# Asymptotic growth rate of solutions to level-set forced mean curvature flows
with evolving spirals

Hiroyoshi Mitake  Graduate School of Mathematical Sciences, University of
Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
<EMAIL_ADDRESS>and  Hung V. Tran  Department of Mathematics,
University of Wisconsin Madison, Van Vleck hall, 480 Lincoln drive, Madison,
WI 53706, USA<EMAIL_ADDRESS>
###### Abstract.

Here, we study a level-set forced mean curvature flow with evolving spirals
and the homogeneous Neumann boundary condition, which appears in a crystal
growth model. Under some appropriate conditions on the forcing term, we prove
that the solution is globally Lipschitz. We then study the large time average
of the solution and deduce the asymptotic growth rate of the crystal. Some
large time behavior results of the solution are obtained.

###### Key words and phrases:

Level-set forced mean curvature flows; Neumann boundary problem; global
Lipschitz regularity; large time behavior; spiral crystal growth; dislocations

###### 2010 Mathematics Subject Classification:

35B40, 49L25, 53E10, 35B45, 35K20, 35K93,

The work of HM was partially supported by the JSPS grants: KAKENHI #22K03382,
#21H04431, #20H01816, #19H00639. The work of HT was partially supported by NSF
CAREER grant DMS-1843320 and a Vilas Faculty Early-Career Investigator Award.

##  1\. Introduction

###  1.1. A spiral crystal growth model

Various spiral patterns are observed in many crystal growth situations in
practice in which the spirals’ centers are often believed to be the locations
of screw dislocations. In this growth mechanism, the crystal surface has
discontinuities in height, which are called steps, along curves that spiral
out from these centers. Atoms bond with the crystal structure with a higher
probability near each step, which results in a surface evolution of this step
with normal velocity

$V=v_{\infty}(\rho_{c}\kappa+1),$ (1.1)

where $\kappa$ is the curvature of the step. The constants $\rho_{c}>0$ and
$v_{\infty}>0$ are the critical radius and the step velocity [1],
respectively. By rescaling, we may assume that $v_{\infty}\rho_{c}=1$, and
thus, (1.1) becomes $V=\kappa+v_{\infty}$. See Figure 1 for an example of a
spiral crystal growth with only one screw dislocation.

Fig. 1. Spiral crystal growth with a screw dislocation.

We now give a more precise description of the spirals and the set up of the
problem following [21, 25, 26]. Let $\Omega\subset\mathbb{R}^{2}$ be a bounded
domain with smooth boundary. Let $N\in\mathbb{N}$ and
$a_{1},\ldots,a_{N}\in\Omega$ be the centers of the spirals with $a_{i}\neq
a_{j}$ for $i\neq j$. Fix $r>0$ sufficiently small so that
$\overline{B}(a_{j},r)\subset\Omega$ for $1\leqslant j\leqslant N$, and
$\overline{B}(a_{i},r)\cap\overline{B}(a_{j},r)=\emptyset$ for $i\neq j$. Let

$\displaystyle
W:=\Omega\setminus\bigcup_{j=1}^{N}B(a_{j},r)\subset\mathbb{R}^{2},$
$\displaystyle\theta(x):=\sum_{j=1}^{N}m_{j}\arg(x-a_{j}),$

where $m_{j}\in\mathbb{Z}\setminus\\{0\\}$ is given for $1\leqslant j\leqslant
N$. The constant $m_{j}=m_{j}^{+}-m_{j}^{-}$ with
$m_{j}^{\pm}\in\mathbb{N}\cup\\{0\\}$ is the strength of the spiral center
$a_{j}$, which is the difference between the strength $m_{j}^{+}$ of
counterclockwise rotating spirals and $m_{j}^{-}$ of clockwise rotating
spirals. Note that $\theta$ is a multi-valued function and it is not well
defined at $a_{1},\ldots,a_{N}$, and this is the reason why we consider the
problem in $W$ instead of $\Omega$. Although $\theta$ is multi-valued,
$D\theta$ is single-valued. Indeed, if $f(x)=\arg(x)$, then we can view that
$f(x)=\arctan(x_{2}/x_{1})+2\pi l$ for some $l\in\mathbb{Z}$. Thus,

$f_{x_{1}}=\frac{1}{1+\left(\frac{x_{2}}{x_{1}}\right)^{2}}\cdot\left(-\frac{x_{2}}{(x_{1})^{2}}\right)=\frac{-x_{2}}{|x|^{2}},\quad
f_{x_{2}}=\frac{1}{1+\left(\frac{x_{2}}{x_{1}}\right)^{2}}\cdot\left(\frac{1}{x_{1}}\right)=\frac{x_{1}}{|x|^{2}}.$

Hence, for $a_{j}=(a_{j,1},a_{j,2})$ for $1\leqslant j\leqslant N$.

$D\theta(x)=\sum_{j=1}^{N}\frac{m_{j}(-(x_{2}-a_{j,1}),x_{1}-a_{j,2})}{|x-a_{j}|^{2}}=\sum_{j=1}^{N}\frac{m_{j}(x-a_{j})^{\perp}}{|x-a_{j}|^{2}}.$

As noted, $D\theta$ is well defined and smooth in $W$. For an auxiliary
function $u:W\times[0,\infty)\to\mathbb{R}$, spirals are implicitly defined
as, for $t\geqslant 0$,

$\Gamma_{t}=\\{x\in\overline{W}\,:\,u(x,t)-\theta(x)=2\pi k\text{ for some
}k\in\mathbb{Z}\\}.$ (1.2)

In this paper, we consider a more general setting in which the step velocity
might be non homogeneous, that is, $v_{\infty}=c(x)$ for $x\in W$. Then,
$V=\kappa+c(x)$, and the auxiliary function $u$ satisfied the following level-
set forced mean curvature flow PDE

$\begin{cases}u_{t}=|D(u-\theta)|\left({\rm
div}\,\left(\frac{D(u-\theta)}{|D(u-\theta)|}\right)+c(x)\right)\qquad&\text{
in }W\times(0,\infty),\\\ D(u-\theta)\cdot\mathbf{n}=0\qquad&\text{ on
}\partial W\times(0,\infty),\\\ u(\cdot,0)=u_{0}\qquad&\text{ on
}\overline{W},\end{cases}$ (1.3)

Here, ${\mathbf{n}}$ is the outward unit normal vector to $\partial W$.
Throughout this paper, we assume that $c\in C^{1}(\overline{W},(0,\infty))$
and $u_{0}\in C^{2}(\overline{W})$ with $D(u_{0}-\theta)\cdot\mathbf{n}=0$ on
$\partial W$ for compatibility. The wellposedness of viscosity solutions to
(1.3) was obtained in [21]. The uniqueness of level sets
$\\{\Gamma_{t}\\}_{t>0}$ with respect to an initial curve $\Gamma_{0}$ was
proved in [13].

Our main goals are to study the Lipschitz regularity, the large time average
and behavior of the auxiliary function $u$, and to deduce the asymptotic
growth rate of the crystal.

###  1.2. Main results

We now describe our main results. Denote by

$\begin{cases}C_{0}:=\max\\{-\lambda\,:\,\lambda\ \textrm{is the curvature of
$\partial W$ at $x_{0}$ for $x_{0}\in\partial W$}\\}\in\mathbb{R},\\\
K_{0}:=\min\\{d\,:\,d\ \textrm{is the diameter of an open ball inscribed in}\
W\\}>0.\end{cases}$

In our setting, it is clear that

$C_{0}=\max\left\\{C_{1},\frac{1}{r}\right\\}>0$

where

$C_{1}:=\max\\{-\lambda\,:\,\lambda\ \textrm{is a curvature of
$\partial\Omega$ at $x_{0}$ for $x_{0}\in\partial\Omega$}\\}\in\mathbb{R}.$

In particular, if $\Omega$ is convex, then $C_{0}=1/r$.

We first have the following uniform Lipschitz estimate for $u$, which is
strongly inspired by [16].

