.9).

More precisely, we want to find a solution of equation (1.1), which has the
following asymptotical behavior

$u(t,x)\approx\omega\big{(}\left|x\right|-\rho(t)\big{)},$ (1.13)

where $\omega$ is given by (1.11). The function $\rho(t)$ satisfies

$\rho(t)=\gamma_{n}(t)+h(t),\qquad\text{with}\
h(t)=O\left(\frac{1}{\log\left|t\right|}\right),$ (1.14)

where $h(t)$ is an $C^{1}$ function with respect of $t$ and the function
$\gamma_{n}(t)$ is defined by (1.9). In fact, $\rho(t)$ can be chosen by
solving the following ODE

$\rho^{\prime}(t)+\frac{(n-3)(n-1)^{2}}{2\rho^{3}(t)}=Q\big{(}\rho(t),\rho^{\prime}(t)\big{)},\qquad\text{with}\
Q\big{(}\rho(t),\rho^{\prime}(t)\big{)}=O\left(\frac{1}{\left|t\right|\big{[}\log\left|t\right|\big{]}^{2(p-1)}}\right),$

for all $t\leq 0$ when $n\geq 4$ or $t\geq 0$ when $n=2$, where $p\in(n,n+1]$,
see Section 5.

Our main results can be stated as follows.

######  Theorem 1.1.

When $n=2$, there exists a radial solution $u(t,\left|x\right|)$ of equation
$(\ref{eq1.1})$ with $t\geq 0$, which has the form
$u(t,x)\approx\omega\big{(}\left|x\right|-\rho(t)\big{)}+\phi(t,\left|x\right|)$,
with

$\rho(t)=\sqrt[4]{2t}+h(t).$ (1.15)

Moreover, $h(t)$ is a $C^{1}$ function with the decay of order
$O(\frac{1}{\log\left|t\right|})$ as $t\rightarrow+\infty$ and
$\lim\limits_{t\rightarrow+\infty}\phi(t,\left|x\right|)=0$ uniformly in
$x\in\mathbb{R}^{2}$. ∎

######  Theorem 1.2.

When $n\geq 4$, there exists an ancient solution $u(t,x)$ of equation with
$t\leq 0$, which has the form
$u(t,x)\approx\omega\big{(}\left|x\right|-\rho(t)\big{)}+\phi(t,\left|x\right|)$
with

$\rho(t)=\sqrt[4]{-2(n-3)(n-1)^{2}t}+h(t).$ (1.16)

Moreover, $h(t)$ is a $C^{1}$ function with the decay of order
$O(\frac{1}{\log\left|t\right|})$ as $t\rightarrow-\infty$ and
$\lim\limits_{t\rightarrow-\infty}\phi(t,\left|x\right|)=0$ uniformly in
$x\in\mathbb{R}^{n}$. ∎

######  Remark 1.3.

In this paper, we mainly prove that the results of Theorem 1.2 hold. For
Theorem 1.1, its proof is similar and we just now notice that $t>0$ in this
case $n=2$. When $n=1$ and $3$, it is easily to check that problem (1.1) has a
radical solution of the form

$u(t,\left|x\right|)=\omega(\left|x\right|-c),$

with any constant $c\in\mathbb{R}$, where $\omega(y)$ is the solution of
(1.11). ∎

######  Remark 1.4.

We prove Theorem 1.2 by using Lyapunov-Schmidt reduction method. This method
has been widely applied to solving the existence problem of solutions to
various kinds of equations [6, 13, 16, 17, 19]. Comparing to the previous
works, we are facing three difficulties during the process of dealing with
problem (1.1) as the following:

(1). Firstly, the main term $\omega\big{(}\left|x\right|-\rho(t)\big{)}$ in
(1.14) is not a good approximate solution since it does not satisfy the
boundary conditions in (2.3) and does not provide enough decay of the error.
To modify it, we introduce a cut-off function around the origin and a
correction function, see (2.14) in Section 2. This main idea comes from [37].

(2). Secondly, equation (1.1) is a parabolic equation of fourth order, which
does not satisfy the Maximum Principle since the heat kernel of the biharmonic
parabolic operator is sign-changing, see [21]. To overcome it, we employ the
blow-up technique together with the representation of parabolic kernel in
Section 3.

(3). Lastly, the correlative heat kernel of the linearization operator of the
equation (1.1) is more intricate, which leads to hard task to get suitable a
priori estimates of linearized problem of equation (1.1). To solve it, we
modify the estimate of the error term $E(t,r)$ in (2.18) with an polynomial
decay in Section 2, and perform more delicate calculations in Section 3. ∎

The paper is organized as follows.

$\clubsuit$ In the first part of Section 2, we deduce some estimates of decay
for $\omega$ in (1.11) and its derivatives. After that, an approximate
solution, say $z(t,\left|x\right|)$ with a parameter $\rho(t)$ in (2.14) and
(2.7), will be defined. By the perturbation of
$z(t,\left|x\right|)+\phi(t,\left|x\right|)$, the setting-up of a projected
form of (1.1) will be derived, see (2.16)-(2.17) with the Lagrange multiplier
$c(t)$. With the introduction of a suitable norm, the estimates of the error
will be provided in the last part of Section 2.

$\clubsuit$ Section 3 is devoting to the collection of some results of linear
parabolic equations with a biharmonic operator and then obtain the solvability
of a linear projected problem in (3.60).

$\clubsuit$ In Section 4, we solve the nonlinear problem (2.16)-(2.17) by an
argument of the fixed-point theorem.

$\clubsuit$ In Section 5, in order to obtain a radial solution to (1.1) we
choose a suitable parameter $h(t)$ (in other words, adjusting the parameter
$\rho$ given by (2.7)) such that $c(t)$ equals to zero, in problem
(2.16)-(2.17).

##  2\. The setting-up: ansatz, the nonlinear projected problem for
perturbation term

###  2.1. Some estimates of the basic layer

Before proving the main theorems, we first derive the decay estimates of the
basic layer $\omega$, which is the solution to equation (1.11).

######  Lemma 2.1.

Let $\omega$ be the solution of problem (1.11), then we have

$\displaystyle\left\\{\begin{array}[]{l}\lim\limits_{y\rightarrow+\infty}\frac{\omega(y)-1+\beta
e^{-\sqrt{W^{\prime\prime}(1)}y}}{\beta^{2}e^{-2\sqrt{W^{\prime\prime}(1)}y}}=\frac{W^{(3)}(1)}{6W^{\prime\prime}(1)};\\\\[8.53581pt]
\lim\limits_{y\rightarrow-\infty}\frac{\omega(y)+1-\beta
e^{\sqrt{W^{\prime\prime}(-1)}y}}{\beta^{2}e^{2\sqrt{W^{\prime\prime}(-1)}y}}=-\frac{W^{(3)}(-1)}{6W^{\prime\prime}(-1)};\\\\[8.53581pt]
\lim\limits_{y\rightarrow+\infty}\frac{\omega^{\prime}(y)}{e^{-\sqrt{W^{\prime\prime}(1)}y}}=\beta\sqrt{W^{\prime\prime}(1)}\quad\text{and}\quad\lim\limits_{y\rightarrow-\infty}\frac{\omega^{\prime}(y)}{e^{\sqrt{W^{\prime\prime}(-1)}y}}=\beta\sqrt{W^{\prime\prime}(-1)},\end{array}\right.$
(2.1)

where $W^{\prime\prime}(s)$ and $W^{(3)}(s)$ denote the derivatives of second
and third orders of the function $W(s)$ respectively, and

$\beta:=\exp\Bigg{\\{}\sqrt{W^{\prime\prime}(1)}\int^{1}_{0}\Bigg{[}\frac{1}{\sqrt{2\big{(}W(s)-W(1)\big{)}}}-\frac{1}{\sqrt{W^{\prime\prime}(1)}\,(1-s)}\Bigg{]}\mathrm{d}s\Bigg{\\}}.$
(2.2)

###### Proof.

Since the function $W$ satisfies the conditions in $(\ref{eq1.4})$, especially
$W^{\prime\prime}(1)=W^{\prime}(-1)>0$, the above limits in $(\ref{eq2.0})$
are well-defined and $\beta$ is finite. Recall that the inverse function of
$\omega$ is the function $\lambda(s)$ given by (1.12). Using L’Hospital’s
rule, Taylor’s formula, the function $W\in C^{3,1}_{\text{loc}}(\mathbb{R})$
satisfies (1.2) and (1.12), we find that

$\lim_{s\rightarrow
1^{-}}\frac{1}{1-s}\Bigg{\\{}\lambda(s)+\frac{\ln(1-s)}{\sqrt{W^{\prime\prime}(1)}}-\int^{1}_{0}\frac{1}{\sqrt{2(W(s)-W(1))}}-\frac{1}{\sqrt{W^{\prime\prime}(1)}(1-s)}\mathrm{d}s\Bigg{\\}}=-\frac{W^{(3)}(1)}{6(W^{\prime\prime}(1))^{3/2}}$

and

$\lim_{s\rightarrow(-1)^{+}}\frac{1}{1+s}\Bigg{\\{}\lambda(s)-\frac{\ln(1+s)}{\sqrt{W^{\prime\prime}(-1)}}-\int^{0}_{-1}\frac{1}{\sqrt{2(W(s)-W(1))}}-\frac{1}{\sqrt{W^{\prime\prime}(1)}(1+s)}\mathrm{d}s\Bigg{\\}}=\frac{W^{(3)}(-1)}{6(W^{\prime\prime}(-1))^{3/2}}.$

According to the fact that if $x=\lambda(s)$, then $s=\omega(y)$, using
Taylor’s formula again and the evenness of $W$ in $(-1,1)$, we easily deduce
that the first two limits in (2.12) hold. So we have

$\lim\limits_{y\rightarrow+\infty}\frac{1-\omega(y)}{e^{-\sqrt{W^{\prime\prime}(1)}y}}=\beta\
\ \text{and}\ \
\lim\limits_{y\rightarrow-\infty}\frac{1+\omega(y)}{e^{\sqrt{-W^{\prime\prime}(1)}y}}=\beta.$

Using L’Hospital’s rule again, the condition (1.2) and the equation (1.11), we
have

$\displaystyle\lim\limits_{y\rightarrow+\infty}\frac{\omega^{\prime}(y)}{e^{-\sqrt{W^{\prime\prime}(1)}y}}=\lim\limits_{y\rightarrow+\infty}\Bigg{\\{}\frac{\omega(y)-1}{-\sqrt{W^{\prime\prime}(1)}e^{-\sqrt{W^{\prime\prime}(1)}y}}\times\frac{W^{\prime}(\omega)-W^{\prime}(1)}{\omega(y)-1}\Bigg{\\}}=\beta\sqrt{W^{\prime\prime}(1)}$

and

$\lim\limits_{y\rightarrow-\infty}\frac{\omega^{\prime}(y)}{e^{\sqrt{W^{\prime\prime}(-1)}y}}=\lim\limits_{y\rightarrow-\infty}\Bigg{\\{}\frac{\omega(y)+1}{\sqrt{W^{\prime\prime}(-1)}e^{\sqrt{W^{\prime\prime}(-1)}y}}\times\frac{W^{\prime}(\omega)-W^{\prime}(-1)}{\omega(y)+1}\Bigg{\\}}=\beta\sqrt{W^{\prime\prime}(-1)}.$

∎

Next we consider the kernel of a fourth order linear operator. The main result
is stated as the following.

######  Lemma 2.2.

Let $W(s)$ be a smooth function satisfying the conditions in (1.2). Then we
have that any solution of the homogeneous problem

$\big{[}\,\partial_{xx}-W^{\prime}(\omega(x))\,\big{]}^{2}\varphi=0,\quad\left|\varphi\right|\leq
1,\quad\left|\varphi_{xx}\right|\leq 1\qquad\text{in}\ \mathbb{R},$

has the form

$\varphi(x)=c\omega^{\prime}(x),$

with some constant $c\in\mathbb{R}$.

For the proof of this lemma, see Lemma 3.6 which is a more general result. ∎

###  2.2. The setting-up of the problem

We will prove that Theorem 1.2 holds. Thus, we always assume that $n\geq 4$
and $t<0$ in the rest of the present paper.

Let $\widetilde{u}(t,\left|x\right|)$ be a solution of (1.1), by a translation
$u(t,\left|x\right|)=\widetilde{u}(t-T,\left|x\right|)$ with some abuse of
notation, then we have that $u(t,r)$ satisfies the following problem

$\left\\{\begin{array}[]{l}u_{t}=-u_{rrrr}-\frac{2(n-1)}{r}u_{rrr}+\Big{(}2W^{\prime\prime}(u)-\frac{(n-1)(n-3)}{r^{2}}\Big{)}u_{rr}+\Big{(}\frac{2(n-1)W^{\prime\prime}(u)}{r}+\frac{(n-1)(n-3)}{r^{3}}\Big{)}u_{r}\\\\[8.53581pt]
\hskip
22.76228pt+W^{\prime\prime\prime}(u)u^{2}_{r}-W^{\prime}(u)W^{\prime\prime}(u),\qquad\
\ \forall\,(t,r)\in(-\infty,-T]\times(0,+\infty),\\\\[8.53581pt]
u_{rrr}(t,0)=u_{r}(t,0)=0,\hskip 66.86414pt\text{for all}\
t\in(-\infty,-T],\end{array}\right.$ (2.3)

where $r=\left|x\right|$ and $T$ is a large positive number whose value can be
adjusted at different steps. For convenience, we denote the right hand side of
the first equation in (2.3) by $F(u)$, that is

$\displaystyle F(u):=$ $\displaystyle-
u_{rrrr}-\frac{2(n-1)}{r}u_{rrr}+\Big{(}2W^{\prime\prime}(u)-\frac{(n-1)(n-3)}{r^{2}}\Big{)}u_{rr}$
(2.4)
$\displaystyle+\Big{(}\frac{2(n-1)W^{\prime\prime}(u)}{r}+\frac{(n-1)(n-3)}{r^{3}}\Big{)}u_{r}+W^{\prime\prime\prime}(u)u^{2}_{r}-W^{\prime}(u)W^{\prime\prime}(u).$

Our purpose is to find a solution of (2.3) with the property

$u(t,r)\approx\omega\big{(}r-\rho(t)\big{)},$

where $\omega(y)$ is the solution of problem (1.11). Firstly, we notice that
$\omega\big{(}r-\rho(t)\big{)}$ does not satisfies the boundary conditions in
(2.3). A smooth cut-off function $\chi(r)$ can be defined in the form

$\chi(r)=0,\qquad\text{for}\
r\leq\frac{\delta_{0}}{2}\qquad\text{and}\qquad\chi(r)=1,\qquad\text{for}\
r\geq\delta_{0},$ (2.5)

for some small fixed positive number $\delta_{0}$. We define the first
approximate solution of (2.3) as the following

$\widehat{\omega}(t,r)=\omega\big{(}r-\rho(t)\big{)}\chi(r)+\chi(r)-1.$ (2.6)

Here we assume that the function $\rho(t)$ has the from

$\rho(t)=\gamma_{n}(t)+h(t),$ (2.7)

where the function $h(t)=O((\log|t|)^{-1})$ as $t\rightarrow-\infty$ and the
function $\gamma_{n}(t)$ is defined in (1.8)-(1.9), i.e.

$\gamma_{n}(t):=\sqrt[4]{-2(n-1)^{2}(n-3)t},\qquad t<0,$ (2.8)

which is a radial solution of Willmore flow equation (1.7) with $n\geq 4$.
More precisely, we assume that $h(t)$ satisfies the following constraint

$\sup_{t\leq-1}\left|h(t)\right|+\sup_{t\leq-1}\left\\{\frac{\left|t\right|}{\log\left|t\right|}\left|h^{\prime}(t)\right|\right\\}\leq
1.$ (2.9)

Secondly, by using (1.11), we have that

$\displaystyle-\partial_{t}\widehat{\omega}(t,r)+F(\widehat{\omega}(t,r))$
$\displaystyle=\rho^{\prime}(t)\omega^{\prime}\big{(}r-\rho(t)\big{)}-\frac{(n-3)(n-1)}{r^{2}}\omega^{\prime\prime}\big{(}r-\rho(t)\big{)}$
(2.10)
$\displaystyle\quad+\frac{(n-1)(n-3)}{r^{3}}\omega^{\prime}\big{(}r-\rho(t)\big{)},\qquad\text{for}\
r>\delta_{0},$

where the operator $F(u)$ is defined by (2.4). By (2.8), we find that the
second term in the right hand side of equality (2.10), that is

$-\frac{(n-1)(n-3)}{r^{2}}\omega^{\prime\prime}\big{(}r-\rho(t)\big{)},$

has a slow decay of order $O\left(\left|t\right|^{-\frac{1}{2}}\right)$ as $t$
goes to negative infinity. However, it is not enough to solve equation (2.10)
since this term is much bigger than other terms in (2.10). To cancel it and
improve the approximate solution, inspired by [37], we define a correction
function

$\widetilde{\omega}(y):=-\omega^{\prime}(y)\int_{0}^{y}\Bigg{[}(\omega^{\prime}(\hat{y}))^{-2}\int_{-\infty}^{\hat{y}}\frac{s(\omega^{\prime}(s))^{2}}{2}\mathrm{d}s\Bigg{]}{\mathrm{d}}\hat{y},$
(2.11)

Then we have that

$\displaystyle
L^{*}(\widetilde{\omega}):=\big{[}-\partial_{yy}+W^{\prime\prime}(\omega(y))\big{]}\widetilde{\omega}(y)=\frac{1}{2}y\omega^{\prime}(y),\qquad(L^{*})^{2}\big{[}\widetilde{\omega}(y)\big{]}=-\omega^{\prime\prime}(y),\qquad\forall\,y\in\mathbb{R},$
(2.12)

and $\widetilde{\omega}(y)$ is an odd function with exponential decay such
that

$\int_{\mathbb{R}}\omega^{\prime}(y)\widetilde{\omega}(y){\mathrm{d}}y=0\quad\text{and}\quad\left|\widetilde{\omega}(y)\right|\leq
Ce^{-\frac{3\alpha}{4}\left|y\right|},\qquad\text{for}\ y\in\mathbb{R}.$
(2.13)

At last, we define an approximate solution of problem (2.3) as the following

$z(t,r):=\omega\big{(}r-\rho(t)\big{)}\chi(r)+\chi(r)-1+\frac{(n-1)(n-3)}{r^{2}}\widetilde{\omega}\big{(}r-\rho(t)\big{)}\chi(r),$
(2.14)

where the cut-off function $\chi(r)$ and the function $\rho(t)$ are given by
(2.5) and (2.7).

We will look for a solution of equation (2.3) of the form

$u(t,r)=z(t,r)+\phi(t,r),$ (2.15)

where $\phi$ is a small perturbation term. This can be done by using the
Lyapunov-Schmidt reduction method in two steps.

(1). The first step (see Sections 3-4) is solving the following projected
version of problem (2.3) in terms of $\phi(t,r)$:

$\displaystyle\phi_{t}=L[\phi]+E(t,r)+N(\phi)-c(t)\partial_{r}\widehat{\omega}(t,r)\qquad\text{in}\
(-\infty,-T]\times(0,\infty),$ (2.16)

and

$\int_{0}^{\infty}\phi(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r=0,\qquad\text{for
all}\ t<-T,$ (2.17)

where the function $\widehat{\omega}(t,r)$ is defined by (2.6), the error term
$E(t,r)$ and nonlinear term $N(\phi)$ are defined respectively by

$E(t,r):=F\big{(}z(t,r)\big{)}-\frac{\partial z(t,r)}{\partial t}$ (2.18)

and

$N(\phi):=F\big{(}z(t,r)+\phi(t,r)\big{)}-F\big{(}z(t,r)\big{)}-F^{\prime}\big{(}z(t,r)\big{)}[\phi].$
(2.19)

In the above, $F(u)$ is defined by (2.4) and the linear operator
$L[\phi]:=F^{\prime}\big{(}z(t,r)\big{)}[\phi]$ is defined as follows

$\displaystyle F^{\prime}\big{(}z(t,r)\big{)}[\phi]:=$
$\displaystyle-\phi_{rrrr}-\frac{2(n-1)}{r}\phi_{rrr}+\Bigg{[}2W^{\prime\prime}\big{(}z(t,r)\big{)}-\frac{(n-1)(n-3)}{r^{2}}\Bigg{]}\phi_{rr}$
(2.20)
$\displaystyle-\big{(}W^{\prime\prime}\big{(}z(t,r)\big{)}\big{)}^{2}\phi+\Bigg{[}\frac{2(n-1)W^{\prime\prime}\big{(}z(t,r)\big{)}}{r}-\frac{(3-n)(n-1)}{r^{3}}\Bigg{]}\phi_{r}$
$\displaystyle+2W^{\prime\prime\prime}\big{(}z(t,r)\big{)}\phi_{r}z_{r}-W^{\prime\prime\prime}\big{(}z(t,r)\big{)}W^{\prime}\big{(}z(t,r)\big{)}\phi+W^{(4)}\big{(}z(t,r)\big{)}\left|z_{r}\right|^{2}\phi$
$\displaystyle+2W^{\prime\prime\prime}\big{(}z(t,r)\big{)}z_{rr}\phi+2\frac{n-1}{r}W^{\prime\prime\prime}\big{(}z(t,r)\big{)}z_{r}\phi.$

(2). The second step is to choose the function $c(t)$ in such a way that
$\phi$ satisfies the orthogonality condition (2.17), namely the following
equality holds:

$\displaystyle
c(t)\int_{0}^{\infty}\partial_{r}\widehat{\omega}(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle=\int_{0}^{\infty}\Big{[}\omega^{\prime\prime\prime}\big{(}r-\rho(t)\big{)}+\frac{n-1}{r}\omega^{\prime\prime}\big{(}r-\rho(t)\big{)}-W^{\prime\prime}\big{(}z(t,r)\big{)}\omega^{\prime}\big{(}r-\rho(t)\big{)}\Big{]}$
$\displaystyle\qquad\qquad\times\left(-\phi_{rr}-\frac{n-1}{r}\phi_{r}+W^{\prime\prime}\big{(}z(t,r)\big{)}\phi\right)r^{n-1}{\mathrm{d}}r$
$\displaystyle\quad+\int_{0}^{\infty}\left[\partial_{rr}z(t,r)+\frac{n-1}{r}\partial_{r}z(t,r)-W^{\prime}\big{(}z(t,r)\big{)}\right]\phi\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle\quad+\int_{0}^{\infty}\phi(t,r)\partial_{t}\big{[}\omega^{\prime}\big{(}r-\rho(t)\big{)}\big{]}r^{n-1}{\mathrm{d}}r\
+\
\int_{0}^{\infty}\big{(}E(t,r)+N(\phi)\big{)}\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r,$
(2.21)

for all $t<-T$. Later on, in Section 5, we will choose $h(t)$ such that
$c(t)=0$. This means that the function $u$ in (2.15) will exactly solve (2.3).