###### Theorem 1.1.

Assume that there exists $\delta>0$ such that

$c(x)^{2}-2|Dc(x)|-2C_{0}|c(x)|-\frac{8C_{0}}{K_{0}}\geqslant\delta\qquad\text{
for all }x\in W.$ (1.4)

Let $u$ be the unique viscosity solution to (1.3). Then, there exists a
constant $L>0$ depending only on $\|u_{0}\|_{C^{2}(\overline{W})}$,
$\|D\theta\|_{C^{1}(\overline{W})}$, $\|c\|_{C^{1}(\overline{W})}$, the
constants $C_{0}$, $K_{0}$, and $\delta$ such that

$\|u_{t}\|_{L^{\infty}(W\times[0,\infty))}+\|Du\|_{L^{\infty}(W\times[0,\infty))}\leqslant
L.$ (1.5)

###### Remark 1.

In case that $c(x)=c>0$ is a constant force term, then (1.4) is equivalent to
the requirement that

$c>C_{0}\left(1+\left(1+\frac{8}{C_{0}K_{0}}\right)^{1/2}\right).$

Note that $C_{0}\geqslant 1/r$, which is quite large for $r>0$ small. Besides,
$K_{0}$ can be quite small if some of the balls $\\{B(a_{j},r)\\}$ are close
to each other or close to $\partial\Omega$. In such situations, we would need
to require $c$ to be very large in order to have uniform Lipschitz estimate
for $u$, which implies that (1.4) is rather restrictive.

###### Example 1.2.

We consider the case that $\Omega=B(0,R)$, $W=B(0,R)\setminus B(0,r)$ for
given $R\geqslant 2r>0$ and $c(x)=c>0$ is a constant force term. Then,
$C_{0}=1/r$, $K_{0}=R-r\geqslant r$, and (1.4) is equivalent to the
requirement that

$c>4C_{0}=\frac{4}{r}.$

Next, we study the large time average of the auxiliary function $u$.

###### Theorem 1.3.

Let $u$ be the unique viscosity solution to (1.3). For each $t\geqslant 0$,
denote by

$S(t)=\max_{x\in\overline{W}}u(x,t).$

Then, there exists a constant $S=S_{c}\in\mathbb{R}$ such that

$\lim_{t\to\infty}\frac{S(t)}{t}=S_{c}.$

If we assume furthermore (1.4), then

$\lim_{t\to\infty}\frac{u(x,t)}{t}=S_{c}\qquad\text{ uniformly for
}x\in\overline{W}.$ (1.6)

We now use the formulation described in [25] to deduce back the asymptotic
growth rate of the crystal from the auxiliary function $u$. Let
$k(x,t)\in\mathbb{Z}$ be such that

$-\pi\leqslant u(x,t)-\left(\Theta(x)+2\pi k(x,t)\right)<\pi,$ (1.7)

where $\Theta(x)=\sum_{j=1}^{N}m_{j}\Theta_{j}(x)$, and
$\Theta_{j}:\overline{W}\to[0,2\pi)$ is the principal value of
$\arg(x-a_{j})$. Denote by $h_{0}>0$ the unit height of steps. We define the
height of the crystal at $(x,t)\in\overline{W}\times[0,\infty)$ by

$h(x,t)=\frac{h_{0}}{2\pi}\left[\Theta(x)+2\pi k(x,t)+\pi\,{\rm
sign}\left(u(x,t)-\left(\Theta(x)+2\pi k(x,t)\right)\right)\right].$ (1.8)

###### Theorem 1.4.

Let $u$ be the unique viscosity solution to (1.3). Let $S_{c}$ be defined as
in Theorem 1.3. Let $h:\overline{W}\times[0,\infty)\to\mathbb{R}$ be the
height of the crystal defined by (1.7)–(1.8). Then,

$\lim_{t\to\infty}\frac{\max_{x\in\overline{W}}h(x,t)}{t}=\frac{h_{0}}{2\pi}S_{c}.$
(1.9)

If we assume furthermore (1.4), then

$\lim_{t\to\infty}\frac{h(x,t)}{t}=\frac{h_{0}}{2\pi}S_{c}\qquad\text{
uniformly for }x\in\overline{W}.$ (1.10)

In general, without assuming (1.4), we do not have uniform control on the
asymptotic growth rate of the crystal. Nevertheless, having the asymptotic
growth rate of the tip (the highest point) of the crystal in (1.9) might be
enough for practical purposes.

###### Definition 1.5.

Let $S_{c}$ be as in Theorem 1.3. We say that $S_{c}$ is the asymptotic growth
rate of $u$ solving (1.3). We also say that $h_{0}S_{c}/(2\pi)$ is the
asymptotic growth rate of the corresponding crystal whose height is defined by
(1.7)–(1.8).

We next study the asymptotic growth rate $S_{c}$ of $u$, or equivalently,
$h_{0}S_{c}/(2\pi)$ of $h$. Various interesting numerical results on the
asymptotic growth rate were obtained in [25, 26].

###### Lemma 1.6.

Assume that $N=1$, $m_{1}=1$, $a_{1}=0$, $W=B(0,R)\setminus B(0,r)$ for given
$0<r<R$, and $c\in C^{1}(\overline{W},(0,\infty))$. Then,

$\min_{x\in\overline{W}}\frac{c(x)}{|x|}\leqslant
S_{c}\leqslant\max_{x\in\overline{W}}\frac{c(x)}{|x|}.$

In Propositions 3.1–3.3 in Section 3, we give various situations when $N=1,2$
in which $S_{c}=0$.

As the asymptotic growth rate of $u$ is $S_{c}$, it is natural to consider the
following ergodic problem

$\begin{cases}-|D(v-\theta)|\left({\rm
div}\,\left(\frac{D(v-\theta)}{|D(v-\theta)|}\right)+c(x)\right)=-S_{c}\qquad&\text{
in }W,\\\ D(v-\theta)\cdot\mathbf{n}=0\qquad&\text{ on }\partial
W.\end{cases}$ (1.11)

We have the following large time behavior result for (1.3) in the case that
$S_{c}=0$.

###### Theorem 1.7.

Let $u$ be the unique viscosity solution to (1.3). Assume (1.4) and $S_{c}=0$.
Then, there exists a viscosity solution $v$ to (1.11) with $S_{c}=0$ such that

$\lim_{t\to\infty}\|u(\cdot,t)-v\|_{L^{\infty}(W)}=0.$

The proof of this theorem follows that of [12, Theorem 1.2] or [16, Theorem
1.3] which uses a Lyapunov function, and is hence omitted. It is worth
emphasizing that the condition $S_{c}=0$ in Theorem 1.7 is essential for the
argument; and if $S_{c}\neq 0$, then the large time behavior for (1.3) is
rather open. Next, we give a result along the line of Lemma 1.6. For a given
angle $\varphi$, denote by $\mathcal{R}_{\varphi}$ a linear transformation
given by rotating vectors through an angle of $\varphi$ counterclockwise in
$\mathbb{R}^{2}$. Then, the matrix of $\mathcal{R}_{\varphi}$ is given by

$\mathcal{R}_{\varphi}=\begin{pmatrix}\cos\varphi&-\sin\varphi\\\
\sin\varphi&\cos\varphi\end{pmatrix}.$

###### Proposition 1.8.

Assume that $N=1$, $m_{1}=1$, $a_{1}=0$, $W=B(0,R)\setminus B(0,r)$ for given
$0<r<R$, and $c(x)=c_{0}|x|$ for $x\in\overline{W}$ for some $c_{0}>0$. Assume
further that $u_{0}(x)=g(x/|x|)$ for all $x\in\overline{W}$ for a given $g\in
C^{2}(\mathbb{R}^{2})$ with $\|Dg\|_{L^{\infty}}<1$. Then, we have

$u(x,t)=g\left(\mathcal{R}_{-c_{0}t}\frac{x}{|x|}\right)+c_{0}t\qquad\text{
for all }(x,t)\in\overline{W}\times[0,\infty).$

In particular, $u(x,t)-c_{0}t$ does not converge as $t\to\infty$ if $g$ is not
constant.

Thus, in general, it is striking that solutions to (1.3) do not converge to
solutions of the ergodic problem (1.11) after appropriate normalizations as
time tends to infinity.

###  1.3. Relevant literature

We give a non exhaustive list of related works to our paper. We refer the
reader to [2, 3, 6] for the wellposedness of viscosity solutions of level-set
mean curvature flows. There have been many important papers using the level-
set forced mean curvature flow PDE to study spiral crystal growths [21, 13,
25, 26, 23, 24, 27]. For Neumann boundary problems similar to this context,
see [15, 11, 22, 20, 18, 16, 17]. Besides the spiral crystal growth model, the
birth and spread crystal growth model has also been studied extensively [9,
10, 8, 14, 7]. We emphasize that it is rather important to study the
asymptotic growth rates and the shapes of the crystals in these models as time
tends to infinity. For related results concerning large time averages and
large time behaviors, see [12, 5, 19, 4].

### Organization of the paper

The paper is organized as follows. In Section 2, we obtain the Lipschitz
regularity of the solution of (1.3) and give the proof of Theorem 1.1 under
the additional assumption (1.4). We then give the proof of Theorems 1.3–1.4
and study the properties of the asymptotic growth rate and the large time
behavior of the solution in Section 3. In Section 4, we give an example of a
non-uniformly Lipschitz continuous solution of (1.3), which shows that (1.4)
is needed in general if we want to have uniform estimates.

### Notations

For $1\leqslant j\leqslant N$, denote by $\theta_{j}(x)=\arg(x-a_{j})$ and
$\Theta_{j}$ the principle value of $\theta_{j}$. For a given smooth function
$\phi$, we write $\phi_{k}=\phi_{x_{k}}$, $\phi_{kl}=\phi_{x_{k}x_{l}}$. When
there is no confusion, we use the Einstein summation convention. For
$x=(x_{1},x_{2})\in\mathbb{R}^{2}$, we write $x^{\perp}=(-x_{2},x_{1})$. For a
given angle $\varphi$, denote by $\mathcal{R}_{\varphi}$ a linear
transformation given by rotating vectors through an angle of $\varphi$
counterclockwise in $\mathbb{R}^{2}$. Note that
$x^{\perp}=\mathcal{R}_{\pi/2}\,x$ for $x\in\mathbb{R}^{2}$.

### Acknowledgement

The authors would like to extend their sincere appreciation to Professor
Yoshikazu Giga for his insights and comments. Additionally, the authors are
grateful to Professor Takeshi Ohtsuka for sending them the references [23, 24]
and his suggestions.

##  2\. Lipschitz regularity

To obtain Lipschitz estimates, we consider the following approximation, for
$\varepsilon\in(0,1)$, $T>0$,

$\begin{cases}u^{\varepsilon}_{t}=\sqrt{\varepsilon^{2}+|D(u^{\varepsilon}-\theta)|^{2}}\left({\rm
div}\,\left(\frac{D(u^{\varepsilon}-\theta)}{\sqrt{\varepsilon^{2}+|D(u^{\varepsilon}-\theta)|^{2}}}\right)+c(x)\right)\quad&\text{
in }W\times(0,T],\\\ D(u^{\varepsilon}-\theta)\cdot\mathbf{n}=0\quad&\text{ on
}\partial W\times[0,T],\\\ u^{\varepsilon}(x,0)=u_{0}(x)\quad&\text{ on
}\overline{W}.\end{cases}$ (2.1)

The following result on a priori estimates on the gradient of
$u^{\varepsilon}$ implies right away Theorem 1.1.

###### Theorem 2.1 (A priori estimates).

Assume that $\partial W$ is smooth and $c\in C^{\infty}(\overline{W})$. Assume
(1.4). For each $\varepsilon\in(0,1)$ and $T>0$, assume that
$u^{\varepsilon}\in C^{\infty}(\overline{W}\times(0,T])\cap
C^{1}(\overline{W}\times[0,T])$ is the unique solution of (2.1). Then, there
exists a constant $L>0$ depending only on $\|u_{0}\|_{C^{2}(\overline{W})}$,
$\|D\theta\|_{C^{1}(\overline{W})}$, $\|c\|_{C^{1}(\overline{W})}$, the
constants $C_{0}$, $K_{0}$, and $\delta$ from (1.4) such that

$\|u^{\varepsilon}_{t}\|_{L^{\infty}(W\times[0,\infty))}+\|Du^{\varepsilon}\|_{L^{\infty}(W\times[0,\infty))}\leqslant
L.$ (2.2)

The proof below follows the ideas in [16], which uses the classical Bernstein
method. We also refer the reader to [17, 24].

###### Proof.

Firstly, let

$M=4\lVert
D^{2}(u_{0}-\theta)\rVert_{L^{\infty}(\overline{W})}+\lVert(1+|D(u_{0}-\theta)|^{2})^{1/2}c\rVert_{L^{\infty}(\overline{W})}.$
(2.3)

We see that $u_{0}\pm Mt$ are a supersolution and a subsolution to (2.1),
respectively. By the comparison of principle, we yield that, for $s\geqslant
0$,

$u_{0}-Ms\leqslant u^{\varepsilon}(\cdot,s)\leqslant u_{0}+Ms.$ (2.4)

Moreover, for each $s\geqslant 0$, $(x,t)\mapsto u^{\varepsilon}(x,t+s)$ is a
solution to (2.1) with initial data $u^{\varepsilon}(x,s)$. By using the
comparison principle once more and (2.4), we deduce that, for $t,s\geqslant
0$,

$\|u^{\varepsilon}(\cdot,t+s)-u^{\varepsilon}(\cdot,t)\|_{L^{\infty}(W)}\leqslant\|u^{\varepsilon}(\cdot,s)-u^{\varepsilon}(\cdot,0)\|_{L^{\infty}(W)}\leqslant
Ms.$

Therefore, we obtain that

$\|u^{\varepsilon}_{t}\|_{L^{\infty}(W\times[0,\infty))}\leqslant M.$ (2.5)

We next show the boundedness of $Du^{\varepsilon}$. Denote by

$w^{\varepsilon}=\sqrt{\varepsilon^{2}+|D(u^{\varepsilon}-\theta)|^{2}}.$

We only need to show that

$\max_{\overline{W}\times[0,T]}w^{\varepsilon}\leqslant C$ (2.6)

for some positive constant $C$ depending only on
$\|u_{0}\|_{C^{2}(\overline{W})}$, $\|D\theta\|_{C^{1}(\overline{W})}$,
$\|c\|_{C^{1}(\overline{W})}$, the constants $C_{0}$, $K_{0}$, and $\delta$
from (1.4). The crucial point here is $C$ does not depend on $T>0$ and
$\varepsilon\in(0,1)$. Pick
$(x_{0},t_{0})\in\operatorname{argmax}_{\overline{W}\times[0,T]}w^{\varepsilon}$.
If $t_{0}=0$, then

$\max_{\overline{W}\times[0,T]}w^{\varepsilon}\leqslant
w^{\varepsilon}(x_{0},0)\leqslant\|D(u_{0}-\theta)\|_{L^{\infty}(\overline{W})}+1,$

and (2.6) holds true.

We consider the case $t_{0}>0$. Write $u=u^{\varepsilon}$, $w=w^{\varepsilon}$
from now on for clarity. Denote by

$b^{ij}=\delta_{ij}-\frac{(u-\theta)_{i}(u-\theta)_{j}}{\varepsilon^{2}+|D(u-\theta)|^{2}}.$

Differentiate (2.1) with respect to $x_{k}$ and multiply the result by
$(u-\theta)_{k}$ to get

$(u-\theta)_{k}(u-\theta)_{kt}-(D_{p}b^{ij}\cdot
D(u-\theta)_{k})(u-\theta)_{k}(u-\theta)_{ij}-b^{ij}(u-\theta)_{k}(u-\theta)_{kij}\\\
-(u-\theta)_{k}c_{k}w-c\frac{(u-\theta)_{k}(u-\theta)_{lk}(u-\theta)_{l}}{w}=0.$

In the above, we use the fact that $\theta_{kt}=0$ as $\theta$ is independent
of $t$. Substituting $ww_{t}=(u-\theta)_{k}(u-\theta)_{kt}$,
$ww_{k}=(u-\theta)_{l}(u-\theta)_{kl}$, and
$ww_{ij}=(u-\theta)_{kij}(u-\theta)_{k}+b^{kl}(u-\theta)_{ki}(u-\theta)_{lj}$,
we yield

$ww_{t}-w(D_{p}b^{ij}\cdot
Dw)(u-\theta)_{ij}-wb^{ij}w_{ij}+b^{ij}b^{kl}(u-\theta)_{ki}(u-\theta)_{lj}\\\
-wD(u-\theta)\cdot Dc-cD(u-\theta)\cdot Dw=0.$ (2.7)

There are two cases to be considered $x_{0}\in W$ and $x_{0}\in\partial W$.