###  2.3. Estimates of the error terms

We will establish some estimates for the error term $E(t,r)$ in (2.18). By
Taylor’s formula, the definitions in (2.4) and (2.14), we have that

$\displaystyle E(t,r)=$ $\displaystyle F\big{(}z(t,r)\big{)}-\frac{\partial
z(t,r)}{\partial t}$ $\displaystyle=$ $\displaystyle
F\big{(}\widehat{\omega}(t,r)+\widetilde{z}(t,r)\big{)}-\frac{\partial\big{[}\widehat{\omega}(t,r)+\widetilde{z}(t,r)\big{]}}{\partial
t}$ $\displaystyle=$ $\displaystyle
F\big{(}\widehat{\omega}(t,r)\big{)}+F^{\prime}\big{(}\widehat{\omega}(t,r)\big{)}\big{[}\widetilde{z}(t,r)\big{]}+F^{\prime\prime}\big{(}\widehat{\omega}(t,r)+\theta\widetilde{z}(t,r)\big{)}\big{[}\widetilde{z}(t,r),\widetilde{z}(t,r)\big{]}$
$\displaystyle-\frac{\partial\widetilde{z}(t,r)}{\partial
t}-\frac{\partial\widehat{\omega}(t,r)}{\partial t}$ $\displaystyle:=$
$\displaystyle E_{1}(t,r)+E_{2}(t,r),$

where $\theta\in(0,1)$. In the above, the operators $F(u)$ and
$F^{\prime}(u)[v]$ are given by (2.4) and (2.20) respectively, and the
function $\widehat{\omega}(t,r)$ is given by (2.6) and $\widetilde{z}(t,r)$ is
defined by

$\displaystyle\widetilde{z}(t,r):=\frac{(n-1)(n-3)}{r^{2}}\widetilde{\omega}\big{(}r-\rho(t)\big{)}\chi(r),$
(2.22)

with $\rho(t)$ and $\chi(r)$ given by (2.7) and (2.5) respectively. The
operator $F^{\prime\prime}(u)[v_{1},v_{2}]$ is defined as the following

$\displaystyle F^{\prime\prime}(u)[v_{1},v_{2}]:=$
$\displaystyle\Delta\big{[}W^{\prime\prime\prime}(u)v_{1}v_{2}\big{]}\ -\
\left\\{W^{\prime\prime\prime}(u)W^{\prime\prime}(u)\ +\
W^{(4)}(u)\big{[}-\Delta u+W^{\prime}(u)\big{]}\right\\}v_{1}v_{2}$ (2.23)
$\displaystyle\ +\ W^{\prime\prime\prime}(u)\Big{\\{}\big{[}\Delta
v_{1}-W^{\prime\prime}(u)v_{1}\big{]}v_{2}\ +\ \big{[}\Delta
v_{2}-W^{\prime\prime}(u)v_{2}\big{]}v_{1}\Big{\\}}.$

The terms $E_{1}(t,r)$ and $E_{2}(t,r)$ have the following explicit forms

$E_{1}(t,r):=F\big{(}\widehat{\omega}(t,r)\big{)}-\frac{\partial\widehat{\omega}(t,r)}{\partial
t}+\frac{(n-1)(n-3)}{r^{2}}\partial_{rr}\widehat{\omega}(t,r)+\frac{(n-1)(n-3)^{2}}{2r^{3}}\partial_{r}\widehat{\omega}(t,r),$
(2.24)

and

$\displaystyle E_{2}(t,r):=$
$\displaystyle-\left[\frac{(n-1)(n-3)}{r^{2}}\partial_{rr}\widehat{\omega}(t,r)+\frac{(n-1)(n-3)^{2}}{2r^{3}}\partial_{r}\widehat{\omega}(t,r)\right]+F^{\prime}\big{(}\widehat{\omega}(t,r)\big{)}\big{[}\widetilde{z}(t,r)\big{]}$
$\displaystyle+F^{\prime\prime}\big{(}\widehat{\omega}(t,r)+\theta\widetilde{z}(t,r)\big{)}\big{[}\widetilde{z}(t,r),\widetilde{z}(t,r)\big{]}-\frac{\partial\widetilde{z}(t,r)}{\partial
t}.$ (2.25)

The main result is given by the following lemma.

######  Lemma 2.3.

Let $\alpha:=\sqrt{W^{\prime\prime}(1)}$, $p\in(n,n+1]$ and $T>1$, we set

$\Phi(t,r):=\left\\{\begin{array}[]{l}\frac{\log\left|t\right|}{\left|t\right|^{\frac{1}{2}}}\frac{1}{\big{(}1+\left|r-\gamma_{n}(t)-\frac{1}{4\alpha}\log\left|t\right|\right|\big{)}^{p}},\qquad\text{if}\
r\in\left[\delta_{0},+\infty\right);\\\\[11.38109pt]
\frac{\log\left|t\right|}{\left|t\right|^{\frac{1}{2}}}\overline{\chi}_{\left\\{\frac{\delta_{0}}{2}\leq
r<\delta_{0}\right\\}},\hskip 66.29497pt\text{if}\
r\in\left(0,\delta_{0}\right);\end{array}\right.$ (2.26)

where the function $\gamma_{n}(t)$ is defined by (2.8) and $\delta_{0}$ is a
small positive number given in (2.5). Here $\overline{\chi}_{A}$ is the
characteristic function of the set $A$. Then there exists constant $C>0$ which
depends only on $\delta_{0}$, $\alpha$, and $n$, such that

$|E(t,r)|\leq C\frac{\Phi(t,r)}{\log\left|t\right|},$

for all $(t,r)\in(-\infty,-T]\times(0,+\infty)$.

###### Proof.

We first estimate the term $E_{1}(t,r)$ in (2.24). By the definitions of
$F(u)$ and $\widehat{\omega}(t,r)$ in (2.4) and (2.6), lemma 2.1, we have that

$\displaystyle E_{1}(t,r)=$
$\displaystyle\left[\rho^{\prime}(t)+\frac{(n-1)^{2}(n-3)}{2r^{3}}\right]\omega^{\prime}\big{(}r-\rho(t)\big{)}\overline{\chi}_{\left\\{r\geq\delta_{0}\right\\}}+O\left(\frac{1}{[\gamma_{n}(t)]^{2}}\right)\overline{\chi}_{\left\\{\frac{\delta_{0}}{2}<r<\delta_{0}\right\\}},$

where $\overline{\chi}_{A}$ is the characteristic function of the set $A$.
Furthermore, using the assumption of $\rho(t)$ in (2.7)-(2.9), the definition
of $\gamma_{n}(t)$ in (2.8) and Lemma 2.1, we have that

$\displaystyle\left|E_{1}(t,r)\right|\leq$ $\displaystyle
C\Bigg{[}\frac{1}{\left|t\right|^{\frac{3}{4}}}\overline{\chi}_{\left\\{r\geq\delta_{0}\right\\}}+\frac{1}{r^{3}}\overline{\chi}_{\left\\{r\geq\frac{\gamma_{n}(t)}{2}\right\\}}+\frac{1}{r^{3}}\overline{\chi}_{\left\\{\frac{\gamma_{n}(t)}{2}\geq
r\geq\delta_{0}\right\\}}\Bigg{]}e^{-\alpha\left|r-\gamma_{n}(t)\right|}\,+\,\frac{C}{\left|t\right|^{\frac{1}{2}}}\overline{\chi}_{\left\\{\frac{\delta_{0}}{2}<r<\delta_{0}\right\\}}$
$\displaystyle\leq$ $\displaystyle
C\frac{e^{-\frac{3\alpha}{4}\left|r-\gamma_{n}(t)-\frac{1}{3\alpha}\log\left|t\right|\right|}}{\left|t\right|^{\frac{1}{2}}}\overline{\chi}_{\left\\{r\geq\delta_{0}\right\\}}+\frac{C}{r^{3}}e^{-\frac{3\alpha}{4}\left|r-\gamma_{n}(t)\right|}\overline{\chi}_{\left\\{\frac{\gamma_{n}(t)}{2}\geq
r\geq\delta_{0}\right\\}}+\frac{C}{\left|t\right|^{\frac{1}{2}}}\overline{\chi}_{\left\\{\frac{\delta_{0}}{2}<r<\delta_{0}\right\\}}$
$\displaystyle\leq$ $\displaystyle
C\frac{\Phi(t,r)}{\log\left|t\right|},\qquad\text{ for all}\
(t,r)\in(-\infty,-T]\times(0,+\infty),$ (2.27)

where $C$ is a positive constant only depending on $\delta_{0}$, $\alpha$ and
$n$. Here we used the fact that there exits a positive constant $C>0$ only
depending on $\delta_{0}$, $\alpha$ and $n$ such that

$\frac{1}{r^{3}}e^{-\frac{3\alpha}{4}\left|r-\gamma_{n}(t)\right|}\overline{\chi}_{\left\\{\frac{\gamma_{n}(t)}{2}\geq
r\geq\delta_{0}\right\\}}\leq
Ce^{-\frac{\alpha}{4}\gamma_{n}(t)}\quad\text{and}\quad
e^{-\frac{3\alpha}{4}\left|x\right|}\leq\frac{C}{\big{(}1+\left|x\right|\big{)}^{p}},$
(2.28)

for all $x\in\mathbb{R}$ and $p\in(n,n+1]$, where $C>0$ does not depend on $x$
and $t$.

Next we consider the term $E_{2}(t,r)$ in (2.25). Using the definitions of
linear operators $F^{\prime}(u)[v]$ and $F^{\prime\prime}(u)[v,v]$ in (2.20)
and (2.23), Lemma 2.1 and estimate in (2.13), we derive that

$\displaystyle E_{2}(t,r)=$
$\displaystyle\Bigg{\\{}F^{\prime}\big{(}\omega\big{(}r-\rho(t)\big{)}\big{)}\big{[}\widetilde{z}(t,r)\big{]}-\frac{(n-1)(n-3)}{r^{2}}\omega^{\prime\prime}\big{(}r-\rho(t)\big{)}+\frac{(n-1)(n-3)^{2}}{2r^{3}}\omega^{\prime}\big{(}r-\rho(t)\big{)}$
$\displaystyle+F^{\prime\prime}\big{(}\omega\big{(}r-\rho(t)\big{)}+\theta\widetilde{z}(t,r)\big{)}\big{[}\widetilde{z}(t,r),\widetilde{z}(t,r)\big{]}+\rho^{\prime}(t)\omega^{\prime}\big{(}r-\rho(t)\big{)}\frac{(n-1)(n-3)}{r^{2}}\Bigg{\\}}\overline{\chi}_{\left\\{r\geq\delta_{0}\right\\}}$
$\displaystyle+O\left(\frac{1}{[\gamma_{n}(t)]^{2}}\right)\overline{\chi}_{\left\\{\frac{\delta_{0}}{2}<r<\delta_{0}\right\\}}$
$\displaystyle=$
$\displaystyle\Bigg{\\{}\left[-\Big{(}\partial_{rr}-W^{\prime\prime}\big{(}\omega\big{(}r-\rho(t)\big{)}\big{)}\Big{)}^{2}\widetilde{\omega}\big{(}r-\rho(t)\big{)}-\omega^{\prime\prime}\big{(}r-\rho(t)\big{)}\right]\frac{(n-1)(n-3)}{r^{2}}$
$\displaystyle+O\left(\frac{1}{r^{3}}+\frac{\rho^{\prime}(t)}{r^{2}}\right)e^{-\frac{3\alpha}{4}\left|r-\rho(t)\right|}\Bigg{\\}}\overline{\chi}_{\left\\{r\geq\delta_{0}\right\\}}\
+\
O\left(\frac{1}{[\gamma_{n}(t)]^{2}}\right)\overline{\chi}_{\left\\{\frac{\delta_{0}}{2}<r<\delta_{0}\right\\}}$
$\displaystyle=$ $\displaystyle
O\left(\frac{1}{r^{3}}+\frac{\rho^{\prime}(t)}{r^{2}}\right)e^{-\frac{3\alpha}{4}\left|r-\rho(t)\right|}\overline{\chi}_{\left\\{r\geq\delta_{0}\right\\}}\
+\
O\left(\frac{1}{[\gamma_{n}(t)]^{2}}\right)\overline{\chi}_{\left\\{\frac{\delta_{0}}{2}<r<\delta_{0}\right\\}},$

where we used the equalities in (2.12) and the fact that the function
$\widetilde{\omega}(t,r)$ in (2.11) and its derivatives are all exponentially
decaying.

By the same argument in (2.27) and the equality in (2.28), we can get that

$|E_{2}(t,r)|\leq C\frac{\Phi(t,r)}{\log\left|t\right|},\qquad\text{ for all}\
(t,r)\in(-\infty,-T]\times(0,+\infty),$

where $C$ is a positive constant only depending on $\alpha$ and $n$.

Eventually, combining the above estimates of the terms $E_{1}(t,r)$ and
$E_{2}(t,r)$, we can obtain the desired results. ∎

######  Remark 2.4.

According to the above proof of Lemma 2.3, it is easy to find that the error
term $E(t,r)$ has an exponentially decaying in space variable

$\left|E(t,r)\right|\leq\frac{C}{\left|t\right|^{\frac{1}{2}}e^{\frac{\alpha}{2}\left|r-\gamma_{n}(t)-\frac{1}{4\alpha}\log\left|t\right|\right|}},$
(2.29)

for all $r\geq\delta_{0}$, where $C>0$ does only depend on $n$ and $\alpha$.
Our goal is to solve nonlinear problem (2.16)-(2.17). Hence, according to
(2.29), we may consider the following linear parabolic problem of fourth order

$-\partial_{t}\varphi+F^{\prime}(z(t,r))[\varphi]=f(t,r),\qquad(t,r)\in(-\infty,-T)\times(0,+\infty),$
(2.30)

with the function $f(t,r)$ satisfying

$\sup_{(t,r)\in(-\infty,-T)\times(0,+\infty)}\frac{\left|f(t,r)\right|}{\left|t\right|^{\frac{1}{2}}e^{\frac{\alpha}{2}\left|r-\gamma_{n}(t)-\frac{1}{4\alpha}\log\left|t\right|\right|}}<+\infty,$

where the linear operator $F^{\prime}(z(t,r))$ is defined in (2.20). However,
the heat kernel of linear parabolic operator
$-\partial_{t}+F^{\prime}(z(t,r))$ would not has the exponential decay in
(2.29). As far as we known, it can satisfy the polynomial decay. Hence, we
provide an estimate of the error term $E(t,r)$ with the polynomial decay in
the above lemma. ∎

##  3\. The Linear Problem

In this section, firstly, we will obtain the solvability of a class of
semilinear biharmonic parabolic equations by applying some properties of
biharmonic heat kernel and fixed-point arguments, which is given in
Proposition 3.4. Secondly, we will prove that the linear projected problem
(3.60) is solvable by using Proposition 3.4 and a priori estimate in Lemma
3.5. The main result is given by Proposition 3.9.

###  3.1. A few results of linear parabolic equations with a biharmonic
operator

We first collect some known results for homogeneous biharmonic parabolic
equation, from [21, 23, 24]. Its solution can be represented by the
convolution of a biharmonic heat kernel and initial function.

######  Proposition 3.1.

We consider the following Cauchy problem for the biharmonic heat equation:

$\left\\{\begin{array}[]{l}u_{t}+(-\Delta)^{2}u=0\qquad\text{in}\
\mathbb{R}^{n+1}_{+}:=(0,+\infty)\times\mathbb{R}^{n},\\\\[5.69054pt]
u(0,x)=u_{0}(x)\hskip 8.5359pt\qquad\text{in}\
\mathbb{R}^{n},\end{array}\right.$ (3.1)

where $n\geq 1$ and $u_{0}\in C^{1}(\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n})$. Then (3.1) admits a unique global in time
solution explicitly given by

$u(t,x)=\int_{\mathbb{R}^{n}}u_{0}(y)p_{n}(t,x-y){\mathrm{d}}y,\qquad
p_{n}(t,x):=\bar{\alpha}_{n}t^{-n/4}f_{n}\Big{(}\frac{\left|x\right|}{t^{1/4}}\Big{)},\qquad\forall\,(t,x)\in\mathbb{R}^{n+1}_{+}.$

Here $p_{n}(t,x)$ is called the biharmonic heat kernel and $\bar{\alpha}_{n}$
denotes a suitable positive normalization number which depends on $n$ and
satisfies

$\bar{\alpha}_{n}t^{-n/4}\int_{\mathbb{R}^{n}}f_{n}\Big{(}\frac{\left|y\right|}{t^{1/4}}\Big{)}{\mathrm{d}}y=1,\qquad\text{for
all}\ t>0.$

Moreover, the function $f_{n}$ is given by

$f_{n}(s):=s^{1-n}\int_{0}^{+\infty}e^{-\varrho^{4}}(s\varrho)^{\frac{n}{2}}J_{\frac{n-2}{2}}(s\varrho){\mathrm{d}}\varrho,\qquad\text{for}\
s>0,$

where $J_{\nu}$ denotes the $\nu$-th Bessel function of the first kind. There
exist two constants $K_{n}$ and $\mu_{n}$ depending on $n$ such that

$\left|f_{n}(s)\right|\leq K_{n}\exp(-\mu_{n}s^{4/3}),\qquad\text{for all}\
s\geq 0.$ (3.2)

For the derivative of $f_{n}$, the following formula holds

$f^{\prime}_{n}(s)=-sf_{n+2}(s).$ (3.3)

∎

Next we will use the above results to study the following inhomogeneous
problem:

$\left\\{\begin{array}[]{l}u_{t}+(-\Delta)^{2}u=f(t,x)\qquad\text{in}\
(t_{0},t_{1})\times\mathbb{R}^{n},\\\\[5.69054pt] u(t_{0},x)=u_{0}(x)\hskip
48.36958pt\text{in}\ \mathbb{R}^{n},\end{array}\right.$ (3.4)

where $f(t,x)\in L^{\infty}((t_{0},t_{1})\times\mathbb{R}^{n})$ and $u_{0}\in
C^{1}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})$.

######  Definition 3.2.

Assume that $t_{1}\in(t_{0},+\infty)$. We say that $u\in
L^{\infty}((t_{0},t_{1})\times\mathbb{R}^{n})$ is a mild solution of (3.4) if

$u(t,x)=\Gamma_{t-t_{0}}[u_{0}](x)+\int^{t}_{t_{0}}\Gamma_{t-\tau}[f(\tau,\cdot)](x){\mathrm{d}}\tau,\qquad\text{for}\
(t,x)\in(t_{0},t_{1})\times\mathbb{R}^{n}.$ (3.5)

Here $\Gamma_{t}$ is a linear operator defined by

$\Gamma_{t}[u](x):=\Big{(}p_{n}(t,\cdot)\ast
u\Big{)}(x)=\int_{\mathbb{R}^{n}}p_{n}(t,x-y)u(y){\mathrm{d}}y,$

where the biharmonic heat kernel $p_{n}(t,x)$ is given by Proposition 3.1. In
addition, when $t_{1}=+\infty$, a mild solution also can be defined by (3.5)
provided $u(t,x)\in L^{\infty}_{loc}(t_{0},+\infty)\times
L^{\infty}(\mathbb{R}^{n})$.

Notice that it is not hard to verify that $u$ is a mild solution of (3.4) if
and only if $u$ solves it in the pointwise sense. We give some regularity
estimates for mild solutions.

######  Proposition 3.3.

Let $u$ be a mild solution of problem (3.4). Assume that $t_{1}<+\infty$. Then
the following estimates are valid:

  * •

If $u_{0}\in L^{\infty}(\mathbb{R}^{n})\cap C^{1}(\mathbb{R}^{n})$, then for
every $t_{*}\in(t_{0},t_{1})$, $t_{*}>0$, $\vartheta\in(0,4)$ and
$\theta\in(0,1)$, it holds that

$\displaystyle\sup_{t\in(t_{*},t_{1})}\|u(t,\cdot)\|_{C^{\vartheta}(\mathbb{R}^{n})}+\sup_{x\in\mathbb{R}^{n}}\|u(\cdot,x)\|_{C^{\theta}(t_{*},t_{1})}\leq
C_{1}\Big{(}\|f\|_{L^{\infty}((t_{*},t_{1})\times\mathbb{R}^{n})}+\|u_{0}\|_{L^{\infty}(\mathbb{R}^{n})}\Big{)},$
(3.6)

for some positive constant $C_{1}$ depending on
$n,\theta,t_{1}-t_{0},t_{1}-t_{*},\vartheta$ and $t_{*}$.