Case 1: $x_{0}\in W$. We follow the computations of [8, Lemma 4.1]. At
$(x_{0},t_{0})$, we have $w_{t}\geqslant 0$, $Dw=0$, $D^{2}w\leqslant 0$, and
thus

$wD(u-\theta)\cdot Dc\geqslant b^{ij}b^{kl}(u-\theta)_{ki}(u-\theta)_{lj}.$

We then use the Cauchy-Schwarz inequality

$(\textrm{tr}\alpha\beta)^{2}\leqslant\textrm{tr}(\alpha^{2})\textrm{tr}(\beta^{2})$

for all $\alpha,\beta\in\mathcal{S}^{2}$, and put
$\alpha=A^{\frac{1}{2}}BA^{\frac{1}{2}}$, $\beta=I_{2}$, where $A=(b^{ij})$,
$B=((u-\theta)_{kl})$, $I_{2}$ the identity matrix of size $2$ to get
$\textrm{tr}(AB)^{2}\geqslant(\textrm{tr}AB)^{2}/\textrm{tr}(I_{2}).$

Therefore, at $(x_{0},t_{0})$,

$\displaystyle\left|Dc(x_{0})\right|w^{2}$ $\displaystyle\geqslant
wD(u-\theta)\cdot Dc\geqslant
b^{ij}b^{kl}(u-\theta)_{ki}(u-\theta)_{lj}=\textrm{tr}(AB)^{2}$
$\displaystyle\geqslant\frac{(\textrm{tr}AB)^{2}}{\textrm{tr}(I_{2})}=\frac{1}{2}\left(u_{t}-c(x_{0})w\right)^{2}.$

Since $\frac{1}{2}c(x)^{2}-\left|Dc(x)\right|\geqslant\frac{\delta}{2}>0$ by
(1.4), we imply that at $(x_{0},t_{0})$,

$\delta w^{2}\leqslant 2u_{t}c(x_{0})w\quad\Longrightarrow\quad
w\leqslant\frac{2M\|c\|_{L^{\infty}(\overline{W})}}{\delta},$

which gives (2.6).

Case 2: $x_{0}\in\partial W$. As $\partial W$ is $C^{2}$, we assume that
$\mathbf{n}$ is defined as a $C^{1}$ function in a neighborhood of $\partial
W$. Note that the Neumann boundary condition $D(u-\theta)\cdot{\mathbf{n}}=0$
gives $\left(D^{2}(u-\theta){\mathbf{n}}+D{\mathbf{n}}D(u-\theta)\right)\cdot
v=0$ for all $v\in\mathbb{R}^{2}$ perpendicular to ${\mathbf{n}}$ on $\partial
W\times[0,T]$. Thus, on $\partial W\times[0,T]$,

$\frac{\partial
w}{\partial\mathbf{n}}=\frac{D^{2}(u-\theta)D(u-\theta)}{w}\cdot{\mathbf{n}}=-\frac{D{\mathbf{n}}D(u-\theta)\cdot
D(u-\theta)}{w}\leqslant C_{0}\frac{|D(u-\theta)|^{2}}{w}.$

We note that $C_{0}\geqslant 1/r$ and at $(x_{0},t_{0})$,

$\frac{\partial w}{\partial{\mathbf{n}}}\leqslant
C_{0}\frac{|D(u-\theta)|^{2}}{w}<C_{0}w.$

Pick $x_{c}\in W$ such that $B:=B(x_{c},K_{0}/2)\subset W$ is tangent to
$\partial W$ at $x_{0}$. Consider a multiplier for $w$

$\rho(x)=-\frac{C_{0}}{K_{0}}|x-x_{c}|^{2}+\frac{C_{0}K_{0}}{4}+1\quad\text{
for }x\in\overline{W}.$

Then, $\rho>1$ in $B$, $\rho=1$ on $\partial B$, and $\rho<1$ on
$\overline{W}\setminus\overline{B}$. Besides,
$C_{0}\rho(x_{0})+\frac{\partial\rho}{\partial{\mathbf{n}}}(x_{0})=0$.

Denote by $\psi=\rho w$. Then, at $(x_{0},t_{0})$,

$\frac{\partial\psi}{\partial{\mathbf{n}}}=\frac{\partial(\rho
w)}{\partial{\mathbf{n}}}=\rho\frac{\partial
w}{\partial{\mathbf{n}}}+w\frac{\partial\rho}{\partial{\mathbf{n}}}<w\left(C_{0}\rho+\frac{\partial\rho}{\partial{\mathbf{n}}}\right)=0.$
(2.8)

As noted above, for
$(z,t)\in\left(\overline{W}\setminus\overline{B}\right)\times[0,T]$,

$\psi(z,t)\leqslant w(z,t)\leqslant w(x_{0},t_{0})=\psi(x_{0},t_{0}),$

and, by (2.8),

$\max_{\overline{W}\times[0,T]}\rho w=\max_{\overline{B}\times[0,T]}\rho
w>\psi(x_{0},t_{0})=w(x_{0},t_{0}).$ (2.9)

Let $(x_{1},t_{1})\in\operatorname{argmax}_{\overline{W}\times[0,T]}\rho w$.
Thanks to (2.8)–(2.9), $x_{1}\in B\subset W$. If $t_{1}=0$, then for all
$(x,t)\in\overline{W}\times[0,T]$,

$\displaystyle w(x,t)$ $\displaystyle\leqslant
w(x_{0},t_{0})=\rho(x_{0})w(x_{0},t_{0})\leqslant\rho(x_{1})w(x_{1},0)$
$\displaystyle\leqslant\left(\frac{C_{0}K_{0}}{4}+1\right)\left(\|D(u_{0}-\theta)\|_{L^{\infty}(\overline{W})}+1\right),$

which gives the desired result. We now consider the case that $t_{1}>0$. At
this point $(x_{1},t_{1})$, we have $\psi_{t}\geqslant 0,$ $D\psi=0,$
$D^{2}\psi\leqslant 0$. Consequently, as $\psi_{t}=\rho w_{t}$,
$D\psi=wD\rho+\rho Dw$, and
$\psi_{ij}=w_{ij}\rho+w_{i}\rho_{j}+w_{j}\rho_{i}+w\rho_{ij}$, we have at
$(x_{1},t_{1})$,

$w_{t}\geqslant-\frac{\rho_{t}}{\rho}w=0,\quad Dw=-\frac{w}{\rho}D\rho,\quad
w_{ij}=\frac{1}{\rho}(\psi_{ij}-w_{i}\rho_{j}-w_{j}\rho_{i}-w\rho_{ij}).$

We use (2.7) and the above to yield that, at $(x_{1},t_{1})$,

$\frac{w^{2}}{\rho}(D_{p}b^{ij}\cdot
D\rho)(u-\theta)_{ij}+\frac{w}{\rho}b^{ij}(w_{i}\rho_{j}+w_{j}\rho_{i}+w\rho_{ij})\\\
+b^{ij}b^{kl}(u-\theta)_{ki}(u-\theta)_{lj}-wD(u-\theta)\cdot
Dc+\frac{cw}{\rho}D(u-\theta)\cdot D\rho\leqslant 0.$

By direct computations

$b^{ij}_{p_{l}}=-\frac{\delta_{il}(u-\theta)_{j}}{\varepsilon^{2}+|D(u-\theta)|^{2}}-\frac{\delta_{jl}(u-\theta)_{i}}{\varepsilon^{2}+|D(u-\theta)|^{2}}+\frac{2(u-\theta)_{i}(u-\theta)_{j}(u-\theta)_{l}}{(\varepsilon^{2}+|D(u-\theta)|^{2})^{2}},$

and hence,

$\displaystyle w(D_{p}b^{ij}\cdot D\rho)(u-\theta)_{ij}$ $\displaystyle=\,$
$\displaystyle
w\left(-\frac{\rho_{i}(u-\theta)_{j}(u-\theta)_{ij}}{\varepsilon^{2}+|D(u-\theta)|^{2}}-\frac{\rho_{j}(u-\theta)_{i}(u-\theta)_{ij}}{\varepsilon^{2}+|D(u-\theta)|^{2}}+\frac{2(u-\theta)_{i}(u-\theta)_{j}(u-\theta)_{l}\rho_{l}(u-\theta)_{ij}}{(\varepsilon^{2}+|D(u-\theta)|^{2})^{2}}\right)$
$\displaystyle=\,$ $\displaystyle-2Dw\cdot D\rho+\frac{2(D(u-\theta)\cdot
D\rho)(D(u-\theta)\cdot Dw)}{w^{2}}.$

Therefore,

$\displaystyle w(D_{p}b^{ij}\cdot
D\rho)(u-\theta)_{ij}+b^{ij}w_{i}\rho_{j}+b^{ij}w_{j}\rho_{i}$
$\displaystyle=\ $ $\displaystyle\frac{2(D(u-\theta)\cdot
D\rho)(D(u-\theta)\cdot
Dw)}{w^{2}}-\frac{(u-\theta)_{i}(u-\theta)_{j}w_{i}\rho_{j}}{w^{2}}-\frac{(u-\theta)_{i}(u-\theta)_{j}w_{j}\rho_{i}}{w^{2}}$
$\displaystyle=\ $ $\displaystyle 0.$

Thus, at $(x_{1},t_{1})$,

$\frac{\rho_{ij}}{\rho}b^{ij}w^{2}+b^{ij}b^{kl}(u-\theta)_{ki}(u-\theta)_{lj}-wD(u-\theta)\cdot
Dc+\frac{cw}{\rho}D(u-\theta)\cdot D\rho\leqslant 0.$ (2.10)

Using the Cauchy-Schwarz type inequality as in the above, we deduce that

$\displaystyle\frac{1}{2}(u_{t}-c(x_{1})w)^{2}$ $\displaystyle\leqslant
b^{ij}b^{kl}(u-\theta)_{ki}(u-\theta)_{lj}$
$\displaystyle\leqslant-\frac{w^{2}}{\rho}b^{ij}\rho_{ij}+wD(u-\theta)\cdot
Dc-\frac{cw}{\rho}D(u-\theta)\cdot D\rho$
$\displaystyle\leqslant\frac{2C_{0}}{K_{0}}\frac{w^{2}}{\rho}\left(2-\frac{|D(u-\theta)|^{2}}{\varepsilon^{2}+|D(u-\theta)|^{2}}\right)+|Dc|w^{2}+C_{0}|c|w^{2}$
$\displaystyle\leqslant\left(\frac{4C_{0}}{K_{0}}+|Dc(x_{1})|+C_{0}|c(x_{1})|\right)w^{2}.$

Combine the above with (1.4) that

$\frac{1}{2}c(x)^{2}-|Dc(x)|-C_{0}|c(x)|-\frac{4C_{0}}{K_{0}}\geqslant\frac{\delta}{2}>0\quad\quad\text{for
all}\ x\in\overline{W},$

we see that
$w(x_{1},t_{1})\leqslant\frac{2M\|c\|_{L^{\infty}(\overline{W})}}{\delta}$.
Thus,

$w(x_{0},t_{0})\leqslant\rho(x_{1})w(x_{1},t_{1})\leqslant\left(\frac{C_{0}K_{0}}{4}+1\right)\frac{2M\|c\|_{L^{\infty}(\overline{W})}}{\delta}.$

∎

##  3\. Large time averages and behaviors

###  3.1. Large time average results

We first prove Theorem 1.3. Our approach is similar to that in [8].

###### Proof of Theorem 1.3.

We note that $u\in C(\overline{W}\times[0,\infty))$ and in light of (2.4),

$u_{0}(x)-Mt\leqslant u(x,t)\leqslant u_{0}(x)+Mt\qquad\text{ for all
}(x,t)\in\overline{W}\times[0,\infty),$

where $M$ is the constant defined in (2.3). Thus, $S\in C([0,\infty))$, and
$|S(t)|\leqslant\|u_{0}\|_{L^{\infty}(\overline{W})}+Mt$ for $t\geqslant 0$.

Fix $s\geqslant 0$. By the definition of $S$, $u(x,s)\leqslant
u_{0}(x)+\|u_{0}\|_{L^{\infty}(\overline{W})}+S(s)$ for $x\in\overline{W}$.
Then, by the comparison principle for (1.3), we imply that, for
$x\in\overline{W}$ and $t\geqslant 0$,

$u(x,s+t)\leqslant u(x,t)+\|u_{0}\|_{L^{\infty}(\overline{W})}+S(s).$

In particular,

$S(s+t)\leqslant S(t)+S(s)+\|u_{0}\|_{L^{\infty}(\overline{W})}.$

Thus, $t\mapsto S(t)+\|u_{0}\|_{L^{\infty}(\overline{W})}$ is subadditive. By
the Fekete lemma, there exists $S=S_{c}\in\mathbb{R}$ such that

$\lim_{t\to\infty}\frac{S(t)+\|u_{0}\|_{L^{\infty}(\overline{W})}}{t}=\lim_{t\to\infty}\frac{S(t)}{t}=S.$

In fact,

$S=\inf_{t>0}\frac{S(t)+\|u_{0}\|_{L^{\infty}(\overline{W})}}{t},$

which yields that $|S|\leqslant M$.

Let us assume further that (1.4) holds. Then, by Theorem 1.1,
$\|Du\|_{L^{\infty}(\overline{W}\times[0,\infty))}\leqslant L$. In particular,
for $(x,t)\in\overline{W}\times[0,\infty)$,

$|u(x,t)-S(t)|\leqslant C.$

Hence, uniformly for $x\in\overline{W}$,

$\lim_{t\to\infty}\frac{u(x,t)}{t}=\lim_{t\to\infty}\frac{S(t)}{t}=S,$

which gives (1.6). ∎

###### Proof of Theorem 1.4.

Notice first that $\|\Theta\|_{L^{\infty}(\overline{W})}\leqslant
2\pi\sum_{j=1}^{N}|m_{j}|$. Thus, for $(x,t)\in\overline{W}\times(0,\infty)$,

$-\frac{C}{t}\leqslant\frac{u(x,t)}{t}-\frac{2\pi
k(x,t)}{t}\leqslant\frac{C}{t},$

and

$-\frac{C}{t}\leqslant\frac{h(x,t)}{t}-\frac{h_{0}k(x,t)}{t}\leqslant\frac{C}{t},$

where $C=(2\pi+h_{0})\left(\sum_{j=1}^{N}|m_{j}|+1\right)$. ∎

###  3.2. Properties of the asymptotic growth rate

We next provide further analysis on the properties of the asymptotic growth
rate $S_{c}$ of $u$.

###### Proof of Lemma 1.6.

It is enough to obtain the lower bound of $S_{c}$ as the proof for the upper
bound of $S_{c}$ follows analogously. Set
$\gamma=\min_{x\in\overline{W}}c(x)/|x|>0$. We claim that

$\phi(x,t)=-\|u_{0}\|_{L^{\infty}(\overline{W})}+\gamma t$

is a viscosity subsolution to (1.3), which will automatically give us that
$S_{c}\geqslant\gamma$ by the comparison principle.