  * •

If $u_{0}\in C^{\vartheta}(\mathbb{R}^{n})$ for some $\vartheta\in(0,4)$ and
$t_{0}>0$, then it holds

$\displaystyle\sup_{t\in(t_{0},t_{1})}\|u(t,\cdot)\|_{C^{\vartheta}(\mathbb{R}^{n})}+\sup_{x\in\mathbb{R}^{n}}\|u(\cdot,x)\|_{C^{\frac{\vartheta}{4}}(t_{0},t_{1})}\leq
C_{2}\Big{(}\|f\|_{L^{\infty}((t_{0},t_{1})\times\mathbb{R}^{n})}+\|u_{0}\|_{C^{\vartheta}(\mathbb{R}^{n})}\Big{)},$
(3.7)

for some positive constant $C_{2}$ depending on $t_{1}-t_{0},n$ and
$\vartheta$.

###### Proof.

According to the formula (3.3) and some direct computations, we can derive
some estimates for the derivatives of biharmonic heat kernel $p_{n}(t,x)$ as
follows:

  * •

$\left|\partial_{t}p_{n}(t,x)\right|\leq
C_{n}\Big{(}t^{-1}\left|p_{n}(t,x)\right|+t^{-1}\left|x\right|^{2}\left|p_{n+2}(t,x)\right|\Big{)}$;

  * •

$\left|\nabla_{x}p_{n}(t,x)\right|\leq
C_{n}\left|x\right|\left|p_{n+2}(t,x)\right|$;

  * •

$\left|\nabla^{2}_{x}p_{n}(t,x)\right|\leq
C_{n}\Big{(}\left|p_{n+2}(t,x)\right|+\left|x\right|^{2}\left|p_{n+4}(t,x)\right|\Big{)}$;

  * •

$\left|\nabla^{3}_{x}p_{n}(t,x)\right|\leq
C_{n}\Big{(}\left|x\right|\left|p_{n+4}(t,x)\right|+\left|x\right|^{3}\left|p_{n+6}(t,x)\right|\Big{)}$;

  * •

$\left|\nabla^{4}_{x}p_{n}(t,x)\right|\leq
C_{n}\Big{(}\left|p_{n+4}(t,x)\right|+\left|x\right|^{2}\left|p_{n+6}(t,x)\right|+\left|x\right|^{4}\left|p_{n+8}(t,x)\right|\Big{)}$,

where $C_{n}>0$ only depends on $n$.

We decompose the mild solution in (3.5) as follows

$u(t,x)=U_{1}(t,x)+U_{2}(t,x),$

where

$U_{1}(t,x)=\Gamma_{t-t_{0}}[u_{0}](x),\quad
U_{2}(t,x)=\int^{t}_{t_{0}}\Gamma_{t-\tau}[f(\tau,\cdot)](x){\mathrm{d}}\tau.$

The analysis will begin with the first term $U_{1}(t,x)$. For any $t>t_{0}$
and $x\in\mathbb{R}^{n}$, using the above estimates and (3.2), we derive that

$\displaystyle\left|\partial_{t}U_{1}(t,x)\right|=\left|\int_{\mathbb{R}^{n}}\partial_{t}p_{n}(t-t_{0},x-y)u_{0}(y){\mathrm{d}}y\right|\leq\frac{C_{n}}{t-t_{0}}\|u_{0}\|_{L^{\infty}(\mathbb{R}^{n})}.$

Moreover, using (3.2), we have that

$\left|U_{1}(t,x)\right|\leq\int_{\mathbb{R}^{n}}\left|p_{n}(t-t_{0},x-y)u_{0}(y)\right|{\mathrm{d}}y\leq
C_{n}\|u_{0}\|_{L^{\infty}(\mathbb{R}^{n})}.$

For any $\vartheta\in(0,4)$ and $t,t^{1}\in(t_{0},t_{1})$, using the fact
$p_{n}(t,x)=t^{-n/4}p_{n}(\frac{x}{t^{1/4}},1)$,

$\displaystyle\frac{\left|U_{1}(t,x)-U_{1}(t^{1},x)\right|}{\left|t-t^{1}\right|^{\frac{\vartheta}{4}}}$
$\displaystyle\leq\left|\int_{\mathbb{R}^{n}}\frac{u_{0}\big{(}(t-t_{0})^{1/4}y\big{)}\,-\,u_{0}\big{(}(t^{1}-t_{0})^{1/4}y\big{)}}{\left|t-t^{1}\right|^{\frac{\vartheta}{4}}}p_{n}(1,y){\mathrm{d}}y\right|$
(3.8)
$\displaystyle\leq\|u_{0}\|_{C^{\vartheta}(\mathbb{R}^{n})}\int_{\mathbb{R}^{n}}\left|p_{n}(1,y)\right|\left|y\right|^{\vartheta}{\mathrm{d}}y\leq
C_{n}\|u_{0}\|_{C^{\vartheta}(\mathbb{R}^{n})}.$

Here we have used the inequality
$a^{\vartheta/4}-b^{\vartheta/4}\leq(a-b)^{\vartheta/4}$ for any $a\geq b\geq
0$ and $\vartheta\in(0,4)$ in the second inequality, $C_{n}>0$ only depends on
$n$.

Moreover, for the derivatives of $U_{1}$ with respect to $x$, by (3.2), we
have that

$\displaystyle\left|\nabla_{x}U_{1}(t,x)\right|$
$\displaystyle=\left|\int_{\mathbb{R}^{n}}\nabla_{x}p_{n}(t-t_{0},x-y)u_{0}(y){\mathrm{d}}y\right|$
$\displaystyle\leq
C_{n}\|u_{0}\|_{L^{\infty}(\mathbb{R}^{n})}\int_{\mathbb{R}^{n}}\left|y\right|\left|p_{n+2}(y,t-t_{0})\right|{\mathrm{d}}y$
$\displaystyle\leq\frac{C_{n}}{(t-t_{0})^{1/4}}\|u_{0}\|_{L^{\infty}(\mathbb{R}^{n})},$

where $C_{n}$ does only depend on $n$. Similarly, we can get that

$\displaystyle\left|\nabla^{2}_{x}U_{1}(t,x)\right|\leq$
$\displaystyle\frac{C_{n}}{(t-t_{0})^{1/2}}\|u_{0}\|_{L^{\infty}(\mathbb{R}^{n})},\qquad\left|\nabla^{3}_{x}U_{1}(t,x)\right|\leq\frac{C_{n}}{(t-t_{0})^{3/4}}\|u_{0}\|_{L^{\infty}(\mathbb{R}^{n})},$

and

$\left|\nabla^{4}_{x}U_{1}(t,x)\right|\leq\frac{C_{n}}{(t-t_{0})}\|u_{0}\|_{L^{\infty}(\mathbb{R}^{n})}.$

Thus for any $x,y\in\mathbb{R}^{n}$ and $\theta\in(0,1)$, we have

$[U_{1}]_{\theta}(t):=\frac{\left|U_{1}(t,x)-U_{1}(t,y)\right|}{\left|x-y\right|^{\theta}}\leq\left\\{\begin{array}[]{l}2\|U_{1}(t,\cdot)\|_{L^{\infty}(\mathbb{R}^{n})},\
\quad\ \quad\text{if}\ \left|x-y\right|\geq 1,\\\\[5.69054pt]
\|\nabla_{x}U_{1}(t,\cdot)\|_{L^{\infty}(\mathbb{R}^{n})},\qquad\text{if}\
\left|x-y\right|\leq 1.\end{array}\right.$

Hence, for any $t_{*}>t_{0}$, we have

$\sup_{t\in(t_{*},t_{1})}[U_{1}]_{\theta}(t)\leq\frac{C_{n}}{(t_{*}-t_{0})^{1/4}}\|u_{0}\|_{L^{\infty}(\mathbb{R}^{n})}.$

By similar arguments, we can obtain the following estimates

$\sup_{t\in(t_{*},t_{1})}[\nabla
U_{1}]_{\theta}(t)\leq\frac{C_{n}}{(t_{*}-t_{0})^{1/2}}\|u_{0}\|_{L^{\infty}(\mathbb{R}^{n})},\qquad\sup_{t\in(t_{*},t_{1})}[\nabla^{2}U_{1}]_{\theta}(t)\leq\frac{C_{n}}{(t_{*}-t_{0})^{3/4}}\|u_{0}\|_{L^{\infty}(\mathbb{R}^{n})},$

and

$\sup_{t\in(t_{*},t_{1})}[\nabla^{3}U_{1}]_{\theta}(t)\leq\frac{C_{n}}{(t_{*}-t_{0})}\|u_{0}\|_{L^{\infty}(\mathbb{R}^{n})}.$

Combining the above estimates, we have that for any $\vartheta\in(0,4)$,
$t_{*}>t_{0}$ and $\theta\in(0,1)$,

$\sup_{t\in(t_{*},t_{1})}\|U_{1}(t,\cdot)\|_{C^{\vartheta}(\mathbb{R}^{n})}+\sup_{x\in\mathbb{R}^{n}}\|U_{1}(\cdot,x)\|_{C^{\theta}(t_{*},t_{1})}\leq
C_{n}\big{[}1+\varrho(t_{*}-t_{0})\big{]}\|u_{0}\|_{L^{\infty}(\mathbb{R}^{n})}$
(3.9)

where
$\varrho(t_{*}-t_{0})=\frac{1}{(t_{*}-t_{0})}+\frac{1}{(t_{*}-t_{0})^{1/4}}+\frac{1}{(t_{*}-t_{0})^{\theta}}$.

Using the formula

$U_{1}(t,x)=\int_{\mathbb{R}^{n}}p_{n}(t-t_{0},y)u_{0}(x-y){\mathrm{d}}y$

and (3.8), for $\vartheta\in(0,4)$, we have

$\sup_{t\in(t_{0},t_{1})}\|U_{1}(t,\cdot)\|_{C^{\vartheta}(\mathbb{R}^{n})}+\sup_{x\in\mathbb{R}^{n}}\|U_{1}(\cdot,x)\|_{C^{\frac{\vartheta}{4}}(t_{0},t_{1})}\leq
C_{n}\|u_{0}\|_{C^{\vartheta}(\mathbb{R}^{n})},$ (3.10)

where $C_{n}$ is a positive constant which depends on $n,\vartheta$.

Next we will estimate $U_{2}(t,x)$. First we find that

$\displaystyle\left|U_{2}(t,x)\right|\leq$ $\displaystyle
C_{n}\int_{t_{0}}^{t}\int_{\mathbb{R}^{n}}\left|p_{n}(t-\tau,x-y)f(\tau,y)\right|{\mathrm{d}}y{\mathrm{d}}\tau$
$\displaystyle\leq$ $\displaystyle
C_{n}(t_{1}-t_{0})\|f\|_{L^{\infty}}((t_{1},t)\times\mathbb{R}^{n}).$

For any $t,t^{1}\in(t_{0},t_{1})$, by the previous estimate of
$\partial_{t}p_{n}(t,x)$, we have

$\left|p_{n}(t,x)-p_{n}(t^{1},x)\right|\leq C_{n}\left|\log t-\log
t^{1}\right|\Big{(}\left|p_{n}\big{(}(1-\theta_{1})t+\theta_{1}t^{1},x\big{)}\right|+\left|x\right|^{2}\left|p_{n+2}\big{(}(1-\theta_{1})t+\theta_{1}t^{1},x\big{)}\right|\Big{)},$

for some $\theta_{1}\in(0,1)$. Using the above inequality and (3.2), for
$\theta\in(0,1)$, we have

$\displaystyle\frac{\left|U_{2}(t,x)-U_{2}(t^{1},x)\right|}{\left|t-t^{1}\right|^{\theta}}\leq$
$\displaystyle\frac{\left|\int_{t_{0}}^{t}\int_{\mathbb{R}^{n}}\Big{[}p_{n}(t-\tau,x-y)-p_{n}(t^{1}-\tau,x-y)\Big{]}f(y,\tau){\mathrm{d}}y{\mathrm{d}}\tau\right|}{\left|t-t^{1}\right|^{\theta}}$
$\displaystyle+\frac{\left|\int^{t}_{t^{1}}\int_{\mathbb{R}^{n}}p_{n}(t^{1}-\tau,x-y)f(\tau,y){\mathrm{d}}y{\mathrm{d}}\tau\right|}{\left|t-t^{1}\right|^{\theta}}$
$\displaystyle\leq$ $\displaystyle
C_{n}\Bigg{(}1+\log\frac{t_{1}-t_{0}}{\left|t-t^{1}\right|}\Bigg{)}\left|t-t^{1}\right|^{1-\theta}\|f\|_{L^{\infty}((t_{1},t)\times\mathbb{R}^{n})},$

where $C_{n}$ is a positive constant which only depends on $n$.

Second, we study the derivatives of $U_{2}(t,x)$ with respect to $x$. By some
direct computations and (3.2), we have

$\displaystyle\left|\nabla_{x}U_{2}(t,x)\right|$
$\displaystyle=\left|\int_{t_{0}}^{t}\int_{\mathbb{R}^{n}}\nabla_{x}p_{n}(t-\tau,x-y)f(\tau,y){\mathrm{d}}y{\mathrm{d}}\tau\right|$
$\displaystyle\leq
C_{n}\|f\|_{L^{\infty}((t_{0},t)\times\mathbb{R}^{n})}\int_{t_{0}}^{t}\int_{\mathbb{R}^{n}}\left|y\right|\left|p_{n+2}(t-\tau,y)\right|{\mathrm{d}}y{\mathrm{d}}\tau$
$\displaystyle\leq
C_{n}\|f\|_{L^{\infty}((t_{0},t)\times\mathbb{R}^{n})}\int_{t_{0}}^{t}(t-\tau)^{-1/4}{\mathrm{d}}\tau$
$\displaystyle\leq
C_{n}(t-t_{0})^{3/4}\|f\|_{L^{\infty}((t_{0},t)\times\mathbb{R}^{n})}.$

By the same arguments as above, we can derive that

$\left|\nabla^{2}_{x}U_{2}(t,x)\right|\leq
C_{n}(t-t_{0})^{1/2}\|f\|_{L^{\infty}((t_{0},t)\times\mathbb{R}^{n})},\qquad\left|\nabla^{3}_{x}U_{2}(t,x)\right|\leq
C_{n}(t-t_{0})^{1/4}\|f\|_{L^{\infty}((t_{0},t)\times\mathbb{R}^{n})},$

and for any $\theta\in(0,1)$,

$\displaystyle\frac{\left|\nabla^{3}_{x}U_{2}(t,x_{1})-\nabla^{3}_{x}U_{2}(t,x_{2})\right|}{\left|x_{1}-x_{2}\right|^{\theta}}$
$\displaystyle=\alpha_{n}\frac{\left|\int_{t_{0}}^{t}\int_{\mathbb{R}^{n}}(t-\tau)^{-\frac{n+3}{4}}\Big{[}\nabla^{3}_{x}f_{n}\Big{(}\frac{\left|x_{1}-y\right|}{(t-\tau)^{1/4}}\Big{)}-\nabla^{3}_{x}f_{n}\Big{(}\frac{\left|x_{2}-y\right|}{(t-\tau)^{1/4}}\Big{)}\Big{]}f(y,\tau){\mathrm{d}}y{\mathrm{d}}\tau\right|}{\left|x_{1}-x_{2}\right|^{\theta}}$
$\displaystyle\leq
C_{n}\frac{\left|\int_{t_{0}}^{t}\int_{\mathbb{R}^{n}}(t-\tau)^{-\frac{3}{4}}\Big{[}\nabla^{3}_{x}f_{n}\Big{(}\left|y-\frac{x_{1}}{(t-\tau)^{1/4}}\right|\Big{)}-\nabla^{3}_{x}f_{n}\Big{(}\left|y-\frac{x_{2}}{(t-\tau)^{1/4}}\right|\Big{)}\Big{]}f\Big{(}(t-\tau)^{1/4}y,\tau\Big{)}{\mathrm{d}}y{\mathrm{d}}\tau\right|}{\left|x_{1}-x_{2}\right|^{\theta}}$
$\displaystyle\leq
C_{n}(t-t_{0})^{(1-\theta)/4}\|f\|_{L^{\infty}((t_{0},t)\times\mathbb{R}^{n})}.$

Combing the above estimates of $U_{2}$, for any $\vartheta\in(0,4)$ and
$\theta\in(0,1)$, we have that

$\sup_{t\in(t_{0},t_{1})}\|U_{2}(t,\cdot)\|_{C^{\vartheta}(\mathbb{R}^{n})}+\sup_{x\in\mathbb{R}^{n}}\|U_{2}(\cdot,x)\|_{C^{\theta}(t_{0},t_{1})}\leq
C_{n}\Big{[}\upsilon(t_{1}-t_{0})+(t_{1}-t_{0})^{1-\theta}\Big{]}\|f\|_{L^{\infty}((t_{0},t)\times\mathbb{R}^{n})}.$
(3.11)

On the other hand, for $\vartheta\in(0,4)$, there holds

$\sup_{t\in(t_{0},t_{1})}\|U_{2}(t,\cdot)\|_{C^{\vartheta}(\mathbb{R}^{n})}+\sup_{x\in\mathbb{R}^{n}}\|U_{2}(\cdot,x)\|_{C^{\frac{\vartheta}{4}}(t_{0},t_{1})}\leq
C_{n}\upsilon(t_{1}-t_{0})\|f\|_{L^{\infty}((t_{0},t)\times\mathbb{R}^{n})},$
(3.12)

where $\upsilon(t-t_{0})=(t-t_{0})+(t-t_{0})^{1-\frac{\vartheta}{4}}$ and
$C_{n}$ is a positive constant which depends on $n$.

The Proposition 3.3 follows from (3.9), (3.10), (3.11) and (3.12). ∎

We will apply the above proposition to study the solvability of a class of
semilinear parabolic equations. Let $u_{0}\in C^{1}(\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n})$, and we consider the initial value problem

$\left\\{\begin{array}[]{l}u_{t}+(-\Delta)^{2}u=G[\Delta u,\nabla
u,u,t,x]\qquad\text{in}\ (t_{0},t_{1})\times\mathbb{R}^{n},\\\\[5.69054pt]
u(t_{0},x)=u_{0}(x)\hskip 95.3169pt\text{in}\
\mathbb{R}^{n},\end{array}\right.$ (3.13)

where
$G[p,\vec{q},s,t,x]:\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}\times[t_{0},+\infty)\times\mathbb{R}^{n}\rightarrow\mathbb{R}$
is a measurable function satisfying

  1. (1)

For every $M>0$ and $T>t_{0}$, there exists $C_{T,M}>0$ such that
$\Big{|}G[p,\vec{q},s,t,x]\Big{|}\leq C_{T,M}$ for all $x\in\mathbb{R}^{n}$,
$t\in[t_{0},T]$ and $p,\left|\vec{q}\right|,s\in[-M,M]$.

  2. (2)

There is a constant $\sigma>0$ in such a way that

$\Big{|}G[p_{1},\vec{q}_{1},s_{1},t,x]-G[p_{2},\vec{q}_{2},s_{2},t,x]\Big{|}\leq\sigma\Big{(}\left|p_{1}-p_{2}\right|+\left|\vec{q}_{1}-\vec{q}_{2}\right|+\left|s_{1}-s_{2}\right|\Big{)},$
(3.14)

for all $x\in\mathbb{R}^{n},t\geq t_{0}$, $p_{1},p_{2}\in\mathbb{R}$,
$\vec{q}_{1},\vec{q}_{2}\in\mathbb{R}^{n}$ and $s_{1},s_{2}\in\mathbb{R}$.

In particular, we can take

$G[\Delta u,\nabla u,u,t,x]=a(t,x)\Delta
u+\sum_{i=1}^{n}b^{i}(t,x)\nabla_{x_{i}}u+c(t,x)u+g(t,x),$ (3.15)

where those functions $a(t,x)$, $b^{1}(t,x),\cdots,b^{n}(t,x)$, $c(t,x)$ are
all belonging to $L^{\infty}([t_{0},+\infty),C^{1}(\mathbb{R}^{n}))$ and $g\in
L^{\infty}([t_{0},+\infty)\times\mathbb{R}^{n})$.

Assume that $u(t,x)$ satisfies that

$\Delta u,\ \left|\nabla u\right|,\ u\in
L^{\infty}((t_{0},t_{1})\times\mathbb{R}^{n}),$

then by (3.5), $u$ is a mild solution of (3.13) if and only if

$u(t,x)=\Gamma_{t-t_{0}}[u_{0}](x)+\int^{t}_{t_{0}}\Gamma_{t-\tau}\Big{[}G[\Delta
u,\nabla
u,u,\tau,\cdot](\tau,\cdot)\Big{]}(x){\mathrm{d}}\tau\qquad\text{for}\
(t,x)\in(t_{0},t_{1})\times\mathbb{R}^{n}.$ (3.16)

Notice that $G[\Delta u,\nabla u,u,t,x]\in
L^{\infty}((t_{0},t_{1})\times\mathbb{R}^{n})$ by the assumptions on $u$ and
$G$. Thus (3.16) is well defined by Proposition 3.3.

Now we devote to the study of the solvability of problem (3.13). For
convenience, we define a map $\mathcal{N}_{G,u_{0}}$ from
$L^{\infty}[(t_{0},t_{1});C^{2}(\mathbb{R}^{n})]$ to itself as follows

$\mathcal{N}_{G,u_{0}}[u](t,x):=\Gamma_{t-t_{0}}[u_{0}](x)+\int^{t}_{t_{0}}\Gamma_{t-\tau}\Big{[}G[\Delta
u,\nabla u,u](\cdot,\tau)\Big{]}(x){\mathrm{d}}\tau\qquad\text{for}\
(x,t)\in\mathbb{R}^{n}\times(t_{0},t_{1}),$

for $u_{0}\in C^{1}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})$ and $u\in
L^{\infty}[(t_{0},t_{1});C^{2}(\mathbb{R}^{n})]$. The main result is the
following:

######  Proposition 3.4.

Given any $u_{0}\in C^{1}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})$,
there exists a unique mild solution $u$ to problem (3.13) with
$t_{1}=+\infty$. Moreover, if $u_{0}\in C^{2}(\mathbb{R}^{n})$, then the
solution $u$ has the property: for $s>t_{0}$, there exists $C$ independent of
$t$ such that

$\sup_{t\in(t_{0},s)}\|u(t,\cdot)\|_{C^{2}(\mathbb{R}^{n})}\leq
C\Big{(}\Big{[}(s-t_{0})+(s-t_{0})^{1/2}\Big{]}\|G(0,0,0,x,t)\|_{L^{\infty}((t_{0},s)\times\mathbb{R}^{n})}+\|u_{0}\|_{C^{2}(\mathbb{R}^{n})}\Big{)}.$
(3.17)

###### Proof.

Notice that this proposition is equivalent to that problem (3.13) has a unique
solution $u$ for any $t_{1}\in(t_{0},+\infty)$. According to the definition in
(3.5), $u$ is a mild solution of (3.13) if and only if $u$ is a fixed point of
the map $\mathcal{N}_{G,u_{0}}$. Thus we will study the existence and
uniqueness of fixed point for the map $\mathcal{N}_{G,u_{0}}$.