Let us now prove this claim. It is obvious that $\phi(\cdot,0)\leqslant
u_{0}$. Next, note that
$D(\phi-\theta)(x,t)=-D\theta(x)=-{x^{\perp}}/{|x|^{2}}$, and hence,

${\rm div}\,\left(\frac{D(\phi-\theta)}{|D(\phi-\theta)|}\right)=-{\rm
div}\,\left(\frac{x^{\perp}}{|x|}\right)=0.$

Thus,

$\displaystyle\phi_{t}-|D(\phi-\theta)|\left({\rm
div}\,\left(\frac{D(\phi-\theta)}{|D(\phi-\theta)|}\right)+c(x)\right)=\gamma-\frac{c(x)}{|x|}\leqslant
0.$

The proof is complete. ∎

###### Remark 2.

Let us note that in the situation of Lemma 1.6, it is also possible to
construct subsolutions and supersolutions to (1.3) of the form

$\phi(x,t)=\varphi\left(\frac{x}{|x|}\right)+\gamma t,$

where $\varphi$ is a smooth function to be decided. Nevertheless, the lower
and upper bounds of $S_{c}$ are unchanged, regardless of the specific choice
of subsolutions and supersolutions in this form.

###### Remark 3.

In the situation of Lemma 1.6, if we assume further that $c(x)=c_{0}|x|$ for
all $x\in\overline{W}$ for some given constant $c_{0}>0$, then
$\phi(x,t)=\alpha+c_{0}t$ is a viscosity solution to (1.3) with
$\phi(x,0)=\alpha$ for any $\alpha\in\mathbb{R}$. Moreover, in this particular
case, we have that $S_{c}=c_{0}$, which is independent of $r$, the radius of
the deleted hole.

Next, we consider the situation where there are two spiral centers which
rotate in different directions of the same strength. This was first studied in
[1] in which the two spirals are called an opposite rotating pair. If the two
spiral centers are close to each other, that is, $|a_{1}-a_{2}|\leqslant
2/\max_{\overline{W}}c$, then they are called an inactive pair, in which case
$S_{c}=0$. (see Proposition 3.2). If the two spiral centers are far away from
each other, [1] gives some heuristic explanations that this opposite rotating
pair actually accelerates the growth rate, which is faster than that of a
single spiral. There has not been a rigorous proof of this point in the
literature yet. We refer the reader to [25, 26] for numerical results.

We give several results along this line. The first one is a toy model case
(see Figure 2).

###### Proposition 3.1.

Assume that $N=2$, $a_{1}=(l,0)$, $a_{2}=(-l,0)$ with $0<r<l$, and

$\big{[}B(a_{1},r)\cup B(a_{2},r)\cup((-l,l)\times(-r,r))\big{]}\cap
W=\emptyset.$

Assume furthermore that $m_{1}=-m_{2}$. Then, $S_{c}=0$.

$W$

Fig. 2. An example of $W$ in Proposition 3.1.

We now consider the situation where the two spiral centers are close to each
other (see Figure 3).

###### Proposition 3.2.

Assume that $N=2$, $a_{1}=(l,0)$, $a_{2}=(-l,0)$ with

$0<r<l\leqslant R_{0}:=\frac{1}{\max_{\overline{W}}c},$

and $m_{1}=-m_{2}$. Pick $\theta\in(0,\pi/2]$ such that
$R_{0}=(l-r\cos\theta)/\sin\theta$. Denote by
$A=(-l,0)+r(\cos\theta,\sin\theta)$, $B=(l,0)+r(-\cos\theta,\sin\theta)$,
$C=(-l,0)+r(\cos\theta,-\sin\theta)$, $D=(l,0)+r(-\cos\theta,-\sin\theta)$.
Let $\overset{\frown}{AB}$ be an arc of a circle of radius $R_{0}$ that is
perpendicular to both $\partial B(a_{1},r)$ and $\partial B(a_{2},r)$. Let
$\overset{\frown}{CD}$ be an arc of a circle of radius $R_{0}$ that is
perpendicular to both $\partial B(a_{1},r)$ and $\partial B(a_{2},r)$. Assume
furthermore that $\overset{\frown}{AB},\overset{\frown}{CD}\subset W$. Then,
$S_{c}=0$.

When $c(\cdot)$ is constant, Proposition 3.2 was already announced in [23].

$A$$B$$C$$D$$0$$R_{0}$

Fig. 3. An example of $W$ in Proposition 3.2 with $\theta=\pi/2$.

By using a similar argument as that in the proof of Proposition 3.2, we obtain
the following rather surprising result with $N=1$ (see Figure 4).

###### Proposition 3.3.

Assume that $N=1$, $m_{1}=1$. Let $R_{0}=1/{\max_{\overline{W}}c}$. Assume
that there exist two arcs $\overset{\frown}{AB},\overset{\frown}{CD}$ of two
circles of radii $R_{0}$ with $A,B\in\partial B(a_{1},r)$,
$\overset{\frown}{AB},\overset{\frown}{CD}\subset W$ such that they are both
perpendicular to $\partial B(a_{1},r)$ and $\partial\Omega$. Assume
furthermore that $\overset{\frown}{AB},\overset{\frown}{CD}$, $\partial
B(a_{1},r)$, and $\partial\Omega$ separate $W$ into two connected components
$W_{1},W_{2}$ such that $\overset{\frown}{AB},\overset{\frown}{CD}$ are convex
on $\partial W_{1}$, and $\arg(x-a_{1})$ is well defined in each of $W_{1}$
and $W_{2}$. Then, $S_{c}=0$.

We note that Proposition 3.3 is related to Theorem 4.1, in which we give an
example of a non-uniformly Lipschitz continuous solution of (1.3).

###### Remark 4.

It is not hard to construct other examples with $N\geqslant 2$ with general
$m_{1},\ldots,m_{N}\in\mathbb{Z}\setminus\\{0\\}$ that are similar to those in
Propositions 3.2–3.3 with $S_{c}=0$. We thus see that the asymptotic growth
rate $S_{c}$ demonstrates notable sensitivity to the geometric characteristics
of the boundary $\partial\Omega$.

$A$$B$$C$$D$$W_{1}$$W_{2}$$R_{0}$

Fig. 4. An example of $W$ in Proposition 3.3.

###### Proof of Proposition 3.1.

It is very interesting that in this case

$\theta(x):=\sum_{j=1}^{2}m_{j}\arg(x-a_{j})$

is well-defined in $W$ as $m_{1}=-m_{2}$ and $a_{1}=(l,0),a_{2}=(-l,0)$. It is
thus easy to see that

$\varphi^{\pm}(x,t)=\pm(\|u_{0}\|_{L^{\infty}(\overline{W})}+\|\theta\|_{L^{\infty}(\overline{W})})+\theta(x)$

is a supersolution and a subsolution to (1.3), respectively. Therefore, for
$(x,t)\in\overline{W}\times[0,\infty)$,

$-(\|u_{0}\|_{L^{\infty}(\overline{W})}+\|\theta\|_{L^{\infty}(\overline{W})})+\theta(x)\leqslant
u(x,t)\leqslant(\|u_{0}\|_{L^{\infty}(\overline{W})}+\|\theta\|_{L^{\infty}(\overline{W})})+\theta(x).$

We then get the desired result. ∎

We now give a proof of Proposition 3.2.

###### Proof of Proposition 3.2.

Without loss of generality, assume $m_{1}=-m_{2}=1$. Let $W_{1}$ be the region
inside $W$ enclosed by the arcs $\overset{\frown}{AB},\overset{\frown}{CD}$
and $\partial B(a_{1},r),\partial B(a_{2},r)$. Let
$W_{2}=W\setminus\overline{W_{1}}$. Note that

$\theta(x):=\arg(x-a_{1})-\arg(x-a_{2})$

is well-defined in $W_{1}$ and $W_{2}$ separately. Let $v=\theta+2\pi k$ be
such that, with an appropriate choice of branch cuts, we have for
$w=v-\theta$, then

$w(x)=\begin{cases}2\pi\qquad&\text{ for }x\in W_{1},\\\ 0\qquad&\text{ for
}x\in W_{2}.\end{cases}$

See Figure 5.

$A$$B$$C$$D$$W_{1}$$W_{2}$$w=2\pi$$w=0$

Fig. 5. Values of $w$ in $W$ in Proposition 3.2.