Using Proposition 3.3 and the assumptions on $G$, we have that
$\mathcal{N}_{G,u_{0}}$ is an operator from the space
$L^{\infty}[(t_{0},t_{1});C^{2}(\mathbb{R}^{n})]$ to itself. We claim that the
map $\mathcal{N}_{G,u_{0}}$ is a contraction when $T_{1}:=t_{1}-t_{0}$ is
small enough.

Indeed, by Proposition 3.1, the assumptions on $G$ in (3.14) and some direct
computations, we have that for any $u,w\in
L^{\infty}[(t_{0},t_{1});C^{2}(\mathbb{R}^{n})]$,

$\displaystyle\big{|}\mathcal{N}_{G,u_{0}}[u](t,x)-\mathcal{N}_{G,u_{0}}[w](t,x)\big{|}$
$\displaystyle\leq\int^{t}_{t_{0}}\int_{\mathbb{R}^{n}}\left|p_{n}(t-\tau,x-y)\right|\cdot\Big{|}G[\Delta
u,\nabla u,u](\tau,y)-G[\Delta w,\nabla
w,w](\tau,y)\Big{|}{\mathrm{d}}y{\mathrm{d}}\tau$ $\displaystyle\leq\sigma
T_{1}C_{n}\Big{(}\|\Delta u-\Delta
w\|_{L^{\infty}((t_{0},t_{1})\times\mathbb{R}^{n})}+\|\nabla u-\nabla
w\|_{L^{\infty}((t_{0},t_{1})\times\mathbb{R}^{n})}+\|u-w\|_{L^{\infty}((t_{0},t_{1})\times\mathbb{R}^{n})}\Big{)},$

for a.e. $x\in\mathbb{R}^{n}$, $t\in(t_{0},t_{1})$, where $C_{n}$ is a
positive number which only depends on $n$. Hence, we can choose
$T_{1}\leq\frac{1}{2\sigma C_{n}}$ such that

$\displaystyle\big{\|}\mathcal{N}_{G,u_{0}}[u](t,x)-\mathcal{N}_{G,u_{0}}[w](t,x)\big{\|}_{L^{\infty}((t_{0},t_{1})\times\mathbb{R}^{n})}$
$\displaystyle\leq\frac{1}{2}\Big{(}\|\Delta u-\Delta
w\|_{L^{\infty}((t_{0},t_{1})\times\mathbb{R}^{n})}+\|\nabla u-\nabla
w\|_{L^{\infty}((t_{0},t_{1})\times\mathbb{R}^{n})}+\|u-w\|_{L^{\infty}((t_{0},t_{1})\times\mathbb{R}^{n})}\Big{)}.$

Thus, the map $\mathcal{N}_{G,u_{0}}$ is a contraction.

Next we consider the following iterative sequence

$u_{m}(t,x)=(\mathcal{N}_{G,u_{0}})^{m}[0](t,x),$

by Banach Fixed Theorem, then there is $u\in
L^{\infty}[(t_{0},t_{1});C^{2}(\mathbb{R}^{n})]$ such that $u_{m}(x,t)$
converges to $u(x,t)$ in the space
$L^{\infty}[(t_{0},t_{1});C^{2}(\mathbb{R}^{n})]$ and
$\mathcal{N}_{G,u_{0}}(u)=u$, which is the unique solution of problem (3.13)
with $t_{1}=t_{0}+\frac{1}{2\sigma C_{n}}$.

Let $u^{1}(t,x)$ denote the mild solution of (3.13), which can be obtained by
the above arguments. We consider the following initial problem:

$\left\\{\begin{array}[]{l}u_{t}+(-\Delta)^{2}u=G[\Delta u,\nabla
u,u,x,t]\hskip 28.45274pt\text{in}\ \big{(}t_{0}+\frac{1}{2\sigma
C_{n}},t_{0}+\frac{1}{\sigma C_{n}}\big{)}\times\mathbb{R}^{n},\\\\[5.69054pt]
u\big{(}t_{0}+\frac{1}{2\sigma
C_{n}},x\big{)}=u^{1}\big{(}t_{0}+\frac{1}{2\sigma
C_{n}},x\big{)}\qquad\text{in}\ \mathbb{R}^{n}.\end{array}\right.$

Repeating the previous arguments, we can get that the above equation has a
unique solution $u^{2}(t,x)$. Thus we can extend the time interval of
existence to $\big{(}t_{0},t_{0}+\frac{l}{2\sigma C_{n}}\big{)}$ with any
positive integer $l\geq 1$ step by step. In this process, we used the fact
that the initial function is belonging to $C^{1}(\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n})$ on each step. Indeed, the solutions are well-
defined bounded continuous functions of $t$ and $x$, and are continuously
differentiable in $x$, due to the estimate (3.7) with $\vartheta=1$ in
Proposition 3.3. The estimate (3.17) comes from Proposition 3.3. ∎

###  3.2. A priori estimate of a linear problem involving the operator $L$

Recall the functions $\rho$, $\gamma_{n}$ and $h$ in (2.7)-(2.9). By defining

$\|\psi\|_{C_{\Phi}\left((t_{1},t_{2})\times(0,\infty)\right)}:=\left\|\frac{\psi}{\Phi}\right\|_{L^{\infty}((t_{1},t_{2})\times(0,\infty))},$

we choose a set consisting of continuous functions as the following

$\displaystyle C_{\Phi}((t_{1},t_{2})\times(0,\infty))$
$\displaystyle:=\left\\{\psi\ :\ \psi(t,r)=0,\
\forall\,(t,r)\in(t_{1},t_{2})\times\left(0,\frac{\delta_{0}}{2}\right),\
\text{and}\
\|\psi\|_{C_{\Phi}((t_{1},t_{2})\times(0,\infty))}<+\infty\right\\},$ (3.18)

where $t_{1}<t_{2}<0$, $\delta_{0}>0$ is a small fixed number, the functions
$\Phi(t,r)$ and is given by (2.26).

Let us consider the following Cauchy problem:

$\left\\{\begin{array}[]{l}\psi_{t}=L[\psi]+g(t,r)\qquad\ \text{in}\
(s,-T]\times(0,\infty),\\\\[5.69054pt] \psi(s,r)=0,\hskip
52.63777pt\forall\,r\in(0,+\infty),\\\\[5.69054pt]
\partial^{j}_{r}\psi(t,0)=0,\hskip 44.10185pt\forall\,t\in(s,-T]\ \text{and}\
j=1,3,\end{array}\right.$ (3.19)

where $g\in C_{\Phi}\left((-\infty,-T]\times(0,\infty)\right)$, $T>0$ and
$s+1<-T$. Notice that problem (3.19) has a unique solution $\psi^{s}(t,r)$.
Indeed, by the definition of the operator $L[\psi]$ in (2.20), we can rewrite
$L[\psi]$ as follows

$\displaystyle L[\psi]=$
$\displaystyle-(\Delta)^{2}\psi+2W^{\prime\prime}\big{(}z(t,r)\big{)}\Delta\psi+2W^{\prime\prime\prime}\big{(}z(t,r)\big{)}\Delta
z(t,r)\psi+W^{(4)}\big{(}z(t,r)\big{)}\left|\nabla z(t,r)\right|^{2}\psi$
$\displaystyle+2W^{\prime\prime\prime}\big{(}z(t,r)\big{)}\nabla
z(t,r)\cdot\nabla\psi-\Big{[}\big{|}W^{\prime\prime}\big{(}z(t,r)\big{)}\big{|}^{2}-W^{\prime}\big{(}z(t,r)\big{)}W^{\prime\prime\prime}\big{(}z(t,r)\big{)}\Big{]}\psi,$

where we recall that approximate solution

$z(t,r)=\left(\omega\big{(}r-\rho(t)\big{)}+\frac{(n-1)(n-3)}{r^{2}}\widetilde{\omega}\big{(}r-\rho(t)\big{)}\right)\chi\big{(}r\big{)}+\chi\big{(}r\big{)}-1,$

and the functions $\chi(r)$ and $\rho(t)$ are given by (2.5) and (2.7)
respectively. According to Lemma 2.1 and the assumptions on $W(s)$ in (1.2),
those coefficients in front of $\Delta\psi$, $\nabla\psi$ and $\psi$ are all
smooth and bounded for $x\in\mathbb{R}^{n}$ and $t<-2$. Thus we know that
problem (3.19) is uniquely solvable by applying Proposition 3.4, and denote
this solution by $\psi^{s}(t,r)$. Moreover, we have that $\psi^{s}(t,r)=0$
when $r\in(0,\frac{\delta_{0}}{2})$. Indeed, we consider

$\psi^{e}(t,r):=\left\\{\begin{array}[]{l}\psi^{s}(t,r),\qquad\text{if}\
r\in(0,\frac{\delta_{0}}{2}];\\\\[5.69054pt] 0,\hskip 46.94687pt\text{if}\
r\in(\frac{\delta_{0}}{2},+\infty),\end{array}\right.$

which satisfies that

$\left\\{\begin{array}[]{l}\psi_{t}=-(\Delta)^{2}\psi+4\Delta\psi-4\psi\qquad\
\text{in}\ (s,-T]\times(0,+\infty),\\\\[5.69054pt] \psi(s,r)=0,\hskip
88.2037pt\forall\,r\in(0,+\infty),\\\\[5.69054pt]
\partial^{j}_{r}\psi(t,0)=0,\hskip 80.52135pt\forall\,t\in(s,-T]\ \text{and}\
j=1,3,\end{array}\right.$

where $n\geq 4$ and we have used the fact that $z(t,r)=-1$ when
$r\in(0,\frac{\delta_{0}}{2}]$. According to the proposition 3.4, the above
equation has a unique mild solution $\psi(t,r)=0$. Hence $\psi^{e}(t,r)=0$,
that is $\psi^{s}(t,r)=0$ when $r\in(0,\frac{\delta_{0}}{2}]$.

We next establish a priori estimate for the solutions of problem (3.19).

######  Lemma 3.5.

Let $g\in C_{\Phi}((s,-T)\times(0,\infty))$ and $\psi^{s}$ be a solution of
problem $(\ref{eq3.5})$ which satisfies the orthogonality condition

$\int_{0}^{\infty}\psi^{s}(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r=0,\qquad
s<t<-T.$ (3.20)

Then there exists a uniform constant $T_{0}>T$ such that for any
$t\in(s,-T_{0}]$, the following estimate is valid

$\sum_{l=0}^{3}\|\partial_{r}^{l}\psi^{s}\|_{C_{\Phi}((s,t)\times(0,\infty))}\leq
C\|g\|_{C_{\Phi}((s,t)\times(0,\infty))},$ (3.21)

where $C>0$ is a uniform positive constant independent of $t$ and $s$.

The proofs of this lemma consist of tedious analysis, which will be given in
the sequel.

####  3.2.1. Proof of Lemma 3.5

Here are the details.

###### Proof.

We prove the above lemma by contradiction. Assume that there exist two
sequences $\\{s_{i}\\}$ and $\\{t_{i}\\}$ such that $s_{i}+1<t_{i}<0$ and
$s_{i}\rightarrow-\infty$, $t_{i}\rightarrow-\infty$ when the sub-index $i$
goes to infinity. We assume that: for each given $i$, there exists $g^{i}\in
C_{\Phi}((s_{i},t_{i})\times(0,\infty))$, such that

$\displaystyle\sum_{l=0}^{3}\big{\|}\partial_{r}^{l}\psi^{i}\big{\|}_{C_{\Phi}((s_{i},t_{i})\times(0,\infty))}=1,$
(3.22)

and

$\|g^{i}\|_{C_{\Phi}((s_{i},t_{i})\times(0,\infty))}\rightarrow
0,\qquad\text{as}\ i\rightarrow+\infty,$ (3.23)

where $\psi^{i}$ is the solution of problem (3.19)-(3.20) with $g=g^{i}$,
$s=s_{i}$ and $-T=t_{i}$.

Assertion: _For any $R>0$, we have that_

$\lim_{i\rightarrow\infty}\sum_{l=0}^{3}\left\|\frac{\partial^{l}_{r}\psi^{i}}{\Phi}\right\|_{L^{\infty}\big{(}A^{(s_{i},t_{i})}_{R}\big{)}}=0,$
(3.24)

_where $\Phi(t,r)$ is given by $(\ref{eq2.13})$ and $A^{(s_{i},t_{i})}_{R}$ is
defined by_

$A^{(s_{i},t_{i})}_{R}:=\Big{\\{}(t,r)\in(s_{i},t_{i})\times(0,\infty):\
\left|r-\gamma_{n}(t)\right|<R+1\Big{\\}}$

_and the function $\gamma_{n}(t)$ is defined by (2.8). _

The proof of (3.24) will be provided in Section 3.2.2. We first accept the
validity of (3.24). According to (3.22), we have that there exists
$l\in\\{0,1,2,3\\}$ such that

$\big{\|}\partial_{r}^{l}\psi^{i}\big{\|}_{C_{\Phi}((s_{i},t_{i})\times(0,\infty))}\geq\frac{1}{4}.$

Without loss of generality, we assume that
$\|\psi^{i}\|_{C_{\Phi}((s_{i},t_{i})\times(0,\infty))}\geq\frac{1}{4}$. For
the other situations, the following arguments can be adapted in a simple way.

Recall the definition of $C_{\Phi}((s_{i},t_{i})\times(0,\infty))$ in (3.18),
then we derive that

$\|\psi^{i}\|_{C_{\Phi}((s_{i},t_{i})\times(0,\infty))}=\sup_{(t,r)\in(s_{i},t_{i})\times(0,\infty)}\frac{\left|\psi^{i}(t,r)\right|}{\Phi(t,r)}\geq\frac{1}{4}.$

Thus, there exists $(\bar{t}_{i},\bar{r}_{i})\in(s_{i},t_{i})\times(0,\infty)$
such that

$\frac{\left|\psi^{i}(\bar{t}_{i},\bar{r}_{i})\right|}{\Phi(\bar{t}_{i},\bar{r}_{i})}\geq\frac{1}{5}.$

Furthermore, by (3.24), we have

$\lim_{i\rightarrow+\infty}\left|\bar{r}_{i}-\gamma_{n}(\bar{t}_{i})\right|=+\infty.$
(3.25)

Let us define

$\phi^{i}(\nu,y):=\frac{\psi^{i}\big{(}\nu+\bar{t}_{i},y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}}{\Phi\big{(}\bar{t}_{i},\bar{y}_{i}+\gamma_{n}(\bar{t}_{i})\big{)}},\qquad\text{where}\
\bar{y}_{i}=\bar{r}_{i}-\gamma_{n}(\bar{t}_{i}),$

where the functions $\gamma_{n}(t)$ and $\Phi(t,r)$ are given by (2.8) and
(2.26) respectively. Thus $\phi^{i}(\nu,y)$ satisfies the following problem

$\displaystyle\partial_{\nu}\phi^{i}=$
$\displaystyle-\phi_{yyyy}^{i}-\frac{2(n-1)}{y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})}\phi_{yyy}^{i}+\Bigg{[}2W^{\prime\prime}\big{(}\bar{z}(\nu,y)\big{)}-\frac{(n-3)(n-1)}{\big{(}y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}^{2}}\Bigg{]}\phi^{i}_{yy}$
$\displaystyle-\Big{(}W^{\prime\prime}\big{(}\bar{z}(\nu,y)\big{)}\Big{)}^{2}\phi^{i}+\Bigg{[}\frac{2(n-1)W^{\prime\prime}\big{(}\bar{z}(\nu,y)\big{)}}{y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})}-\frac{(3-n)(n-1)}{\big{(}y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}^{3}}\Bigg{]}\phi^{i}_{y}$
$\displaystyle+2W^{\prime\prime\prime}\big{(}\bar{z}(\nu,y)\big{)}\bar{z}_{y}\phi^{i}_{y}-W^{\prime\prime\prime}\big{(}\bar{z}(\nu,y)\big{)}W^{\prime}\big{(}\bar{z}(\nu,y)\big{)}\phi+W^{(4)}\big{(}\bar{z}(\nu,y)\big{)}\left|\bar{z}_{y}\right|^{2}\phi^{i}$
$\displaystyle+2W^{\prime\prime\prime}\big{(}\bar{z}(\nu,y)\big{)}\bar{z}_{yy}\phi^{i}+\frac{2(n-1)}{y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})}W^{\prime\prime\prime}\big{(}\bar{z}(\nu,y)\big{)}\bar{z}_{y}\phi^{i}+\phi^{i}_{y}\partial_{\nu}\gamma_{n}(\nu+\bar{t}_{i})$
$\displaystyle+\frac{g^{i}\big{(}\nu+\bar{t}_{i},y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}}{\Phi\big{(}\bar{t}_{i},\bar{y}_{i}+\gamma_{n}(\bar{t}_{i})\big{)}}\qquad\text{in}\
B^{(s_{i},t_{i})}_{i},$ (3.26)

with the conditions

$\displaystyle\left|\phi^{i}(0,0)\right|\geq\frac{1}{5},\qquad\phi^{i}(s_{i}-\bar{t}_{i},y)=0,\qquad\text{for}\
y\in\big{(}-\bar{y}_{i}-\gamma_{n}(\nu+\bar{t}_{i}),+\infty\big{)},$ (3.27)

where

$\bar{z}(\nu,y):=z\big{(}\nu+\bar{t}_{i},y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}$

and the set $B^{(s_{i},t_{i})}_{i}$ is defined by

$B^{(s_{i},t_{i})}_{i}:=(s_{i}-\bar{t}_{i},0]\times(-\bar{y}_{i}-\gamma_{n}(\nu+\bar{t}_{i}),+\infty).$

Since $\lim\limits_{i\rightarrow+\infty}\left|\bar{y}_{i}\right|=+\infty$ due
to (3.24), we have that $\lim\limits_{i\rightarrow+\infty}\bar{y}_{i}=+\infty$
or $-\infty$ up to a subsequence. Moreover, notice that there exist two cases:

Case $1$, $\lim\limits_{i\rightarrow\infty}s_{i}-\bar{t}_{i}=-\infty$.

Case $2$, $\lim\limits_{i\rightarrow\infty}s_{i}-\bar{t}_{i}=-\varsigma$ for
some positive constant $\varsigma>0$.
Next we will prove that Lemma 3.5 holds, and the process will be divided into
the following two steps.

Step one. We first consider Case $1$, and then have that
$\phi^{i}\rightarrow\phi$ locally uniformly. Moreover,
$\left|\phi(0,0)\right|>\frac{1}{5}$ and $\phi$ satisfies

$\phi_{\nu}=-\phi_{yyyy}+2W^{\prime\prime}(1)\phi_{yy}-\big{[}W^{\prime\prime}(1)\big{]}^{2}\phi\qquad\text{in}\
(-\infty,0]\times\mathbb{R}.$ (3.28)

By the assumption on $\psi^{i}$ in (3.22), we have that for all $(\nu,y)\in
B^{j}_{i}$,

$\left|\phi^{i}(\nu,y)\right|=\left|\frac{\psi^{i}\big{(}\nu+\bar{t}_{i},y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}}{\Phi\big{(}\bar{t}_{i},\bar{y}_{i}+\gamma_{n}(\bar{t}_{i})\big{)}}\right|\leq\left|\frac{\Phi\big{(}\nu+\bar{t}_{i},y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}}{\Phi\big{(}\bar{t}_{i},\bar{y}_{i}+\gamma_{n}(\bar{t}_{i})\big{)}}\right|,$
(3.29)

where the sets $B_{i}^{j}$ with $j=0,1$, are defined by

$\displaystyle B_{i}^{0}:$
$\displaystyle=(s_{i}-\bar{t}_{i},0]\times\big{(}-\bar{y}_{i}-\gamma_{n}(\nu+\bar{t}_{i}),\
\delta_{0}-\gamma_{n}(\nu+\bar{t}_{i})-\bar{y}_{i}\big{]},$ (3.30)
$\displaystyle B_{i}^{1}:$
$\displaystyle=(s_{i}-\bar{t}_{i},0]\times\big{(}\delta_{0}-\gamma_{n}(\nu+\bar{t}_{i})-\bar{y}_{i},\
+\infty\big{)}.$

Next we estimate the right hand side of the above inequality (3.29). There
exist the following different situations:
(1). When $j=1$ and
$\bar{r}_{i}=\bar{y}_{i}+\gamma_{n}(\bar{t}_{i})>\delta_{0}$, by the
definitions of $\gamma_{n}(t)$ and $\Phi(t,r)$ in (2.8) and (2.26), we have
that

$\displaystyle\left|\frac{\Phi\big{(}\nu+\bar{t}_{i},y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}}{\Phi\big{(}\bar{t}_{i},\bar{y}_{i}+\gamma_{n}(\bar{t}_{i})\big{)}}\right|$
$\displaystyle\leq\frac{\hbar(\nu+\bar{t}_{i})}{\hbar(\bar{t}_{i})}\left(\frac{1+\left|\bar{y}_{i}-\frac{1}{3\alpha}\log\left|\bar{t}_{i}\right|\right|}{1+\left|y+\bar{y}_{i}-\frac{1}{3\alpha}\log\left|\nu+\bar{t}_{i}\right|\right|}\right)^{p}$
$\displaystyle\leq
C\frac{\hbar(\nu+\bar{t}_{i})}{\hbar(\bar{t}_{i})}\Big{(}1+\left|y\right|+\log\Big{[}\frac{\left|\nu+\bar{t}_{i}\right|}{\left|\bar{t}_{i}\right|}\Big{]}\Big{)}^{p}$
$\displaystyle\leq C\left(\left|y\right|+1\right)^{p},$

where the function $\hbar(t)$ is defined by

$\hbar(t):=\left|t\right|^{-\frac{1}{2}}\log\left|t\right|,$ (3.31)

and $C$ is a positive constant which depends on $n$ and $\alpha$.

(2). When $j=1$ and $\frac{\delta_{0}}{2}<\bar{r}_{i}\leq\delta_{0}$, by the
same argument as above, we have that

$\displaystyle\left|\frac{\Phi\big{(}\nu+\bar{t}_{i},y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}}{\Phi\big{(}\bar{t}_{i},\bar{y}_{i}+\gamma_{n}(\bar{t}_{i})\big{)}}\right|\leq
C.$

(3). When $j=0$, by the definitions of $\gamma_{n}(t)$ and $\Phi(t,r)$ in
(2.8) and (2.26), we have that

$\displaystyle\left|\frac{\Phi\big{(}\nu+\bar{t}_{i},y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}}{\Phi\big{(}\bar{t}_{i},\bar{y}_{i}+\gamma_{n}(\bar{t}_{i})\big{)}}\right|$
$\displaystyle\leq\frac{\hbar(\nu+\bar{t}_{i})}{\hbar(\bar{t}_{i})}\left(1+\left|\bar{y}_{i}-\frac{\alpha}{3}\log\left|\bar{t}_{i}\right|\right|\right)^{p}$
$\displaystyle\leq C(1+\left|y\right|)^{p},$

where we have used the fact that
$\left|\bar{y}_{i}-\frac{\alpha}{4}\log\left|t_{i}\right|\right|\leq\left|y\right|$
since $(\nu,y)\in B_{i}^{0}$ given by (3.30).

Combining above estimates, we derive that

$\left|\phi^{i}(\nu,y)\right|\leq C(\left|y\right|+1)^{p},$ (3.32)

for all $(\nu,y)\in B_{i}^{j}$ with $j=0,1$, where $p\in(n,n+1)$ and $C$ is a
positive constant only depending on $\alpha$ and $n$. Similarly, we have that

$\left|\phi^{i}_{y}(\nu,y)\right|\leq
C(\left|y\right|+1)^{p},\quad\left|\phi^{i}_{yy}(\nu,y)\right|\leq
C(\left|y\right|+1)^{p}\quad\text{and}\quad\left|\phi^{i}_{yyy}(\nu,y)\right|\leq
C(\left|y\right|+1)^{p},$ (3.33)

for all $(\nu,y)\in B_{i}^{j}$ and $C$ is a positive constant which only
depends on $\alpha$ and $n$. Notice that

$\cup_{j=0}^{1}B_{i}^{j}=\big{(}s_{i}-\bar{t}_{i},0\big{]}\times\big{(}-\bar{y}_{i}-\gamma_{n}(\nu+\bar{t}_{i}),\,+\infty\big{)}$

due to (3.30). Owing to (3.32) and (3.33), we have that

$\phi^{i}\rightarrow\phi\qquad\mbox{in}\
L_{\text{loc}}^{2}\big{[}(-\bar{\varsigma},0);H^{2}_{\text{loc}}(\mathbb{R})\big{]},$

where $\bar{\varsigma}=\varsigma$ or $\infty$. Moreover, we note that there
holds that

$\lim\limits_{i\rightarrow+\infty}-\gamma_{n}(\nu+\bar{t}_{i})-\bar{y}_{i}=M,$

up to a subsequence, where $M=-\infty$ or $M\in(-\infty,0]$. For the second
case, that is $M\in(-\infty,0]$, we can get that $\phi$ satisfies that

$\phi_{\nu}=-\phi_{yyyy}+2W^{\prime\prime}(1)\phi_{yy}-\big{[}W^{\prime\prime}(1)\big{]}^{2}\phi\qquad\text{in}\
(-\infty,0]\times[0,+\infty).$

Then, we can consider the function

$\widetilde{\phi}(\nu,x):=\phi(\nu,x)\quad\text{when}\ x\geq
0,\quad\text{and}\quad\widetilde{\phi}(\nu,x):=\phi(\nu,-x)\quad\text{when}\
x<0,$

which is the even extension of $\phi$ with respect to $x$. It is easy to check
that $\widetilde{\phi}$ satisfies equation (3.28).

For Case $2$: $\lim\limits_{i\rightarrow\infty}s_{i}-\bar{t}_{i}=-\varsigma$
for some constant $\varsigma>0$, we have $\phi^{i}\rightarrow\phi$ locally
uniformly, and $\phi$ satisfies

$\left\\{\begin{array}[]{l}\phi_{\nu}=-\phi_{yyyy}+2W^{\prime\prime}(1)\phi_{yy}-\big{[}W^{\prime\prime}(1)\big{]}^{2}\phi\qquad\text{in}\
(-\varsigma,0]\times\mathbb{R},\\\\[8.53581pt]
\phi(-\varsigma,y)=0,\qquad\text{for all}\ y\in\mathbb{R}.\end{array}\right.$

By Proposition 3.4, the above equation has a unique solution $\phi\equiv 0$.
However, we have that

$\left|\phi(0,0)\right|=\lim\limits_{i\rightarrow\infty}\left|\phi^{i}(0,0)\right|\geq\frac{1}{5}$

in (3.26), which leads to a contradiction. Hence the Case $2$ cannot happen.

Step two. We claim that

$\phi(\nu,y)\equiv 0,$ (3.34)

for all $(\nu,y)\in(-\infty,0]\times\mathbb{R}$. This result contradicts with
$\left|\phi(0,0)\right|>\frac{1}{5}$, thus we can derive that Lemma 3.5 holds.

In the rest part of this proof, the job is to show that the above conclusion
in (3.34) holds. Recall that $\alpha=\sqrt{W^{\prime\prime}(1)}>0$, we then
consider the following parabolic equation

$u_{\nu}=-u_{yyyy}+2\alpha^{2}u_{yy},\qquad\forall\,(\nu,y)\in(0,+\infty)\times\mathbb{R}.$

Using the Fourier transformation for $y$, the above equation has a formula
solution

$\mathbf{Q}(\nu,y)=\frac{1}{2\pi}\int_{\mathbb{R}}\exp\Big{\\{}-\nu\big{(}\left|\xi\right|^{4}+2\alpha^{2}\left|\xi\right|^{2}\big{)}\Big{\\}}e^{i\xi
y}{\mathrm{d}}\xi.$ (3.35)

Hence, for any $T>0$ and $f\in L^{\infty}((-T,0)\times\mathbb{R})$, the
following initial value problem

$u_{\nu}=-u_{yyyy}+2\alpha^{2}u_{yy}-\alpha^{4}u+f(\nu,y),\qquad(\nu,y)\in(-T,0)\times\mathbb{R},\qquad
u(-T,y)=0,\qquad\forall\,y\in\mathbb{R},$

has a solution of the form

$u(\nu,y)=\frac{1}{2\pi}\int_{0}^{\nu-
T_{1}}\int_{\mathbb{R}}e^{-\alpha^{4}\tau}\mathbf{Q}(\tau,x)f(\nu-\tau,y-x){\mathrm{d}}x{\mathrm{d}}\tau,\qquad\forall\,(\nu,y)\in(-T,0)\times\mathbb{R}.$
(3.36)

According to formula (3.36) and equation (3.26), we have that

$\phi^{i}(\nu,y)=\int_{0}^{\nu-
s_{i}+\bar{t}_{i}}\int_{\mathbb{R}}e^{-\alpha^{4}\tau}\mathbf{Q}(\tau,x)f_{i}\big{(}\phi^{i}(\nu-\tau,y-x),\,\nu-\tau,y-x\big{)}{\mathrm{d}}x{\mathrm{d}}\tau,$
(3.37)

for all $(\nu,y)\in(s_{i}-\bar{t}_{i},0)\times\mathbb{R}$, where the function
$f_{i}(\phi^{i},\nu,y)$ is defined as follows

$\displaystyle f_{i}(\phi^{i},\nu,y):=$
$\displaystyle-\frac{2(n-1)}{y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})}\phi_{yyy}^{i}-\frac{(n-3)(n-1)}{\big{(}y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}^{2}}\phi^{i}_{yy}$
$\displaystyle+\Bigg{[}\frac{2(n-1)W^{\prime\prime}\big{(}\bar{z}(\nu,y)\big{)}}{y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})}-\frac{(3-n)(n-1)}{\big{(}y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}^{3}}\Bigg{]}\phi^{i}_{y}$
$\displaystyle+2W^{\prime\prime\prime}\big{(}\bar{z}(\nu,y)\big{)}\bar{z}_{y}\phi^{i}_{y}-W^{\prime\prime\prime}\big{(}\bar{z}(\nu,y)\big{)}W^{\prime}\big{(}\bar{z}(\nu,y)\big{)}\phi+W^{(4)}\big{(}\bar{z}(\nu,y)\big{)}\left|\bar{z}_{y}\right|^{2}\phi^{i}$
$\displaystyle+2W^{\prime\prime\prime}\big{(}\bar{z}(\nu,y)\big{)}\bar{z}_{yy}\phi^{i}+2\frac{n-1}{y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})}W^{\prime\prime\prime}\big{(}\bar{z}(\nu,y)\big{)}\bar{z}_{y}\phi^{i}+\phi^{i}_{y}\partial_{\nu}\gamma_{n}(\nu+\bar{t}_{i})$
$\displaystyle+\frac{g^{i}\big{(}\nu+\bar{t}_{i},y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}}{\Phi\big{(}\bar{t}_{i},\bar{y}_{i}+\gamma_{n}(\bar{t}_{i})\big{)}}$
$\displaystyle+2\Big{[}W^{\prime\prime}\big{(}\bar{z}(\nu,y)\big{)}-W^{\prime\prime}(1)\Big{]}\phi^{i}_{yy}-\Big{(}\big{[}W^{\prime\prime}\big{(}\bar{z}(\nu,y)\big{)}\big{]}^{2}-\big{[}W^{\prime\prime}(1)\big{]}^{2}\Big{)}\phi^{i},$

where
$\bar{z}(\nu,y)=z\big{(}\nu+\bar{t}_{i},y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}$,
the function $z(\nu,r)$ is given by (2.14).

Using (3.18), (3.23), (3.32), (3.33), the same arguments in proof of Lemma
3.6, we have that

$\displaystyle\left|f_{i}\big{(}\phi^{i}(\nu,y),\nu,y\big{)}\right|\leq$
$\displaystyle
C(1+\left|y\right|)^{p}\Bigg{[}\exp\big{\\{}-\alpha\left|y+\bar{y}_{i}\right|\big{\\}}+\|g^{i}\|_{C_{\Phi}((s_{i},t_{i})\times(0,+\infty))}$
$\displaystyle\qquad\qquad+\frac{1}{\left[-(\nu+\bar{t}_{i})\right]^{3/4}}+\sum_{l=1}^{3}\big{(}y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{)}^{-l}\Bigg{]}\overline{\chi}_{\left\\{y>\frac{\delta_{0}}{2}-\gamma_{n}(\nu+\bar{t}_{i})-\bar{y}_{i}\right\\}},$

for all
$(\nu,y)\in(s_{i}-\bar{t}_{i},0]\times(-\bar{y}_{i}-\gamma_{n}(\nu+\bar{t}_{i}),+\infty)$,
where $C$ depends on $\alpha$ and $n$, and $\overline{\chi}_{A}$ denotes the
characteristic function of the set $A$.

Moreover a straightforward shift of contour gives that there exists a positive
constant $C$ such that

$\left|\mathbf{Q}(\nu,y)\right|\leq
C\nu^{-\frac{1}{4}}\exp\left\\{-\frac{\left|y\right|}{\nu^{1/4}}\right\\},$

or see $(3.2)$ in [4]. Thus, using (3.37) and the above inequality, we have
that

$\displaystyle\left|\phi^{i}(\nu,x)\right|$ $\displaystyle\leq C\int_{0}^{\nu-
s_{i}+\bar{t}_{i}}\int^{+\infty}_{\frac{\delta_{0}}{2}-\bar{y}_{i}-\gamma_{n}(\nu+\bar{t}_{i})}\tau^{-\frac{1}{4}}\exp\Bigg{\\{}-\frac{\left|x-y\right|}{\tau^{1/4}}-\alpha^{4}\tau\Bigg{\\}}\Bigg{[}\exp\big{\\{}-\alpha\left|y+\bar{y}_{i}\right|\big{\\}}$
(3.38)
$\displaystyle\qquad\quad+\frac{1}{\left[-(\nu+\bar{t}_{i}-\tau)\right]^{3/4}}+\|g^{i}\|_{C_{\Phi}((s_{i},t_{i})\times\mathbb{R})}+\sum_{l=1}^{3}\big{[}y+\bar{y}_{i}+\gamma_{n}(\nu+\bar{t}_{i})\big{]}^{-l}\Bigg{]}\big{(}1+\left|y\right|\big{)}^{p}{\mathrm{d}}y{\mathrm{d}}\tau.$

Thus, by Lebesgue dominated convergence theorem, the facts

$\left|\bar{y}_{i}\right|\rightarrow\infty,\qquad\bar{t}_{i}\rightarrow-\infty\qquad\mbox{and}\qquad\|g^{i}\|_{C_{\Phi}((s_{i},t_{i})\times(0,+\infty))}\rightarrow
0,$

when $i$ goes to infinity in (3.23), we can obtain (3.34). ∎

####  3.2.2. Proof of Assertion

We will prove (3.24) by contradiction arguments in the following five steps.
We assume that (3.24) is not valid. Then there exists a sequence
$\\{i_{m}\in\mathbb{N}\\}$ with $\lim\limits_{m\rightarrow\infty}i_{m}=\infty$
and $R>0$ such that there exists $l\in\\{0,1,2,3\\}$ satisfying

$\left\|\frac{\partial^{l}_{r}\psi^{i_{m}}}{\Phi}\right\|_{L^{\infty}\left(A^{\left(s_{i_{m}},t_{i_{m}}\right)}_{R}\right)}>0.$

Without loss of the generality, we assume that

$\Big{\|}\frac{\psi^{i_{m}}}{\Phi}\Big{\|}_{L^{\infty}\left(A^{\left(s_{i_{m}},t_{i_{m}}\right)}_{R}\right)}>0.$

For the other situations, the following arguments are similar.

Let $(\hat{t}_{i_{m}},r_{i_{m}})\in
A^{(s_{i_{m}},t_{i_{m}})}_{R}=\big{\\{}(t,r)\in(s_{i_{m}},t_{i_{m}})\times\mathbb{R}:\left|r-\gamma_{n}(t)\right|<R+1\big{\\}}$
such that there exist $\delta>0$ and

$\left|\frac{\psi^{i_{m}}(\hat{t}_{i_{m}},r_{i_{m}})}{\Phi(\hat{t}_{i_{m}},r_{i_{m}})}\right|>\delta>0.$
(3.39)

Let us introduce the following change of variables

$\nu=t-\hat{t}_{i_{m}},\quad
y=r-y_{i_{m}}-\rho(t+\hat{t}_{i_{m}})\quad\text{and}\quad
y_{i_{m}}=r_{i_{m}}-\gamma_{n}(\hat{t}_{i_{m}}),$

and set

$\widehat{\psi}^{i_{m}}(\nu,y):=\frac{\psi_{i_{m}}\big{(}\nu+\hat{t}_{i_{m}},\,y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\big{)}\,}{\Phi\big{(}\hat{t}_{i_{m}},y_{i_{m}}+\rho(\hat{t}_{i_{m}})\big{)}},$
(3.40)

where we recall that $\rho(t)=\gamma_{n}(t)+h(t)$ defined in (2.7). Hence,
there exists $y_{0}$ such that

$\lim\limits_{m\rightarrow\infty}y_{i_{m}}=y_{0}\quad\mbox{and}\quad\left|y_{0}\right|<R+2.$

According to (3.19), we have that $\widehat{\psi}^{i_{m}}$ satisfies the
following problem

$\displaystyle\widehat{\psi}^{i_{m}}_{\nu}=$
$\displaystyle-\widehat{\psi}^{i_{m}}_{yyyy}-\frac{2(n-1)}{y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})}\widehat{\psi}^{i_{m}}_{yyy}+\Bigg{[}2W^{\prime\prime}\big{(}\hat{z}(\nu,y)\big{)}-\frac{(n-3)(n-1)}{\big{(}y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\big{)}^{2}}\Bigg{]}\widehat{\psi}^{i_{m}}_{yy}$
$\displaystyle-\Big{(}W^{\prime\prime}\big{(}\hat{z}(\nu,y)\big{)}\Big{)}^{2}\widehat{\psi}^{i_{m}}+\Bigg{[}\frac{2(n-1)W^{\prime\prime}\big{(}\hat{z}(\nu,y)\big{)}}{y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})}-\frac{(3-n)(n-1)}{\big{(}y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\big{)}^{3}}\Bigg{]}\widehat{\psi}^{i_{m}}_{y}$
$\displaystyle+2W^{\prime\prime\prime}\big{(}\hat{z}(\nu,y)\big{)}\hat{z}_{yy}\widehat{\psi}^{i_{m}}+2\frac{n-1}{y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})}W^{\prime\prime\prime}\big{(}\hat{z}(\nu,y)\big{)}\hat{z}_{y}\widehat{\psi}^{i_{m}}$
$\displaystyle+2W^{\prime\prime\prime}\big{(}\hat{z}(\nu,y)\big{)}\hat{z}_{y}\widehat{\psi}^{i_{m}}_{y}-W^{\prime\prime\prime}\big{(}\hat{z}(\nu,y)\big{)}W^{\prime}\big{(}\hat{z}(\nu,y)\big{)}\widehat{\psi}^{i_{m}}+W^{(4)}\big{(}\hat{z}(\nu,y)\big{)}\left|\hat{z}_{y}\right|^{2}\widehat{\psi}^{i_{m}}$
$\displaystyle+\frac{g^{i_{m}}\big{(}\nu+\hat{t}_{i_{m}},\,y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\big{)}}{\Phi\big{(}\hat{t}_{i_{m}},y_{i_{m}}+\rho(\hat{t}_{i_{m}})\big{)}}-\partial_{\nu}\rho(\nu+\hat{t}_{i_{m}})\widehat{\psi}^{i_{m}}_{y}\qquad\text{in}\
\Gamma^{s_{i_{m}},t_{i_{m}}},$ (3.41)

with the conditions

$\displaystyle\left|\widehat{\psi}^{i_{m}}(0,0)\right|\geq\delta,\quad\widehat{\psi}^{i_{m}}(s_{i_{m}}-\hat{t}_{i_{m}},y)=0,\quad
y\in\big{(}-y_{i_{m}}-\rho(\nu+\hat{t}_{i_{m}}),+\infty\big{)},$ (3.42)

where

$\hat{z}(\nu,y)=z\big{(}\nu+\hat{t}_{i_{m}},y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\big{)},$

and

$\Gamma^{s_{i_{m}},t_{i_{m}}}:=\Big{\\{}(\nu,y)\in\big{(}s_{i_{m}}-\hat{t}_{i_{m}},\,0\big{]}\times\big{(}-y_{i_{m}}-\rho(\nu+\hat{t}_{i_{m}}),\,+\infty\big{)}\Big{\\}}.$

As the same as previous part,
$\lim\limits_{m\rightarrow\infty}(s_{i_{m}}-\hat{t}_{i_{m}})=\hat{\varsigma}$,
where $\hat{\varsigma}$ equal to $-\infty$ or a negative number.