Surely, $v$ is discontinuous across the arcs
$\overset{\frown}{AB},\overset{\frown}{CD}$. We now show that $v$ is a
supersolution to (1.3) by proving that $w=v-\theta$ is a supersolution to

$\begin{cases}-|Dw|\left(\operatorname{div}\left(\frac{Dw}{|Dw|}\right)+\frac{1}{R_{0}}\right)=0\qquad&\text{
in }W,\\\ Dw\cdot\mathbf{n}=0\qquad&\text{ on }\partial W.\end{cases}$

We only need to check the supersolution property on the open arcs
$\overset{\frown}{AB},\overset{\frown}{CD}$ and at the boundary points
$A,B,C,D$.

First, let $\phi$ be a smooth test function such that $\phi$ touches $w_{*}$
from below at a point $x_{0}\in\overset{\frown}{AB}\cap W$. Thanks to [9,
Lemma A.1], we get that there exists $s\leqslant 0$ such that

$|D\phi(x_{0})|=|s|\quad\text{ and
}\quad|D\phi(x_{0})|\operatorname{div}\left(\frac{D\phi(x_{0})}{|D\phi(x_{0})|}\right)\leqslant\frac{s}{R_{0}},$

which gives the desired supersolution test.

Second, let $\phi$ be a smooth test function such that $\phi$ touches $w_{*}$
from below at $A$. Parameterize the arc $\overset{\frown}{AB}$ by a smooth
curve $\xi$ such that $\xi(0)=A$ and $\xi(1)=B$. Then we have $\phi(\xi(0))=0$
and $\phi(\xi(t))\leqslant 0$ for all $t\in[0,1]$, which imply

$D\phi(\xi(0))\cdot\dot{\xi}(0)\leqslant 0.$

Therefore, at $A$,

$D\phi(A)\cdot\mathbf{n}\geqslant 0.$

Thus, $v$ is a supersolution to (1.3). By the usual comparison principle, we
yield that, for $(x,t)\in\overline{W}\times[0,\infty)$,

$u(x,t)\leqslant\|u_{0}\|_{L^{\infty}(\overline{W})}+\|v\|_{L^{\infty}(\overline{W})}+v(x).$

The proof is complete. ∎

We give a proof of Proposition 3.3, which is similar to that of Proposition
3.2.

###### Proof of Proposition 3.3.

By our assumptions, we have that

$\theta(x):=\arg(x-a_{1})$

is well-defined in $W_{1}$ and $W_{2}$ separately. Let $v=\theta+2\pi k$ be
such that, with an appropriate choice of branch cuts, we have for
$w=v-\theta$, then

$w(x)=\begin{cases}2\pi\qquad&\text{ for }x\in W_{1},\\\ 0\qquad&\text{ for
}x\in W_{2}.\end{cases}$

See Figure 6.

$A$$B$$C$$D$$W_{1}$$W_{2}$$w=2\pi$$w=0$

Fig. 6. Values of $w$ in $W$ in Proposition 3.3.

Of course, $v$ is discontinuous across the arcs
$\overset{\frown}{AB},\overset{\frown}{CD}$. We now show that $v$ is a
supersolution to (1.3) by proving that $w=v-\theta$ is a supersolution to

$\begin{cases}-|Dw|\left(\operatorname{div}\left(\frac{Dw}{|Dw|}\right)+\frac{1}{R_{0}}\right)=0\qquad&\text{
in }W,\\\ Dw\cdot\mathbf{n}=0\qquad&\text{ on }\partial W.\end{cases}$

We only need to check the supersolution property on the open arcs
$\overset{\frown}{AB},\overset{\frown}{CD}$ and at the boundary points
$A,B,C,D$.

First, let $\phi$ be a smooth test function such that $\phi$ touches $w_{*}$
from below at a point $x_{0}\in\overset{\frown}{AB}\cap W$. Thanks to [9,
Lemma A.1], we get that there exists $s\leqslant 0$ such that

$|D\phi(x_{0})|=|s|\quad\text{ and
}\quad|D\phi(x_{0})|\operatorname{div}\left(\frac{D\phi(x_{0})}{|D\phi(x_{0})|}\right)\leqslant\frac{s}{R_{0}},$

which gives the desired supersolution test.

Second, we need to check the supersolution property at the boundary points
$A,B,C,D$. It is enough to check it at $B$ as the proof is analogous for other
points. let $\phi$ be a smooth test function such that $\phi$ touches $w_{*}$
from below at $B$. Parameterize the arc $\overset{\frown}{AB}$ by a smooth
curve $\eta$ such that $\eta(0)=B$ and $\eta(1)=A$. Then we have
$\phi(\eta(0))=0$ and $\phi(\eta(t))\leqslant 0$ for all $t\in[0,1]$, which
imply

$D\phi(\eta(0))\cdot\dot{\eta}(0)\leqslant 0.$

Therefore, at $B$,

$D\phi(B)\cdot\mathbf{n}\geqslant 0.$

Thus, $v$ is a supersolution to (1.3). By the usual comparison principle, we
yield that, for $(x,t)\in\overline{W}\times[0,\infty)$,

$u(x,t)\leqslant\|u_{0}\|_{L^{\infty}(\overline{W})}+\|v\|_{L^{\infty}(\overline{W})}+v(x).$

The proof is complete. ∎

###  3.3. Large time behaviors

The proof of Theorem 1.7 follows that of [12, Theorem 1.2] or [16, Theorem
1.3] by using a standard Lyapunov function, and is hence omitted. We remark
that assuming both (1.4) and $S_{c}=0$ is really restrictive.

Let us now prove Proposition 1.8.

###### Proof of Proposition 1.8.

This can be done by direct computations. Set

$u(x,t)=g\left(\mathcal{R}_{-c_{0}t}\frac{x}{|x|}\right)+c_{0}t\qquad\text{
for all }(x,t)\in\overline{W}\times[0,\infty).$

Then, for $(x,t)\in\overline{W}\times[0,\infty)$,

$\displaystyle u_{t}$
$\displaystyle=\frac{d}{dt}(\mathcal{R}_{-c_{0}t})Dg\cdot\frac{x}{|x|}+c_{0}$
$\displaystyle=c_{0}\left(1-Dg\cdot\left(\mathcal{R}_{\pi/2-c_{0}t}\frac{x}{|x|}\right)\right)=c_{0}\left(1-Dg\cdot\left(\mathcal{R}_{-c_{0}t}\frac{x^{\perp}}{|x|}\right)\right),$

and

$Du=\frac{\mathcal{R}_{-c_{0}t}}{|x|}\left(Dg\cdot\frac{x^{\perp}}{|x|}\right)\frac{x^{\perp}}{|x|}.$

Therefore, for $(x,t)\in\overline{W}\times[0,\infty)$, we use the assumption
that $\|Dg\|_{L^{\infty}}<1$ to yield

$|D(u-\theta)|=\frac{1}{|x|}\left(1-Dg\cdot\left(\mathcal{R}_{-c_{0}t}\frac{x^{\perp}}{|x|}\right)\right).$

As $c(x)=c_{0}|x|$ and $\operatorname{div}(x^{\perp}/|x|)=0$, we imply the
conclusion. ∎

##  4\. An example of a non-uniformly Lipschitz solution to (1.3)

In this section we construct a non-uniformly Lipschitz continuous solution to
(1.3). Let us consider $c\equiv 1$, in which case (1.3) becomes

$\begin{cases}u_{t}=|D(u-\theta)|\left({\rm
div}\,\left(\frac{D(u-\theta)}{|D(u-\theta)|}\right)+1\right)\qquad&\text{ in
}W\times(0,\infty),\\\ D(u-\theta)\cdot\mathbf{n}=0\qquad&\text{ on }\partial
W\times(0,\infty),\\\ u(\cdot,0)=u_{0}\qquad&\text{ on
}\overline{W},\end{cases}$ (4.1)

for a domain $W\subset\mathbb{R}^{2}$ to be constructed. Fix $r>0$, and take
$R>r$, where $R$ will be chosen later. Let $\bm{a}:=(-r,0)\in\partial B(0,r)$,
take $\overline{\bm{a}}:=(-r,\overline{a}_{2})\in\partial B(0,R)$ with
$\overline{a}_{2}>0$. Note that $(\overline{\bm{a}}-\bm{a})\cdot\bm{a}=0$. Let
$\overline{\bm{b}}:=(-\frac{r}{\sqrt{2}},\frac{r}{\sqrt{2}})$. Take
$\overline{\bm{b}}=(\overline{b}_{1},\overline{b}_{2})\in\partial B(0,R)$ so
that

$(\overline{\bm{b}}-\bm{b})\cdot\bm{b}=0,\quad\overline{b}_{2}>0.$

Set $\overline{\bm{c}}:=(\overline{\bm{a}}+\overline{\bm{b}})/{2}$. Take
$\bm{c}=(c_{1},c_{2})\in\partial B(0,r)$ so that

$(\overline{\bm{c}}-\bm{c})\cdot\bm{c}=0,\quad
c_{1}\in\left(-r,-\frac{r}{\sqrt{2}}\right).$

Note that
$\bm{a},\overline{\bm{a}},\bm{b},\overline{\bm{b}},\bm{c},\overline{\bm{c}}$
are uniquely determined.