Step $\mathbf{1}$. We first consider the case:
$\lim\limits_{m\rightarrow\infty}(s_{i_{m}}-\hat{t}_{i_{m}})=-\infty$. We have
that $\widehat{\psi}^{i_{m}}\rightarrow\widehat{\psi}$ locally uniformly,
$\left|\widehat{\psi}(0,0)\right|>\delta>0$, which $\widehat{\psi}$ satisfies
the following equation

$\displaystyle\widehat{\psi}_{\nu}=$
$\displaystyle-\widehat{\psi}_{yyyy}+2W^{\prime\prime}\big{(}\omega(y+y_{0})\big{)}\widehat{\psi}_{yy}+2W^{\prime\prime\prime}\big{(}\omega(y+y_{0})\big{)}\omega^{\prime}(y+y_{0})\widehat{\psi}_{y}-\Big{[}W^{\prime\prime}\big{(}\omega(y+y_{0})\big{)}\Big{]}^{2}\widehat{\psi}$
(3.43)
$\displaystyle+\Big{[}W^{\prime\prime\prime\prime}\big{(}\omega(y+y_{0})\big{)}\big{(}\omega^{\prime}(y+y_{0})\big{)}^{2}+W^{\prime\prime\prime}\big{(}\omega(y+y_{0})\big{)}\omega^{\prime\prime}(y+y_{0})\Big{]}\widehat{\psi},\qquad\text{in}\
(-\infty,0]\times\mathbb{R}.$

According to (2.26), (3.22) and (3.40), by the definition of $\rho$ in (2.7),
similar as (3.32), we have

$\displaystyle\left|\widehat{\psi}^{i_{m}}(\nu,y)\right|=$
$\displaystyle\left|\frac{\psi_{i_{m}}\big{(}\nu+\hat{t}_{i_{m}},y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\big{)}}{\Phi\big{(}\hat{t}_{i_{m}},y_{i_{m}}+\rho(\hat{t}_{i_{m}})\big{)}}\right|$
$\displaystyle\leq$
$\displaystyle\left|\frac{\Phi\big{(}\nu+\hat{t}_{i_{m}},y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\big{)}}{\Phi\big{(}\hat{t}_{i_{m}},y_{i_{m}}+\rho(\hat{t}_{i_{m}})\big{)}}\right|$
$\displaystyle\leq$ $\displaystyle
C\big{(}\alpha,n,R,\|h\|_{L^{\infty}}\big{)}\big{(}1+\left|y\right|\big{)}^{p},\qquad\forall\,(\nu,y)\in
B_{\hat{t}_{i_{m}},j},$ (3.44)

for all $j=0,1$, where $\rho(t)=\gamma_{n}(t)+h(t)$ in (2.7) and the sets
$B_{\hat{t}_{i_{m}},j}$ are defined as the following

$\displaystyle B_{\hat{t}_{i_{m}},0}$
$\displaystyle:=\Big{\\{}(\nu,y)\in(s_{i_{m}}-\hat{t}_{i_{m}},0]\times\mathbb{R}:0<y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\leq\delta_{0}\Big{\\}},$
$\displaystyle B_{\hat{t}_{i_{m}},1}$
$\displaystyle:=\Big{\\{}(\nu,y)\in(s_{i_{m}}-\hat{t}_{i_{m}},0]\times\mathbb{R}:\delta_{0}\leq
y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})<+\infty\Big{\\}}.$

We notice that

$\Gamma^{s_{i_{m}},t_{i_{m}}}=\cup_{l=0}^{1}B_{\hat{t}_{i_{m}},j}=\big{(}s_{i_{m}}-\hat{t}_{i_{m}},0\big{]}\times\big{(}-y_{i_{m}}-\rho(\nu+\hat{t}_{i_{m}}),\,+\infty\big{)}.$
(3.45)

Similarly, we have that

$\sum_{l=1}^{3}\left|\partial_{y}^{l}\widehat{\psi}^{i_{m}}(\nu,y)\right|\leq
C(1+\left|y\right|)^{p},$ (3.46)

for all $(\nu,y)\in\Gamma^{s_{i_{m}},t_{i_{m}}}$, where $p\in(n,n+1)$ and $C$
depends on $\alpha,R$, $n$ and $\|h\|_{L^{\infty}}$.

Since $\gamma_{n}(\nu+\hat{t}_{i_{m}})\rightarrow+\infty$ as $i$ goes to
infinity, by (3.46), (3.41)-(3.42), and $y_{i_{m}}\rightarrow y_{0}$, we can
get the limiting equation in (3.43). If $\hat{\varsigma}\in(-\infty,0)$, we
have $\hat{\psi}(\hat{\varsigma},y)=0$. According to Proposition 3.4, we have
that equation (3.43) has a unique solution $\hat{\psi}(\nu,y)\equiv 0$, which
contradicts with $\left|\hat{\psi}(0,0)\right|>0$. Hence, it only happens that
$\hat{\varsigma}=-\infty$. Thus, we get the desired result.

Step $\mathbf{2}$. We will prove the following orthogonality condition for
$\widehat{\psi}$

$\int_{\mathbb{R}}\widehat{\psi}(\nu,y)\omega^{\prime}(y+y_{0}){\mathrm{d}}y=0,\qquad\text{for
all}\ \nu\in(-\infty,0].$ (3.47)

In fact, according to (3.20) and (3.40), we get that

$\displaystyle 0$
$\displaystyle=\frac{\left[y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\right]^{1-n}}{\Phi\big{(}\hat{t}_{i_{m}},y_{i_{m}}+\rho(\hat{t}_{i_{m}})\big{)}}\int_{0}^{\infty}\psi^{i_{m}}(\nu+\hat{t}_{i_{m}},r)\omega^{\prime}\big{(}r-\rho(\nu+\hat{t}_{i_{m}})\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle=\int_{-y_{i_{m}}-\rho(\nu+\hat{t}_{i_{m}})}^{\infty}\widehat{\psi}^{i_{m}}(\nu,y)\omega^{\prime}(y+y_{i_{m}})\left[\frac{y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})}{y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})}\right]^{n-1}{\mathrm{d}}y,$

where $\rho(t)=\gamma_{n}(t)+h(t)$ with $\|h(t)\|_{L^{\infty}}<1$ and the
function $\omega$ is given by (1.11). And using (3.2.2), (3.45) and Lemma 2.1,
we also get that

$\displaystyle\left|\widehat{\psi}_{i_{m}}(\nu,y)\omega^{\prime}(y+y_{i_{m}})\right|\leq
C(R,n,\alpha,\|h\|_{L^{\infty}})(1+\left|y\right|)^{p}\exp\big{\\{}-\alpha\left|y+y_{i_{m}}\right|\big{\\}}.$

Since $y_{i_{m}}\rightarrow y_{0}$ and $\hat{t}_{i_{m}}\rightarrow-\infty$, as
$m$ goes to infinity, using Lebesgue Dominated Convergence Theorem, we can
obtain that (3.47) holds.

Step $\mathbf{3}$. In this step, we will prove the following decay of
$\widehat{\psi}(\nu,y)$: there exists a constant $C$ of the form

$C=C\big{(}\alpha,n,R,\|h\|_{L^{\infty}}\big{)}>0$

such that

$\left|\widehat{\psi}(\nu,y)\right|\leq\frac{C}{\left(1+\left|y\right|\right)^{2}},\qquad\forall\,(\nu,y)\in(-\infty,0]\times\mathbb{R}.$
(3.48)

In fact, for any $(\nu,y)\in B_{t_{i_{m}},j}$, by the definition of $\rho(t)$
in (2.7), in view of the proof of (3.2.2), we have

$\left|\frac{g^{i_{m}}\big{(}\nu+\hat{t}_{i_{m}},y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\big{)}}{\Phi\big{(}\hat{t}_{i_{m}},y_{i_{m}}+\rho(\hat{t}_{i_{m}})\big{)}}\right|\leq
C\|g^{i_{m}}\|_{C_{\Phi}((s_{i_{m}},\,t_{i_{m}})\times(0,+\infty))}(1+\left|y\right|)^{p},\
$ (3.49)

for all $(\nu,y)\in\Gamma^{s_{i_{m}},t_{i_{m}}}$ given by (3.45), where $C$
depends on $\alpha,R,n$ and $\|h\|_{L^{\infty}}$.

By formula (3.36), the solution of equation (3.41) has the form

$\widehat{\psi}_{i_{m}}(\nu,y)=\int_{0}^{\nu-
s_{i_{m}}+\hat{t}_{i_{m}}}\int_{\mathbb{R}}e^{-\alpha^{4}\tau}\mathbf{Q}(\tau,x)\hat{f}_{i_{m}}\big{(}\hat{\psi}^{i_{m}}(\nu-\tau,y-x),\nu-\tau,y-x\big{)}{\mathrm{d}}x{\mathrm{d}}\tau,$
(3.50)

for all $(\nu,y)\in(s_{i_{m}}-\hat{t}_{i_{m}},0)\times\mathbb{R}$, where
$\alpha=\sqrt{W^{\prime\prime}(1)}$ and the function $\hat{f}_{i_{m}}$ is
given by

$\displaystyle\hat{f}_{i_{m}}=$ $\displaystyle
2\Big{[}W^{\prime\prime}\big{(}\hat{z}(\nu,y)\big{)}-W^{\prime\prime}(1)\Big{]}\widehat{\psi}^{i_{m}}_{yy}+2W^{\prime\prime\prime}\big{(}\hat{z}(\nu,y)\big{)}\partial_{y}\hat{z}(\nu,y)\widehat{\psi}^{i_{m}}_{y}-\Big{(}\big{[}W^{\prime\prime}\big{(}\hat{z}(\nu,y)\big{)}\big{]}^{2}-\big{[}W^{\prime\prime}(1)\big{]}^{2}\Big{)}\widehat{\psi}^{i_{m}}$
$\displaystyle+\Big{\\{}\partial_{yy}W^{\prime\prime}\big{(}\hat{z}(\nu,y)\big{)}+W^{\prime\prime\prime}\big{(}\hat{z}(\nu,y)\big{)}\big{[}\partial_{yy}\hat{z}(\nu,y)-W^{\prime}\big{(}\hat{z}(\nu,y)\big{)}\big{]}\Big{\\}}\widehat{\psi}^{i_{m}}$
$\displaystyle-\frac{2(n-1)}{y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})}\widehat{\psi}^{i_{m}}_{yyy}-\frac{(n-3)(n-1)}{\big{(}y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\big{)}^{2}}\widehat{\psi}^{i_{m}}_{yy}$
$\displaystyle+\Bigg{[}\frac{2(n-1)W^{\prime\prime}\big{(}\hat{z}(\nu,y)\big{)}}{y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})}-\frac{(3-n)(n-1)}{\big{(}y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\big{)}^{3}}\Bigg{]}\widehat{\psi}^{i_{m}}_{y}$
$\displaystyle+2\frac{n-1}{y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})}W^{\prime\prime\prime}\big{(}\hat{z}(\nu,y)\big{)}\hat{z}_{y}\widehat{\psi}^{i_{m}}+\frac{g^{i_{m}}\big{(}\nu+\hat{t}_{i_{m}},y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\big{)}}{\Phi\big{(}\hat{t}_{i_{m}},y_{i_{m}}+\rho(\hat{t}_{i_{m}})\big{)}}$
$\displaystyle-\partial_{\nu}\rho(\nu+\hat{t}_{i_{m}})\widehat{\psi}^{i_{m}}_{y}$

and
$\hat{z}(\nu,y)=z\big{(}\nu+\hat{t}_{i_{m}},y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\big{)}$.

According to the lemma 2.1, (3.18), (3.49), (3.2.2) and (3.46), we have that

$\displaystyle\left|\hat{f}_{i_{m}}\right|\leq$ $\displaystyle
C(1+\left|y\right|)^{p}\Bigg{[}\exp\big{\\{}-\alpha\left|y+y_{i_{m}}\right|\big{\\}}+\|g^{i_{m}}\|_{C_{\Phi}((s_{i_{m}},t_{i_{m}})\times(0,+\infty))}$
$\displaystyle\qquad\qquad+\frac{1}{\left[-(\nu+\hat{t}_{i_{m}})\right]^{3/4}}+\sum_{l=1}^{3}\left(y+y_{i_{m}}+\rho(\nu+\hat{t}_{i_{m}})\right)^{-l}\Bigg{]}\overline{\chi}_{\left\\{y>\frac{\delta_{0}}{2}-\rho(\nu+\hat{t}_{i_{m}})-y_{i_{m}}\right\\}},$

for all
$(\nu,y)\in\big{(}s_{i_{m}}-\hat{t}_{i_{m}},0\big{)}\times(-\rho(\nu+\hat{t}_{i_{m}})-y_{i_{m}},+\infty)$,
where $C$ depends on $\alpha,R,n$ and $\|h\|_{L^{\infty}}$, and
$\overline{\chi}_{A}$ is the characteristic function of the set $A$.

Thus by (3.50), similar arguments as (3.38) and $\left|y_{i_{m}}\right|\leq
R+2$, let $m$ goes to infinity, and then we can get the desired result since
$\|g^{i_{m}}\|_{C_{\Phi}((s_{i_{m}},\bar{t}_{i_{m}})\times(0,+\infty))}\rightarrow
0$ as index $m$ goes to infinity in (3.23) and the following fact that

$\displaystyle\int_{0}^{\nu+\check{t}_{i_{m}}}\int_{\mathbb{R}}e^{-\alpha^{4}\tau}\left|\mathbf{Q}(\tau,y-x)\right|\big{(}1+\left|x\right|\big{)}^{p}\exp\big{\\{}-\alpha\left|x+y_{i_{m}}\right|\big{\\}}{\mathrm{d}}x{\mathrm{d}}\tau$
$\displaystyle\leq
C\int_{0}^{\nu+\check{t}_{i_{m}}}\int_{\mathbb{R}}\exp\left\\{-\alpha^{4}\tau-\frac{1}{4}\left|x\right|\right\\}\exp\left\\{-\frac{\alpha}{2}\left|y-\tau^{1/4}x\right|\right\\}{\mathrm{d}}x{\mathrm{d}}\tau$
$\displaystyle\leq
C\int_{0}^{\nu+\check{t}_{i_{m}}}\int_{\mathbb{R}}\exp\left\\{-\alpha^{4}\tau-\frac{1}{4}\left|x\right|\right\\}\frac{1}{\left(1+\left|y-\tau^{1/4}x\right|\right)^{2}}{\mathrm{d}}x{\mathrm{d}}\tau$
$\displaystyle\leq
C\int_{0}^{\nu+\check{t}_{i_{m}}}\int_{\mathbb{R}}\exp\left\\{-\alpha^{4}\tau-\frac{1}{4}\left|x\right|\right\\}\left(\frac{1+\left|\tau^{1/4}x\right|}{1+\left|y\right|}\right)^{2}{\mathrm{d}}x{\mathrm{d}}\tau$
$\displaystyle\leq\frac{C}{\left(1+\left|y\right|\right)^{2}},$

where $\check{t}_{i_{m}}:=\hat{t}_{i_{m}}-s_{i_{m}}$ and $C>0$ only depends on
$\alpha$.

Step $\mathbf{4}$. To proceed further, we need the following result. However,
we can not find the proof of this result in references. Here we give a proof
in details.

######  Lemma 3.6.

Considering the Hilbert space

$H=\left\\{\phi(y)\in
H^{2}(\mathbb{R}):\int_{\mathbb{R}}\phi(y)\omega^{\prime}(y){\mathrm{d}}y=0\right\\},$

then the following inequality is valid

$\int_{\mathbb{R}}\left|\phi^{\prime\prime}(y)-W^{\prime\prime}(\omega)\phi(y)\right|^{2}{\mathrm{d}}y\geq
c\int_{\mathbb{R}}\left|\phi(y)\right|^{2}{\mathrm{d}}y,\qquad\forall\,\phi\in
H,$ (3.51)

where $c>0$.

###### Proof.

For any $\phi\in H$, since $\omega^{\prime}>0$, we can set
$\phi(y)=\zeta_{1}(y)\omega^{\prime}(y)$, then

$\displaystyle\int_{\mathbb{R}}\left|\phi^{\prime\prime}(y)-W^{\prime\prime}(\omega)\phi(y)\right|^{2}{\mathrm{d}}y=$
$\displaystyle\int_{\mathbb{R}}\left|\omega^{\prime\prime\prime}(y)\zeta_{1}(y)+2\zeta^{\prime}_{1}(y)\omega^{\prime\prime}(y)+\zeta^{\prime\prime}_{1}(y)\omega^{\prime}(y)-W^{\prime\prime}(\omega)\zeta_{1}(y)\omega^{\prime}(y)\right|^{2}{\mathrm{d}}y$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\left|2\omega^{\prime\prime}(y)\zeta_{1}^{\prime}(y)+\omega^{\prime}(y)\zeta_{1}^{\prime\prime}(y)\right|^{2}{\mathrm{d}}y\geq
0,$

where we have used
$\omega^{\prime\prime\prime}=W^{\prime\prime}(\omega)\omega^{\prime}$ and
integration by parts in the last equality.

Hence we have

$\int_{\mathbb{R}}\left|\phi^{\prime\prime}(y)-W^{\prime\prime}(\omega)\phi(y)\right|^{2}{\mathrm{d}}y=0\quad\text{if
and only if}\quad\
2\omega^{\prime\prime}(y)\zeta_{1}^{\prime}(y)+\omega^{\prime}(y)\zeta_{1}^{\prime\prime}(y)=0.$

All solutions of the last equation have the following form

$\zeta_{1}(x)=c_{1}\int_{0}^{x}\frac{1}{(\omega^{\prime}(y))^{2}}{\mathrm{d}}y+c_{2},\qquad
c_{1},c_{2}\in\mathbb{R}.$

Since

$\phi(x)=\zeta_{1}(x)\omega^{\prime}(x)\in
H^{2}(\mathbb{R})\qquad\mbox{and}\qquad\int_{\mathbb{R}}\phi(y)\omega^{\prime}(y){\mathrm{d}}y=0,$

we have that $c_{1}=0$ and $c_{2}=0$.

Now we assume that there exists a sequence $\\{\phi_{m}\\}\in H$ such that

$\int_{\mathbb{R}}\left|\phi_{m}\right|^{2}{\mathrm{d}}y=1\ \ \text{and}\ \
\int_{\mathbb{R}}\left|\phi^{\prime\prime}_{m}-W^{\prime\prime}(\omega)\phi_{m}\right|^{2}{\mathrm{d}}y\leq\frac{1}{m}.$
(3.52)

Thus, $\phi_{m}\rightharpoonup\phi$ in $H(\mathbb{R})$ and
$\phi_{m}\rightarrow\phi$ in $L^{2}(K)$ for any compact subset
$K\subset\mathbb{R}$, which implies that

$0=\int_{\mathbb{R}}\phi_{m}(y)\omega^{\prime}(y){\mathrm{d}}y\rightarrow\int_{\mathbb{R}}\phi\omega^{\prime}{\mathrm{d}}y=0,$

for some $\phi\in H(\mathbb{R})$. By Fatou’s Lemma, we have

$\int_{\mathbb{R}}\left|\phi^{\prime\prime}-W^{\prime\prime}(\omega)\phi\right|^{2}{\mathrm{d}}y=0.$

Hence $\phi=0$.

On the other hand, we first have

$\displaystyle\big{|}W^{\prime\prime}(1)\big{|}^{2}\int_{\mathbb{R}}\left|\phi_{m}\right|^{2}{\mathrm{d}}y$
$\displaystyle\leq\int_{\mathbb{R}}\big{|}\phi^{\prime\prime}_{m}-W^{\prime\prime}(1)\phi_{m}\big{|}^{2}{\mathrm{d}}y$
$\displaystyle\leq
2\Bigg{\\{}\int_{\mathbb{R}}\big{|}\phi^{\prime\prime}_{m}-W^{\prime\prime}(\omega)\phi_{m}\big{|}^{2}{\mathrm{d}}y+\int_{\mathbb{R}}\Big{|}\big{[}W^{\prime\prime}(1)-W^{\prime\prime}(\omega)\big{]}\phi_{m}\Big{|}^{2}{\mathrm{d}}y\Bigg{\\}}.$

Thus, by (3.52), we get that

$\big{|}W^{\prime\prime}(1)\big{|}^{2}\leq\int_{\mathbb{R}}\Big{|}\big{[}W^{\prime\prime}(1)-W^{\prime\prime}(\omega)\big{]}\phi\Big{|}^{2}{\mathrm{d}}y.$

According to the assumptions on $W$ in (1.2), $W^{\prime\prime}(1)\neq 0$. The
above last inequality implies that $\phi\neq 0$, which is a contradiction. ∎

Step $\mathbf{5}$. We will prove that (3.24) holds. Multiplying (3.43) by
$\hat{\psi}(y)$ and integrating with respect to $y$, and using (3.51), we have

$\displaystyle 0=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}}\big{(}\widehat{\psi}^{2}\big{)}_{\nu}{\mathrm{d}}y+\int_{\mathbb{R}}\left|\widehat{\psi}_{yy}-W^{\prime\prime}\big{(}\omega(y+y_{0})\big{)}\widehat{\psi}(\nu,y)\right|^{2}{\mathrm{d}}y$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}}\big{(}\widehat{\psi}^{2}\big{)}_{\nu}{\mathrm{d}}y+c\int_{\mathbb{R}}\left|\widehat{\psi}(\nu,y)\right|^{2}{\mathrm{d}}y,$

where the constant $c>0$. Then, according to the Gronwall inequality, we get
that

$a(\nu)\geq a(0)e^{-2c\nu},\ \ \ \ \forall\,\nu<0,\ $

where the function

$a(\nu):=\int_{\mathbb{R}}\left|\widehat{\psi}(\nu,y)\right|^{2}{\mathrm{d}}y.$

This is a contradiction since (3.48). The proof of (3.24) is completed.

###  3.3. The projected linear problem involving the operator $L$

We consider the following projection problem:

$\left\\{\begin{array}[]{l}\psi_{t}=L[\psi]+g(t,r)-c(t)\partial_{r}\widehat{\omega}(t,r)\qquad\text{in}\
(s,-T]\times(0,\infty),\\\\[5.69054pt]
\psi(s,r)=0,\qquad\forall\,r\in(0,+\infty),\\\\[5.69054pt]
\partial_{r}^{j}\psi(t,0)=0,\qquad\forall\,t\in(s,-T],\ \
j=1,3,\end{array}\right.$ (3.53)

where the linear operator $L[\psi]=F^{\prime}\big{(}z(t,r)\big{)}[\psi]$ is
defined by (2.20), the function $g\in C_{\Phi}((s,-T)\times(0,+\infty))$ and
$c(t)$ satisfies the following relation

$\displaystyle
c(t)\int_{0}^{\infty}\partial_{r}\widehat{\omega}(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle=\int_{0}^{\infty}\Big{[}\omega^{\prime\prime\prime}\big{(}r-\rho(t)\big{)}+\frac{n-1}{r}\omega^{\prime\prime}\big{(}r-\rho(t)\big{)}-W^{\prime\prime}\big{(}z(t,r)\big{)}\omega^{\prime}\big{(}r-\rho(t)\big{)}\Big{]}$
$\displaystyle\qquad\qquad\times\left[-\psi_{rr}-\frac{n-1}{r}\psi_{r}+W^{\prime\prime}\big{(}z(t,r)\big{)}\psi\right]r^{n-1}{\mathrm{d}}r$
$\displaystyle\quad+\int_{0}^{\infty}\left[\partial_{rr}z(t,r)+\frac{n-1}{r}\partial_{r}z(t,r)-W^{\prime}\big{(}z(t,r)\big{)}\right]\psi\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle\quad+\int_{0}^{\infty}\psi(t,r)\partial_{t}\big{[}\omega^{\prime}\big{(}r-\rho(t)\big{)}\big{]}r^{n-1}{\mathrm{d}}r\,+\,\int_{0}^{\infty}g(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r,$
(3.54)

for all $t\in(s,-T]$, where the functions $\widehat{\omega}(t,r)$ and $z(t,r)$
are defined by (2.6) and (2.14) respectively.

If $\psi$ is a solution of (3.53) and $c(t)$ satisfies (3.54), integration by
parts will imply that $\psi$ satisfies the following the orthogonality
condition

$\int_{0}^{\infty}\psi(t,r)\omega^{\prime}\big{(}r-\rho_{i}(t)\big{)}r^{n-1}{\mathrm{d}}r=0,\qquad\forall\,t\in(s,-T).$
(3.55)

For (3.54), we have the following result:

######  Lemma 3.7.

Let $T>0$ big enough and $\psi,\psi_{r},\psi_{rr}$ and $g\in
C_{\Phi}((s,-T)\times(0,+\infty))$. Then there exists $c(t)$ such that
$(\ref{eq3.20})$ holds. Furthermore the following estimates are valid

$\displaystyle\left|c(t)\right|\leq\frac{C_{0}}{\big{[}\log\left|t\right|\big{]}^{p-1}\left|t\right|^{1/2}}\left\\{\frac{1}{\left|t\right|^{1/4}}\left[\sum_{l=0}^{2}\big{\|}\partial_{r}^{l}\psi\big{\|}_{C_{\Phi}((s,-T)\times(0,\infty))}\right]\
+\ \|g\|_{C_{\Phi}((s,-T)\times(0,\infty))}\right\\},$ (3.56)

and

$\displaystyle\left|\frac{c(t)\partial_{r}\widehat{\omega}(t,r)}{\Phi(t,r)}\right|\,\leq\,$
$\displaystyle\frac{C_{0}}{\left|t\right|^{1/4}}\sum_{l=0}^{2}\big{\|}\partial_{r}^{l}\psi\big{\|}_{C_{\Phi}((s,-T)\times(0,\infty))}\
+\ C_{0}\|g\|_{C_{\Phi}((s,-T)\times(0,\infty))},$ (3.57)

for any $t\in[s,-T]$, where the function $\widehat{\omega}(t,r)$ is given by
(2.6), $p\in(n,n+1]$ and $C_{0}$ is a positive constant which does not depend
on $s,t,T$, $\psi$ and $g$.

###### Proof.

We first consider the left hand side of system $(\ref{eq3.20})$. By Lemma 2.1
and the definition of $\widehat{\omega}(t,r)$ in (2.6), we have that

$\displaystyle\int_{0}^{\infty}\partial_{r}\widehat{\omega}(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle=\int_{\delta_{0}-\rho(t)}^{\infty}\omega^{\prime}(r)\omega^{\prime}(r)\big{(}r+\rho(t)\big{)}^{n-1}{\mathrm{d}}r\
+\ O\left([\gamma_{n}(t)]^{n-2}\right)$
$\displaystyle=\left(\rho(t)\right)^{n-1}\int_{\mathbb{R}}\big{[}\omega^{\prime}(y)\big{]}^{2}dy\
+\ O\left([\gamma_{n}(t)]^{n-2}\right),$

where the functions $\rho(t)$ and $\gamma_{n}(t)$ are given by (2.7) and (2.8)
respectively.

For the first and second terms in the right hand side of (3.54), we can obtain
that

$\displaystyle I(t):=$
$\displaystyle\int_{0}^{\infty}\Big{[}-\omega^{\prime\prime\prime}\big{(}r-\rho_{i}(t)\big{)}-\frac{n-1}{r}\omega^{\prime\prime}\big{(}r-\rho_{i}(t)\big{)}+W^{\prime\prime}\big{(}z(t,r)\big{)}\omega^{\prime}\big{(}r-\rho(t)\big{)}\Big{]}$
$\displaystyle\qquad\qquad\times\left(\psi_{rr}+\frac{n-1}{r}\psi_{r}-W^{\prime\prime}\big{(}z(t,r)\big{)}\psi\right)r^{n-1}{\mathrm{d}}r$
$\displaystyle+\int_{0}^{\infty}\left[\partial_{rr}z(t,r)+\frac{n-1}{r}\partial_{r}z(t,r)-W^{\prime}\big{(}z(t,r)\big{)}\right]\psi\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle\leq$ $\displaystyle
C\sum_{l=0}^{2}\big{\|}\partial_{r}^{l}\psi\big{\|}_{C_{\Phi}((s,-T)\times(0,\infty))}\int_{0}^{\infty}\Bigg{[}\left|\omega^{\prime\prime\prime}\big{(}r-\rho(t)\big{)}-W^{\prime\prime}\big{(}z(t,r)\big{)}\omega^{\prime}\big{(}r-\rho(t)\big{)}\right|$
$\displaystyle\qquad\qquad\qquad\qquad+\left|\omega^{\prime}\big{(}r-\rho(t)\big{)}\right|\left|\partial_{rr}z(t,r)-W^{\prime}\big{(}z(t,r)\big{)}\right|\Bigg{]}\Phi(t,r)r^{n-1}{\mathrm{d}}r$
$\displaystyle\,+\,C\sum_{l=0}^{2}\big{\|}\partial_{r}^{l}\psi\big{\|}_{C_{\Phi}((s,-T)\times(0,\infty))}\int_{0}^{\infty}\Phi(t,r)\sum_{l=1}^{3}\left|\omega^{(l)}\big{(}r-\rho(t)\big{)}\right|r^{n-2}{\mathrm{d}}r,$

due to the facts that the functions $\psi,\psi_{r},\psi_{rr}$ belong to
$C_{\Phi}((s,-T)\times(0,+\infty))$. By similar arguments in the proof of
Lemma 2.3 and the definition of $\Phi(t,r)$ in (2.26), we have that

$\displaystyle I(t)$ $\displaystyle\leq
C\sum_{l=0}^{2}\big{\|}\partial_{r}^{l}\psi\big{\|}_{C_{\Phi}((s,-T)\times(0,\infty))}\Bigg{\\{}\int_{0}^{\infty}\Phi(t,r)\sum_{l=1}^{3}\left|\omega^{(l)}\big{(}r-\rho(t)\big{)}\right|r^{n-2}{\mathrm{d}}r$
$\displaystyle\qquad\qquad+\int_{0}^{\infty}\Phi(t,r)\left|\omega^{\prime}\big{(}r-\rho(t)\big{)}\right|r^{n-3}{\mathrm{d}}r\Bigg{\\}}$
$\displaystyle\leq
C\frac{\big{[}\rho(t)\big{]}^{n-1}}{\left|t\right|^{3/4}\big{[}\log\left|t\right|\big{]}^{p-1}}\left\\{\sum_{l=0}^{2}\big{\|}\partial_{r}^{l}\psi\big{\|}_{C_{\Phi}((s,-T)\times(0,\infty))}\right\\}.$

Similarly, by the definitions in (2.7) and (2.26) again, we have that

$\displaystyle\int_{0}^{\infty}\psi(t,r)\partial_{t}\big{[}\omega^{\prime}\big{(}r-\rho(t)\big{)}\big{]}r^{n-1}{\mathrm{d}}r$
$\displaystyle=-\rho^{\prime}(t)\int_{0}^{\infty}\psi(t,r)\omega^{\prime\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle=O\left\\{\frac{\big{[}\rho(t)\big{]}^{n-1}}{\left|t\right|^{5/4}}\right\\}\|\psi\|_{C_{\Phi}((s,-T)\times(0,\infty))}.$

And, there holds that

$\displaystyle\left|\int_{0}^{\infty}g(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r\right|\leq$
$\displaystyle
C\int_{0}^{\infty}\Phi(t,r)\left|\omega^{\prime}\big{(}r-\rho(t)\big{)}\right|r^{n-1}{\mathrm{d}}r$
$\displaystyle\leq$ $\displaystyle
C\frac{\big{[}\rho(t)\big{]}^{n-1}}{\left|t\right|^{1/2}\big{[}\log\left|t\right|\big{]}^{p-1}}.$

According the above estimates and (3.54), we can get the validity of (3.56).

Using the following fact

$\left|\frac{\partial_{r}\widehat{\omega}(t,r)}{\Phi(t,r)}\right|\leq
C\left|t\right|^{1/2}\big{[}\log\left|t\right|\big{]}^{p-1},\qquad\text{for
all}\ r>0,$

where $\widehat{\omega}(t,r)$ is defined by (2.6) and $C$ is a constant which
does not depend on $t$, we can obtain (3.57). ∎

According to Lemma 3.7, we have

######  Lemma 3.8.

Let $g\in C_{\Phi}((s,-T)\times(0,\infty))$, then there exists a uniform
constant $T_{0}>0$ and a unique solution $\psi^{s}$ of problem
$(\ref{eq3.19})$. Moreover, $\psi^{s}$ satisfies the orthogonality conditions
in $(\ref{eq3.21})$ with $t\in(s,-T_{0})$, and

$\sum_{l=0}^{3}\big{\|}\partial_{r}^{l}\psi^{s}\big{\|}_{C_{\Phi}((s,t)\times(0,\infty))}\leq
C\|g\|_{C_{\Phi}((s,t)\times(0,\infty))},$ (3.58)

where $t\in(s,-T_{0})$ and $C$ is a uniform constant which does not depend on
$t,s$.

###### Proof.

We will use a fixed-point argument to prove this lemma. Let

$X^{s}:=\Big{\\{}\psi\,:\,\sum_{l=0}^{3}\big{\|}\partial_{r}^{l}\psi\big{\|}_{C_{\Phi}((s,s+1)\times(0,\infty))}<+\infty\Big{\\}},$

and $\mathcal{T}^{s}(g)$ be the solution of (3.19) with $-T=s+1$. We consider
the operator $\mathcal{A}^{s}:X^{s}\rightarrow X^{s}$ defined by

$\mathcal{A}^{s}(\psi):=\mathcal{T}^{s}(g-C(\psi)),\qquad
C(\psi)=c(t)\partial_{r}\widehat{\omega}(t,r),$

where $\widehat{\omega}(t,r)$ is given by (2.6) and the function $c(t)$
satisfies (3.54). Hence $c(t)$ depends on $\psi$, and $\psi$ satisfies the
orthogonality conditions in (3.55). By Lemma 3.5, we have that

$\sum_{l=0}^{3}\big{\|}\partial_{r}^{l}\mathcal{A}^{s}(\psi)\big{\|}_{C_{\Phi}((s,s+1)\times(0,\infty))}\leq
C_{1}\|g-C(\psi)\|_{C_{\Phi}((s,s+1)\times(0,\infty))},$ (3.59)

for a uniform constant $C_{1}>0$.

Set

$c:=(C_{0}+C_{1})\|g\|_{C_{\Phi}((s,s+1)\times(0,\infty))}$

and

$X^{s}_{c}:=\big{\\{}\psi:\|\psi\|_{C_{\Phi}((s,s+1)\times(0,\infty))}<2c\big{\\}},$

where $C_{0}$ and $C_{1}$ are given by Lemma 3.7 and (3.59) respectively. It
is easy to derive the following two facts:

(1). By (3.59) and Lemma 3.7, we have

$\displaystyle\sum_{l=0}^{3}\big{\|}\partial^{l}_{r}\mathcal{A}^{s}(\psi)\big{\|}_{C_{\Phi}((s,s+1)\times(0,\infty))}\leq$
$\displaystyle
C_{0}\Big{(}\|C(\psi)\|_{C_{\Phi}((s,s+1)\times(0,\infty))}+\|h\|_{C_{\Phi}((s,s+1)\times(0,\infty))}\Big{)}$
$\displaystyle\leq$
$\displaystyle\frac{CC_{0}}{\sqrt[4]{\left|s+1\right|}}\Big{(}\sum_{l=0}^{2}\big{\|}\partial_{r}^{l}\psi\big{\|}_{C_{\Phi}((s,s+1)\times(0,\infty))}\Big{)}+c,$

where $C$ is given by Lemma 3.7.

(2). For any $\psi_{1},\psi_{2}\in X^{s}_{c}$, by (3.59) and Lemma 3.7 again,
we have

$\displaystyle\sum_{l=0}^{3}\big{\|}\partial^{l}_{r}\left[\mathcal{A}^{s}(\psi_{1})-\mathcal{A}^{s}(\psi_{2})\right]\big{\|}_{C_{\Phi}((s,s+1)\times(0,\infty))}\leq$
$\displaystyle
C_{0}\,\|C(\psi_{1})-C(\psi_{2})\|_{C_{\Phi}((s,s+1)\times(0,\infty))}$
$\displaystyle\leq$ $\displaystyle
C_{0}\,\|C(\psi_{1}-\psi_{2})\|_{C_{\Phi}((s,s+1)\times(0,\infty))}$
$\displaystyle\leq$
$\displaystyle\frac{CC_{0}}{\sqrt[4]{\left|s+1\right|}}\,\|\psi_{1}-\psi_{2}\|_{C_{\Phi}((s,s+1)\times(0,\infty))},$

where $C_{0}$ is given by Lemma 3.7.

Hence, taking $s$ large enough, we have that the operator $\mathcal{A}^{s}$ is
a contraction map from $X^{s}_{c}$ to itself. According to the Banach Fixed
Theorem, we know that there exists a unique $\psi^{s}\in X^{s}_{c}$ such that
$\mathcal{A}^{s}(\psi^{s})=\psi^{s}$, which is a solution to problem (3.53)
with $-T=s+1$.

Next we will extend the solution $\psi^{s}(t,r)$ in $(s,s+1)\times(0,+\infty)$
to $(s,-T_{0})\times(0,+\infty)$, which will satisfy the orthogonality
condition in (3.54) and the a priori estimate in Lemma 3.5. We choose $T_{0}$
large enough such that $\frac{CC_{0}}{\sqrt[4]{T_{0}}}<1$, where $C$ is given
by Lemma 3.7 and $C_{0}$ is given by (3.59). Thus the above fixed-point
argument can be repeated when $s+1\leq-T_{0}$. Hence passing finite steps of
fixed-point arguments, the solution $\psi^{s}(t,r)$ can be extended to
$(s,-T_{0}]$. Moreover the solution $\psi^{s}$ satisfies (3.58) and the
orthogonality condition. ∎

###  3.4. The solvability of a linear projected problem

In this section, we devote to building the solvability of the following linear
parabolic projected problem:

$\left\\{\begin{array}[]{l}\psi_{t}=L[\psi]+g(t,r)-c(t)\partial_{r}\widehat{\omega}(t,r)\qquad\text{in}\
(-\infty,-T]\times(0,\infty),\\\\[5.69054pt]
\int_{\mathbb{R}}r^{n-1}\psi(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}{\mathrm{d}}r=0,\hskip
21.33955pt\text{for all}\ t\in(-\infty,-T],\end{array}\right.$ (3.60)

for a bounded function $g$, and $T>0$ fixed sufficiently large. In the above,
the linear operator $L$ is given by (2.20), the functions $\rho(t)$ and
$\widehat{\omega}(t,r)$ are defined by (2.7) and (2.6) respectively. The
function $c(t)$ solves the following relation:

$\displaystyle
c(t)\int_{0}^{\infty}\partial_{r}\widehat{\omega}(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle=\int_{0}^{\infty}\Big{[}\omega^{\prime\prime\prime}\big{(}r-\rho(t)\big{)}+\frac{n-1}{r}\omega^{\prime\prime}\big{(}r-\rho(t)\big{)}-W^{\prime\prime}\big{(}z(t,r)\big{)}\omega^{\prime}\big{(}r-\rho(t)\big{)}\Big{]}$
$\displaystyle\qquad\qquad\times\left[-\psi_{rr}-\frac{n-1}{r}\psi_{r}+W^{\prime\prime}\big{(}z(t,r)\big{)}\psi\right]r^{n-1}{\mathrm{d}}r$
$\displaystyle\quad+\int_{0}^{\infty}\Bigg{[}\partial_{rr}z(t,r)+\frac{n-1}{r}\partial_{r}z(t,r)-W^{\prime}\big{(}z(t,r)\big{)}\Bigg{]}\psi\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle\quad+\int_{0}^{\infty}\psi(t,r)\partial_{t}[\omega^{\prime}\big{(}r-\rho(t)\big{)}]r^{n-1}{\mathrm{d}}r\,+\,\int_{0}^{\infty}g(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r,$
(3.61)

for all $t<-T$, where $z(t,r)$ is given by (2.14). Indeed, this can be solved
uniquely since if $T$ is taken sufficiently large, the coefficient
$\int_{0}^{\infty}\partial_{r}\widehat{\omega}(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
is strictly positive.

The main result in this section is as follows:

######  Proposition 3.9.

There exist positive constants $T_{1}$ and $C$ such that for any $t\leq-T_{1}$
and $g\in C_{\Phi}((-\infty,t)\times(0,\infty))$, problem (3.60)-(3.61) has a
solution $\psi=\mathcal{A}(g)$, which defines a linear operator of $g$ and
satisfies the following estimate

$\sum_{l=0}^{3}\big{\|}\partial^{l}_{r}\psi\big{\|}_{C_{\Phi}((-\infty,t)\times(0,\infty))}\leq
C\|g\|_{C_{\Phi}((-\infty,t)\times(0,\infty))},\qquad\text{for all}\ t\leq-
T_{1},$ (3.62)

where we denote the $l$-th order derivative of $\psi(t,r)$ with respect of $r$
by $\partial^{l}_{r}\psi$.

$\mathbf{Proof\ of\ Proposition\ \ref{prop2}}$: We choose a sequence
$s_{j}\rightarrow-\infty$. Let $\psi^{s_{j}}$ be the solution to problem
(3.53) with $s=s_{j}$, according to Lemma 3.8. By (3.58), we can find that the
sequence $\\{\psi^{s_{j}}\\}$ converges to $\psi$ (up to subsequence) locally
uniformly in $(-\infty,-T_{1})\times(0,\infty)$. Using (3.21) and Proposition
3.3, we have that $\psi$ is a solution of (3.60) and satisfies (3.62). The
proof is completed.

##  4\. Solving the nonlinear projected problem

In this section, we mainly solve the nonlinear problem (2.16)-(2.17) by using
the fixed-point arguments. According to the result in Proposition 3.9, $\phi$
is a solution of (2.16)-(2.17) if only if $\phi\in
C_{\Phi}((-\infty,-T_{1})\times(0,\infty))$ is a fixed point of the following
operator

$\mathbf{T}(\phi):=\mathcal{A}\big{(}E(t,r)+N(\phi)-c(t)\partial_{r}\widehat{\omega}(t,r)\big{)},$
(4.1)

where $T_{1}$ and $\mathcal{A}$ are given by Proposition 3.9.

Let $T\geq T_{1}$, We define a set

$\Lambda_{T}:=\Big{\\{}h\in C^{1}(-\infty,-T]:\
\|h\|_{\Lambda_{T}}<1\Big{\\}},$ (4.2)

with the norm

$\|h\|_{\Lambda_{T}}:=\sup_{t\leq-T}|h(t)|+\sup_{t\leq-T}\Big{(}\frac{|t|}{\log|t|}|h^{\prime}(t)|\Big{)},$
(4.3)

and also a close domain

$X_{T}:=\left\\{\phi:\phi\in C_{\Phi}((-\infty,-T)\times(0,\infty))\ \
\text{and}\ \
\sum_{l=0}^{2}\big{\|}\partial^{l}_{r}\phi\big{\|}_{C_{\Phi}((-\infty,-T)\times(0,\infty))}\leq\frac{2\widehat{C}}{\log
T}\right\\},$ (4.4)

where the space $C_{\Phi}((-\infty,-T)\times(0,\infty))$ is defined by (3.18)
and $\widehat{C}$ is a fixed constant.

The main result is given by the following proposition

######  Proposition 4.1.

There exists $T_{2}\geq T_{1}$ such that for any given function $h(t)$ with
each $h\in\Lambda_{T_{2}}$, there is a solution $\phi(t,r,h)$ to
$\phi=\mathbf{T}(\phi)$ with respect to $\rho(t)=\gamma_{n}(t)+h(t)$. The
solution $\phi(t,r,h)$ satisfies problem (2.16)-(2.17). Furthermore, the
following estimate holds

$\sum_{l=0}^{2}\big{\|}\partial^{l}_{r}\phi(t,r,h_{1})-\partial^{l}_{r}\phi(t,r,h_{2})\big{\|}_{C_{\Phi}((-\infty,-T_{1})\times(0,\infty))}\leq
C\frac{1}{\log T_{2}}\|h_{1}-h_{2}\|_{\Lambda_{T_{2}}},$ (4.5)

where $T_{1}$ and $\mathbf{T}$ are given by Proposition 3.9 and (4.1)
respectively, $C$ is a positive constant which does not depend on
$h_{1},h_{2}$ and $T_{2}$.

To prove the above proposition, we first prepare two lemmas. We note that the
error term $E(t,r)$ and the nonlinear term $N(\phi)$ in (2.18)-(2.19) are all
dependent of $h$, due to the setting $\rho(t)=\gamma_{n}(t)+h(t)$ in (2.7). So
we denote $E(t,r)$ and $N(\phi)$ by $E(t,r,h)$ and $N(\phi,h)$.