Set $l:=|\bm{c}-\overline{\bm{c}}|$, and note that $l=l(R)$ depends on $R>0$.
By the construction, it is clear to see that $l(R_{1})<l(R_{2})$ if
$0<R_{1}<R_{2}$. Since $l(R)\to 0$ as $R\to r$, $l(R)\to\infty$ as
$R\to\infty$, there exists $R>0$ such that $l(R)=1$. Henceforth, we fix such a
$R>0$. We set

$\left[\overline{\bm{c}},\overline{\bm{b}}\right]:=\left\\{\overline{\bm{d}}(s):=\overline{\bm{c}}+s\left(\overline{\bm{b}}-\overline{\bm{c}}\right)\,:\,s\in[0,1]\right\\}.$

Take $\bm{d}(s)=(d_{1}(s),d_{2}(s))\in\partial B(0,r)$ so that

$(\overline{\bm{d}}(s)-\bm{d}(s))\cdot\bm{d}(s)=0,\quad
d_{1}(s)\in\left(c_{1},-\frac{r}{\sqrt{2}}\right).$

We define a family of arcs by

$\Gamma(s):=\left\\{\overline{\bm{d}}(s)+\mathcal{R}_{\varphi}(\bm{d}(s)-\overline{\bm{d}}(s))\,:\,\varphi\in\left[0,\frac{\pi}{2}\right]\right\\}.$

Recall that

$\mathcal{R}_{\varphi}=\begin{pmatrix}\cos\varphi&-\sin\varphi\\\
\sin\varphi&\cos\varphi\end{pmatrix}.$

It is important to note that $|\bm{d}(s)-\overline{\bm{d}}(s)|>1$ for
$s\in(0,1]$. Set the vector field by

$\bm{n}(x):=\frac{x-\overline{\bm{d}}(s)}{|x-\overline{\bm{d}}(s)|}\qquad\text{
for }x\in\Gamma(s).$

Note that, by construction, it is clear that $\bm{n}$ is smooth. Therefore,
for a fixed $z\in\Gamma(0)$, there exists $t_{0}>0$ such that the initial
value problem of the ordinary differential equation

$\left\\{\begin{array}[]{ll}\dot{\bm{\gamma}}(t)=\bm{n}(\bm{\gamma}(t))&\text{
for}\ t\in(0,t_{0}),\\\ \bm{\gamma}(0)=z\end{array}\right.$

has a solution at least for a short time. Write
$\bm{\gamma}(t)=(\gamma_{1}(t),\gamma_{2}(t))$. For $t_{0}>0$ is small enough,
$\gamma_{2}(t)>0$ for $t\in(0,t_{0})$. Take $s_{0}\in(0,1)$ so that
$\bm{\gamma}(t_{0})\in\Gamma(s_{0})$. Next, we extend
$\bm{\gamma}=(\gamma_{1},\gamma_{2})$ to be a $C^{2}$ curve satisfying there
exist $t_{1}<0<t_{0}<t_{2}$ such that

$\left\\{\begin{array}[]{ll}\gamma_{2}(t)>0\qquad\text{ for all}\
t\in(t_{1},t_{2}),\\\ \gamma_{2}(t_{1})=\gamma_{2}(t_{2})=0,\\\
\dot{\gamma}_{2}(t_{1})=\dot{\gamma}_{2}(t_{2})=0.\end{array}\right.$

Finally, we define $\partial\Omega\subset\mathbb{R}^{2}$ by

$\partial\Omega:=\left\\{(\gamma_{1}(t),\gamma_{2}(t))\,:\,t\in[t_{1},t_{2}]\right\\}\cup\left\\{(\gamma_{1}(t),-\gamma_{2}(t))\,:\,t\in[t_{1},t_{2}]\right\\},$
(4.2)

and we denote by $W$ the domain enclosed by $\partial\Omega\cup\partial
B(0,r)$. Here, we assume $N=1$, $a_{1}=0$, $m_{1}=1$. Then,
$\theta(x)=\arg(x)$.

Fig. 7. The constructed domain $W$.

Now, let $0<\alpha<\beta<2\pi$, and we define $u_{0}\in{\rm
Lip\,}(\overline{W})$ by

$u_{0}(x_{1},x_{2})-\theta(x):=\left\\{\begin{array}[]{ll}\alpha+\frac{s}{s_{0}}(\beta-\alpha)&\text{for}\
(x_{1},x_{2})\in\Gamma(s),\ s\in[0,s_{0}],\\\
\alpha+\frac{s}{s_{0}}(\beta-2\pi-\alpha)&\text{for}\
(x_{1},x_{2})\in\tilde{\Gamma}(s),\ s\in[0,s_{0}],\end{array}\right.$ (4.3)

where we set
$\tilde{\Gamma}(s):=\\{(x_{1},-x_{2})\,:\,(x_{1},x_{2})\in\Gamma(s)\\}$.

We set

$\mathcal{A}:=\bigcup_{0\leq s\leq
s_{0}}\Gamma(s),\qquad\tilde{\mathcal{A}}:=\bigcup_{0\leq s\leq
s_{0}}\tilde{\Gamma}(s).$

Take $x_{1}>r$ such that $\pm(x_{1},0)\in\Omega$. We denote by $B_{1}$ and
$B_{2}$ the connected component in
$\overline{W}\setminus(\mathcal{A}\cup\tilde{\mathcal{A}})$ which include
$(-x_{1},0)$ and $(x_{1},0)$, respectively. We set

$u_{0}(x_{1},x_{2})-\theta(x):=\left\\{\begin{array}[]{ll}\alpha&\text{ for}\
(x_{1},x_{2})\in B_{1}\\\ \beta&\text{ for}\ (x_{1},x_{2})\in
B_{2}.\end{array}\right.$ (4.4)

We note that $\Gamma(0)$ and $\tilde{\Gamma}(0)$ are exactly the two arcs
$\overset{\frown}{AB},\overset{\frown}{CD}$ of two circles of radii $1$
required in Proposition 3.3. For $s\in(0,s_{0}]$, $\Gamma(s)$ is an arc of a
circle of radius greater than $1$. Naturally, under the normal velocity
$V=\kappa+1$, all the arcs $\Gamma(s)$ for $s\in(0,s_{0}]$ converge to
$\Gamma(0)$ as time tends to infinity. See Figure 7. By the above
construction, the result in the following theorem follows immediately.

######  Theorem 4.1.

Assume $N=1$, $a_{1}=0$, $m_{1}=1$. Let $\Omega$ be the set defined by (4.2),
and let $u_{0}\in{\rm Lip\,}(\overline{W})$ be the function defined by (4.3)
and (4.4). Let $u$ be the solution to (4.1). Then,

$\lim_{t\to\infty}(u(x,t)-\theta(x))=\left\\{\begin{array}[]{ll}\alpha&\text{
for}\ (x_{1},x_{2})\in B_{1}\\\ \beta&\text{ for}\ (x_{1},x_{2})\in
W\setminus\overline{B_{1}}.\end{array}\right.$

In particular, $S_{c}=0$.

We skip the proof of this theorem and we refer the reader to [16, Section 6]
for detail on a rather similar result.

## Data availability

Data sharing not applicable to this article as no datasets were generated or
analyzed during the current study.

## Conflict of interest

There is no conflict of interest.

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