######  Lemma 4.2.

Let $h_{1},h_{2}\in\Lambda_{T}$ and $\phi_{1},\phi_{2}\in X_{T}$, then there
exists a constant $C$ depending on $\widehat{C}$ such that

$\displaystyle\big{\|}N(\phi_{1},h_{1})-N(\phi_{2},h_{2})\big{\|}_{C_{\Phi}((-\infty,-T)\times(0,\infty))}$
(4.6) $\displaystyle\leq\frac{C}{\log
T}\Bigg{\\{}\|h_{1}-h_{2}\|_{\Lambda_{T}}+\sum_{l=0}^{2}\big{\|}\partial_{r}^{l}\big{(}\phi_{1}-\phi_{2}\big{)}\big{\|}_{C_{\Phi}((-\infty,-T)\times(0,\infty))}\Bigg{\\}}$

and

$\displaystyle\|E(t,r,h_{1})-E(t,r,h_{2})\|_{C_{\Phi}((-\infty,-T)\times(0,\infty))}\leq$
$\displaystyle\frac{C}{\log T}\|h_{1}-h_{2}\|_{\Lambda_{T}}.$ (4.7)

###### Proof.

By the definition of $N(\phi,h)$ in (2.19) and the smoothness of $W$, we have
that

$\displaystyle\left|N(\phi_{1},h)-N(\phi_{2},h)\right|$
$\displaystyle=\left|\int_{0}^{1}\Big{\\{}F^{\prime}\big{(}z(t,r)+y\phi_{1}+(1-y)\phi_{2}\big{)}[\phi_{1}-\phi_{2}]-F^{\prime}\big{(}z(t,r)\big{)}[\phi_{1}-\phi_{2}]\Big{\\}}{\mathrm{d}}y\right|$
$\displaystyle=\left|\int_{0}^{1}\Big{\\{}F^{\prime\prime}\big{(}z(t,r)+\theta\big{[}y\phi_{1}+(1-y)\phi_{2}\big{]}\big{)}\big{[}y\phi_{1}+(1-y)\phi_{2},\,\phi_{1}-\phi_{2}\big{]}\Big{\\}}{\mathrm{d}}y\right|$
$\displaystyle\leq
C\left(\sum_{l=0}^{2}\big{\|}\partial_{r}^{l}\big{(}\phi_{1}-\phi_{2}\big{)}\big{\|}_{C_{\Phi}((-\infty,T_{1})\times(0,\infty))}\right)\left(\sum_{l=0}^{2}\sum_{i=1}^{2}\big{\|}\partial_{r}^{l}\big{(}\phi_{i}\big{)}\big{\|}_{C_{\Phi}((-\infty,T_{1})\times(0,\infty))}\right)\Phi(t,r),$
(4.8)

and

$\displaystyle\left|N(\phi,h_{1})-N(\phi,h_{2})\right|$
$\displaystyle=\left|\int_{0}^{1}\int_{0}^{1}F^{\prime\prime}\big{(}yz_{1}(t,r)+(1-y)z_{2}(t,r)+s\big{)}\big{[}z_{1}(t,r)-z_{2}(t,r),\,\phi\big{]}{\mathrm{d}}yds\right|$
$\displaystyle\leq
C\|h_{1}-h_{2}\|_{\Lambda_{T_{1}}}\|\phi\|_{C_{\Phi}((-\infty,T_{1})\times(0,\infty))}\Phi(t,r),$
(4.9)

where $z_{i}(t,r)$ is given by (2.14) with $\rho(t)=\gamma_{n}(t)+h_{i}(t)$,
for $i=1,2$. In the above, $C$ is a positive constant independent of
$h,h_{1},h_{2}$ and $\phi,\phi_{1},\phi_{2}$, and the linear operator
$F^{\prime\prime}(u)[v_{1},v_{2}]$ is defined by (2.23). Hence combining (4)
and (4.9), we can obtain that (4.6) holds.

Next we prove that (4.7) also holds. By (2.24) and (2.25), similar arguments
in Lemma 2.3 will imply that

$\displaystyle\left|E_{1}(t,r,h_{1})-E_{1}(t,r,h_{2})\right|$
$\displaystyle\leq
C\Bigg{\\{}\left|\rho^{\prime}_{1}(t)\omega^{\prime}\big{(}r-\rho_{1}(t)\big{)}\,-\,\rho^{\prime}_{2}(t)\omega^{\prime}\big{(}r-\rho_{2}(t)\big{)}\right|$
$\displaystyle\quad\qquad\qquad+\frac{(n-1)^{2}(n-3)}{r^{3}}\left|\omega^{\prime}\big{(}r-\rho_{1}(t)\big{)}\,-\,\omega^{\prime}\big{(}r-\rho_{2}(t)\big{)}\right|\Bigg{\\}}\overline{\chi}_{\left\\{r\geq\delta_{0}\right\\}}$
$\displaystyle\quad+\frac{C}{[\gamma_{n}(t)]^{2}}\Big{\\{}\left|h_{1}(t)-h_{2}(t)\right|+\left|h^{\prime}_{1}(t)-h^{\prime}_{2}(t)\right|\Big{\\}}\overline{\chi}_{\left\\{\frac{\delta_{0}}{2}<r<\delta_{0}\right\\}}$
$\displaystyle\leq C\|h_{1}-h_{2}\|_{\Lambda}\frac{\Phi(t,r)}{\log T},$

and

$\displaystyle\left|E_{2}(t,r,h_{1})-E_{2}(t,r,h_{2})\right|$
$\displaystyle\leq
C\Bigg{\\{}\left|\rho^{\prime}_{1}(t)\widetilde{\omega}^{\prime}\big{(}r-\rho_{1}(t)\big{)}-\rho^{\prime}_{2}(t)\widetilde{\omega}^{\prime}\big{(}r-\rho_{2}(t)\big{)}\right|\,+\,\frac{1}{r^{3}}\sum_{l=0}^{3}\left|\partial_{r}^{l}\omega\big{(}r-\rho_{1}(t)\big{)}-\partial_{r}^{l}\omega\big{(}r-\rho_{2}(t)\big{)}\right|$
$\displaystyle\qquad\qquad\,+\,\frac{1}{r^{3}}\sum_{l=0}^{3}\left|\partial_{r}^{l}\widetilde{\omega}\big{(}r-\rho_{1}(t)\big{)}-\partial_{r}^{l}\widetilde{\omega}\big{(}r-\rho_{2}(t)\big{)}\right|\Bigg{\\}}\overline{\chi}_{\left\\{r\geq\delta_{0}\right\\}}$
$\displaystyle\quad+\frac{C}{[\gamma_{n}(t)]^{2}}\Big{\\{}\left|h_{1}(t)-h_{2}(t)\right|+\left|h^{\prime}_{1}(t)-h^{\prime}_{2}(t)\right|\Big{\\}}\overline{\chi}_{\left\\{\frac{\delta_{0}}{2}<r<\delta_{0}\right\\}}$
$\displaystyle\leq C\|h_{1}-h_{2}\|_{\Lambda}\frac{\Phi(t,r)}{\log T},$

where $\Phi(t,r)$ is defined by (2.26) and $C>0$ is a uniform constant
independent of $h_{1},h_{2}$ and $t$. Using the fact that
$E(t,r)=E_{1}(t,r)+E_{2}(t,r)$ and combining the above estimates, we can get
(4.7). ∎

######  Lemma 4.3.

Let $h_{1},h_{2}\in\Lambda_{T}$, $\phi_{1},\phi_{2}\in X_{T}$, $c(\phi,h,t)$
satisfy equation $(\ref{eq3.20})$ with respect to $\phi$ and
$\rho=\gamma_{n}+h$. Then

$\displaystyle\left|c(\phi_{1},h_{1},t)-c(\phi_{2},h_{2},t)\right|$
$\displaystyle\leq\frac{C}{T^{3/4}\big{[}\log
T\big{]}^{p-1}}\|\phi_{1}-\phi_{2}\|_{C_{\Phi}((-\infty,-T)\times(0,\infty))}$
(4.10) $\displaystyle\quad+\frac{C}{T^{1/2}\big{[}\log
T\big{]}^{p-1}}\|h_{1}-h_{2}\|_{\Lambda_{T}},$

where $C$ depends on $\widehat{C}$ in $(\ref{eq4.9})$, and $p\in(n,n+1]$.

###### Proof.

Proving this lemma just need to do some similar calculations in Lemmas 3.7 and
4.2, we omit it here. ∎

Proof of Proposition 4.1 We consider the operator $\mathbf{T}$ defined by
(4.1) from the domain $X_{T}$ in (4.4) to itself. We will prove $\mathbf{T}$
is a contraction mapping. Thus by fixed-point theorem, the operator
$\mathbf{T}$ has a unique fixed point $\phi$, i.e. $\mathbf{T}(\phi)=\phi$.

For any $\phi_{1},\phi_{2}\in X_{T}$, according to Lemma 2.3, Lemma 4.2 and
Proposition 3.9, we find that

$\sum_{l=0}^{2}\big{\|}\partial^{l}_{r}\mathbf{T}(0)\big{\|}_{C_{\Phi}((-\infty,T_{4})\times(0,\infty))}\leq\frac{\widehat{C}}{\log
T}$

and

$\sum_{l=0}^{2}\big{\|}\partial^{l}_{r}\left[\mathbf{T}(\phi_{1})-\mathbf{T}(\phi_{2})\right]\big{\|}_{C_{\Phi}((-\infty,T)\times(0,\infty))}\leq\frac{\widehat{C}}{\log
T}\|\phi_{1}-\phi_{2}\|_{C_{\Phi}((-\infty,T)\times(0,\infty))}.$

Hence, $\mathbf{T}$ is a contraction mapping in $X_{T}$ for any $T$ large
enough. Hence, according to Banach fixed-point theorem, there exists a unique
$\phi\in X_{T}$ such that $\mathbf{T}(\phi)=\phi$.

Next we will prove the estimate (4.5). Choosing $h_{1},h_{2}\in\Lambda_{T}$,
according to the above proof, we know that there exist
$\phi_{i}=\phi(t,r,h_{i})$, $i=1,2$, are two solutions to problem
(2.16)-(2.17) with $\rho=\gamma_{n}+h_{i}$ respectively.

We note that $\phi_{1}-\phi_{2}$ does not satisfy the orthogonality condition
(2.17). Let us consider a function $\bar{\phi}=\phi_{1}-\bar{\phi}_{2}$, where

$\bar{\phi}_{2}=\phi_{2}-\tilde{c}(t)\partial_{r}\widehat{\omega}_{1}(t,r),$

where
$\widehat{\omega}_{l}(t,r):=\omega\big{(}r-\rho_{1}(t)\big{)}\chi(\gamma_{n}(t)r)$
with $\rho_{1}(t)=\gamma_{n}(t)+h_{1}(t)$, the cut-off function $\chi$ is
defined by (2.5), and $\tilde{c}(t)$ is defined by the following relation

$\tilde{c}(t)\int_{0}^{\infty}\partial_{r}\widehat{\omega}_{1}(t,r)\omega^{\prime}\big{(}r-\rho_{1}(t)\big{)}r^{n-1}{\mathrm{d}}r=\int_{0}^{\infty}\phi_{2}(t,r)\omega^{\prime}\big{(}r-\rho_{1}(t)\big{)}r^{n-1}{\mathrm{d}}r,$

for all $t\leq-T$. According to the proof of Lemma 3.7, the coefficient of
$\tilde{c}(t)$ in the left hand side of above equality is strictly positive,
hence the function $\tilde{c}(t)$ is well-defined.

Thus $\bar{\phi}$ satisfies the following problem

$\left\\{\begin{array}[]{l}\bar{\phi}_{t}=F^{\prime}\big{(}z_{1}(t,r)\big{)}[\bar{\phi}]\,+\,\big{[}E(t,r,h_{1})-E(t,r,h_{2})\big{]}\,+\,\big{[}N(\phi_{1},h_{1})-N(\phi_{2},h_{2})\big{]}\\\\[8.53581pt]
\quad\quad+c_{2}(t)\big{[}\partial_{r}\widehat{\omega}_{2}(t,r)-\partial_{r}\widehat{\omega}_{1}(t,r)\big{]}\,+\,R(h_{1},h_{2})\,-\,\big{[}c_{1}(t)-c_{2}(t)\big{]}\partial_{r}\widehat{\omega}_{1}(t,r)\\\\[8.53581pt]
\quad\quad\ \text{in}\ (-\infty,-T)\times(0,\infty),\\\\[8.53581pt]
\int_{0}^{\infty}\bar{\phi}(t,r)\omega^{\prime}\big{(}r-\rho_{1}(t)\big{)}r^{n-1}{\mathrm{d}}r=0,\qquad\text{for
all}\ t<-T,\end{array}\right.$

where $\widehat{\omega}_{i}(t,r):=\omega(r-\rho_{i}(t))\chi(\gamma_{n}(t)r)$
with $i=1,2$, and the term $R(h_{1},h_{2})$ is defined by

$\displaystyle R(h_{1},h_{2})=$
$\displaystyle\,\tilde{c}^{\prime}(t)\partial_{r}\widehat{\omega}_{1}(t,r)+\tilde{c}(t)\partial_{t}\big{[}\partial_{r}\widehat{\omega}_{1}(t,r)\big{]}$
$\displaystyle+\tilde{c}(t)F^{\prime}(z_{1}(t,r))\big{[}\widehat{\omega}_{1}(t,r)\big{]}+F^{\prime}\big{(}z_{2}(t,r)\big{)}[\phi_{2}]-F^{\prime}\big{(}z_{1}(t,r)\big{)}[\phi_{2}],$

where $z_{i}(t,r)$ is defined by (2.14) with $\rho(t)=\gamma_{n}(t)+h_{i}(t)$
for $i=1,2$.

Recall that the linear operator $L[\phi]=F^{\prime}\big{(}z(t,r)\big{)}[\phi]$
is given by

$\displaystyle L[\phi]:=$
$\displaystyle-\phi_{rrrr}-\frac{2(n-1)}{r}\phi_{rrr}+\Bigg{[}2W^{\prime\prime}\big{(}z(t,r)\big{)}-\frac{(n-3)(n-1)}{r^{2}}\Bigg{]}\phi_{rr}-\big{(}W^{\prime\prime}\big{(}z(t,r)\big{)}\big{)}^{2}\phi$
$\displaystyle+\Bigg{[}\frac{2(n-1)W^{\prime\prime}\big{(}z(t,r)\big{)}}{r}-\frac{(3-n)(n-1)}{r^{3}}\Bigg{]}\phi_{r}+2W^{\prime\prime\prime}\big{(}z(t,r)\big{)}\phi_{r}z_{r}+W^{(4)}\big{(}z(t,r)\big{)}\left|z_{r}\right|^{2}\phi$
$\displaystyle-W^{\prime\prime\prime}\big{(}z(t,r)\big{)}W^{\prime}\big{(}z(t,r)\big{)}\phi+2W^{\prime\prime\prime}\big{(}z(t,r)\big{)}z_{rr}\phi+2\frac{n-1}{r}W^{\prime\prime\prime}\big{(}z(t,r)\big{)}z_{r}\phi,$

where the approximate solution $z(t,r)$ is defined by (2.14).

Hence by Proposition 3.9, the proof of Lemma 3.6, Lemmas 3.8 and 4.2, we have

$\displaystyle\sum_{l=0}^{2}\|\partial^{l}_{r}\bar{\phi}\|_{C_{\Phi}((-\infty,-T)\times(0,\infty))}\leq$
$\displaystyle C\frac{1}{\log
T}\Big{\\{}\|h_{1}-h_{2}\|_{\Lambda_{T}}+\|\phi_{1}-\phi_{2}\|_{C_{\Phi}((-\infty,-T)\times(0,\infty))}\Big{\\}}$
$\displaystyle+C\sup_{t\leq-T}\frac{\left|t\right|^{1/2}}{\left(\log\left|t\right|\right)^{1-p}}\Big{(}\left|\tilde{c}(t)\right|+\left|\tilde{c}^{\prime}(t)\right|\Big{)}.$
(4.11)

By the orthogonality condition (2.17), we have

$\displaystyle\left|\int_{0}^{\infty}\phi_{2}(t,r)\partial_{r}\widehat{\omega}_{1}(t,r)r^{n-1}{\mathrm{d}}r\right|$
$\displaystyle=\left|\int_{0}^{\infty}\phi_{2}(t,r)\Big{[}\partial_{r}\widehat{\omega}_{2}(t,r)-\partial_{r}\widehat{\omega}_{1}(t,r)\Big{]}r^{n-1}{\mathrm{d}}r\right|$
(4.12)
$\displaystyle\leq\frac{C}{\big{[}\log\left|t\right|\big{]}^{p-1}\left|t\right|^{\frac{1}{2}}\log
T}\|h_{1}-h_{2}\|_{\Lambda_{T}}\left|\gamma_{n}(t)\right|^{n-1},$

where we have used the following fact

$\left|\frac{\partial_{r}\widehat{\omega}(t,r)}{\Phi(t,r)}\right|\leq
C\left|t\right|^{1/2}\big{[}\log\left|t\right|\big{]}^{p-1},\qquad\text{for
all}\ r>0,$

where $\widehat{\omega}(t,r)$ is defined by (2.6).

We consider

$\displaystyle\left|\frac{{\mathrm{d}}}{{\mathrm{d}}t}\int_{0}^{\infty}\phi_{2}(t,r)\partial_{r}\widehat{\omega}_{1}(t,r)r^{n-1}{\mathrm{d}}r\right|=\left|\frac{{\mathrm{d}}}{{\mathrm{d}}t}\int_{0}^{\infty}\phi_{2}(t,r)\big{[}\partial_{r}\widehat{\omega}_{2}(t,r)-\partial_{r}\widehat{\omega}_{1}(t,r)\big{]}r^{n-1}{\mathrm{d}}r\right|.$
(4.13)

By the definition of $F^{\prime}\big{(}z(t,r)\big{)}[\phi]$ and integration by
parts, we have

$\displaystyle\int_{0}^{\infty}(\phi_{2})_{t}\big{[}\partial_{r}\widehat{\omega}_{2}(t,r)-\partial_{r}\widehat{\omega}_{1}(t,r)\big{]}r^{n-1}{\mathrm{d}}r$
$\displaystyle=\int_{0}^{\infty}\Big{[}(\phi_{2})_{t}-F^{\prime}\big{(}z_{2}(t,r)\big{)}[\phi]\Big{]}\Big{(}\partial_{r}\widehat{\omega}_{2}(t,r)-\partial_{r}\widehat{\omega}_{1}(t,r)\Big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle\quad+\int_{0}^{\infty}\left(-\phi_{rr}-\frac{n-1}{r}\phi_{r}+W^{\prime\prime}\big{(}z_{2}(t,r)\big{)}\right)\partial_{rr}\big{[}\partial_{r}\widehat{\omega}_{2}(t,r)-\partial_{r}\widehat{\omega}_{1}(t,r)\big{]}r^{n-1}{\mathrm{d}}r$
$\displaystyle\quad+(n-1)\int_{0}^{\infty}\left(-\phi_{rr}-\frac{n-1}{r}\phi_{r}+W^{\prime\prime}\big{(}z_{2}(t,r)\big{)}\right)\partial_{r}\big{[}\partial_{r}\widehat{\omega}_{2}(t,r)-\partial_{r}\widehat{\omega}_{1}(t,r)\big{]}r^{n-2}{\mathrm{d}}r$
$\displaystyle\quad+\int_{0}^{\infty}W^{\prime\prime}\big{(}z_{2}(t,r)\big{)}\Big{[}\phi_{rr}+\frac{n-1}{r}\phi_{r}-W^{\prime\prime}\big{(}z_{2}(t,r)\big{)}\phi\Big{]}\Big{(}\partial_{r}\widehat{\omega}_{2}(t,r)-\partial_{r}\widehat{\omega}_{1}(t,r)\Big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle\quad+\int_{0}^{\infty}\left(\partial_{rr}z_{2}+\frac{n-1}{r}\partial_{r}z_{2}-W^{\prime}\big{(}z_{2}(t,r)\big{)}\right)W^{\prime\prime\prime}\big{(}z_{2}(t,r)\big{)}\phi\Big{(}\partial_{r}\widehat{\omega}_{2}(t,r)-\partial_{r}\widehat{\omega}_{1}(t,r)\Big{)}r^{n-1}{\mathrm{d}}r.$
(4.14)

By the previous fixed-point arguments and the above equality (4.14), we have

$\displaystyle\left|\int_{0}^{\infty}r^{n-1}(\phi_{2})_{t}\Big{[}\partial_{r}\widehat{\omega}_{2}(t,r)-\partial_{r}\widehat{\omega}_{1}(t,r)\Big{]}{\mathrm{d}}r\right|\leq\frac{C}{\log
T}\frac{\|h_{1}-h_{2}\|_{\Lambda_{T}}}{\left|t\right|^{\frac{1}{2}}\big{[}\log\left|t\right|\big{]}^{p-1}}\left|\gamma_{n}(t)\right|^{n-1}.$
(4.15)

By the above estimates (4.12), (4.13), (4.15) and the definition of
$\tilde{c}(t)$, we have

$\left|\tilde{c}(t)\right|+\left|\tilde{c}^{\prime}(t)\right|\leq\frac{C}{\log
T}\frac{\|h_{1}-h_{2}\|_{\Lambda_{T}}}{\left|t\right|^{\frac{1}{2}}\big{[}\log\left|t\right|\big{]}^{p-1}},\qquad\forall\,t<-T.$

Hence

$\sum_{l=0}^{2}\big{\|}\partial^{l}_{r}\bar{\phi}\big{\|}_{C_{\Phi}((-\infty,-T)\times(0,\infty))}\leq\frac{C}{\log
T}\Big{[}\big{\|}\phi_{1}-\phi_{2}\big{\|}_{C_{\Phi}((-\infty,-T)\times(0,\infty))}+\|h_{1}-h_{2}\|_{\Lambda_{T}}\Big{]},$

where $C$ is uniform positive constant independent of $T$.

Eventually, we have that

$\displaystyle\sum_{l=0}^{2}\big{\|}\partial_{r}^{l}[\phi_{1}-\phi_{2}]\big{\|}_{C_{\Phi}((-\infty,-T)\times(0,\infty))}$
$\displaystyle\leq\sum_{l=0}^{2}\big{\|}\partial_{r}^{l}\bar{\phi}\big{\|}_{C_{\Phi}((-\infty,-T)\times(0,\infty))}+C\sup_{t\leq-T}\frac{\left|t\right|^{1/2}}{\big{(}\log\left|t\right|\big{)}^{p-1}}\left|\tilde{c}(t)\right|$
$\displaystyle\leq C\frac{1}{\log