# Global well-posedness of the defocusing, cubic nonlinear wave equation
outside of the ball with radial data

Guixiang Xu  Guixiang Xu
Laboratory of Mathematics and Complex Systems,
Ministry of Education,
School of Mathematical Sciences,
Beijing Normal University,
Beijing, 100875, People’s Republic of China<EMAIL_ADDRESS>and
Pengxuan Yang  Pengxuan Yang
Laboratory of Mathematics and Complex Systems,
Ministry of Education,
School of Mathematical Sciences,
Beijing Normal University,
Beijing, 100875, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

We consider the defocusing, cubic nonlinear wave equation with zero Dirichlet
boundary value in the exterior $\Omega=\mathbb{R}^{3}\backslash\bar{B}(0,1)$.
We make use of the distorted Fourier transform in [35, 50] to establish the
dispersive estimate and the global-in-time (endpoint) Strichartz estimate of
the linear wave equation outside of the ball with radial data. As an
application, we combine the Fourier truncation method as those in [4, 18, 29]
with the energy method to show global well-posedness of radial solution to the
defoucusing, cubic nonlinear wave equation outside of a ball in the Sobolev
space $\left(\dot{H}^{s}_{D}(\Omega)\cap
L^{4}(\Omega)\right)\times\dot{H}^{s-1}_{D}(\Omega)$ with $s>3/4$. To the best
of the author’s knowledge, it is first result about low regularity of
semilinear wave equation with zero Dirichlet boundary value on the exterior
domain.

###### Key words and phrases:

Dispersive estimate; Distorted Fourier transform; Exterior domain; Strichartz
estimate; Wave equation

###### 2010 Mathematics Subject Classification:

Primary: 35L05; Secondary: 35E05, 35L20

##  1\. Introduction

In this paper, we consider the 3D defocusing, cubic wave equation with zero
Dirichlet boundary value outside of the ball

$\begin{cases}\partial^{2}_{t}u-\Delta u+u^{3}=0,&(t,x)\in I\times\Omega,\\\
u(0,x)=u_{0}(x),&x\in\Omega,\\\ \partial_{t}u(0,x)=u_{1}(x),&x\in\Omega,\\\
u(t,x)=0,&x\in\partial\Omega,\end{cases}$ (1.1)

where $0\in I$, the exterior domain
$\Omega=\mathbb{R}^{3}\backslash\bar{B}(0,1)$, the function
$u:I\times\Omega\longrightarrow\mathbb{R}$ and initial data $(u_{0},u_{1})$ is
radial and belongs to the Sobolev space
$\dot{H}^{s}_{D}(\Omega)\times\dot{H}^{s-1}_{D}(\Omega)$. The solution of
equation (1.1) formly enjoys the energy conservation

$\displaystyle E(u)(t)=\int_{\Omega}\left(\frac{1}{2}|\nabla
u(t,x)|^{2}+\frac{1}{2}|\partial_{t}u(t,x)|^{2}+\frac{1}{4}|u(t,x)|^{4}\right)\;dx=E(u)(0),\quad\forall\,t\in
I.$ (1.2)

Typically, we call (1.1) the (conformal) $\dot{H}^{1/2}$-critical wave
equation by the scaling analysis.

There are lots of works focusing on the Cauchy problem of semilinear wave
equation including the energy (sub)-critical wave equation (see [11, 12, 15,
27, 28, 37, 41, 45] and reference therein), and there are also many important
results about the energy-critical wave/Schrödinger equations outside of the
domain (see [6, 7, 14, 17, 31, 35, 36, 42] and reference therein). The energy
conservation law plays a crutial rule in long time dynamics in both of the
Cauchy problem (IVP) and the initial boundary value problem (IBVP) of the
energy critical wave/Schrödinger equations.

By the heat kernel estimate on a convex obstacle in [51] and classical
harmonic analysis theory including the Calderón-Zygnumd theory and Mikhlin’s
multiplier theorem in [21, 38], etc, Killip, Visan and Zhang establish the
boundedness of Riesz transform, the Littlewood-Paley theory of the Sobolev
space on a convex obstacle and the equivalence between
$\dot{H}^{s,p}_{D}(\Omega)$ with $\dot{H}^{s,p}(\mathbb{R}^{d})$ for the sharp
ranges of regularity exponent $s$ and integrability exponent $p$ in [30] (see
also [24]). We will recall more useful properties of the Sobolev space
$\dot{H}^{s,p}_{D}(\Omega)$ in [30, 35] in Section 2.

We now give the definition of local-well-posedness of (1.1) in
$\dot{H}^{s}_{D}(\Omega)\times\dot{H}^{s-1}_{D}(\Omega)$.

######  Definition 1.1.

The equation (1.1) is said to be locally well-posed in
$\dot{H}^{s}_{D}(\Omega)\times\dot{H}^{s-1}_{D}(\Omega)$ if there exists an
open interval $I\subset\mathbb{R}$ containing $0$ such that $(1)$ there is a
unique solution in $\dot{H}^{s}_{D}(\Omega)\times\dot{H}^{s-1}_{D}(\Omega)$;
$(2)$ the solution is continuous in time, that is $(u,\partial_{t}u)\in
C\left(I;\;\dot{H}^{s}_{D}(\Omega)\times\dot{H}^{s-1}_{D}(\Omega)\right)$;
$(3)$ the solution depends continuously on the initial data.

By the Strichartz estimate in Theorem 3.2 and standard Picard fixed point
argument, we can obtain the following local well-posedness result of (1.1) in
$\dot{H}^{s}_{D}(\Omega)\times\dot{H}^{s-1}_{D}(\Omega)$, $\frac{1}{2}\leq
s<\frac{3}{2}$.

######  Theorem 1.2.

The equation (1.1) is locally well-posed in
$\dot{H}^{1/2}_{D}(\Omega)\times\dot{H}^{-1/2}_{D}(\Omega)$ on some interval
$I=\left(-T,T\right)$. Moreover, the regularity of initial data is enough to
give a lower bound on the time of well-posedness, that is, there exists some
positive lifespan
$T=T\big{(}\|(u_{0},u_{1})\|_{\dot{H}^{s}_{D}(\Omega)\times\dot{H}^{s-1}_{D}(\Omega)}\big{)}$
for any $\frac{1}{2}<s<\frac{3}{2}$.

The local well-posedness theory together with the energy conservation law
implies that

######  Theorem 1.3.

The equation (1.1) is globally well-posed in in
$\left(\dot{H}^{1}_{D}(\Omega)\cap L^{4}(\Omega)\right)\times L^{2}(\Omega)$.

As shown in the local/global well-posedness theory in [7, 14, 42], the (local-
in-time) Strichartz estimate is one of useful linear estimates in long time
behavior of the wave equations. In fact, the dispersive estimate and the
(local-in-time) Strichartz estimate for the wave/Schrödinger equations on the
(exterior) domain themselves are extremely complicate, we can refer to [3, 5,
6, 16, 22, 23, 25, 26, 42, 43, 44, 46] and reference therein.

In this paper, we consider the Dirichlet boundary value problem (1.1) with
radial data outside of the ball, there is a crucial analysis tool, that is,
the explicit distorted Fourier transform adapted to the Dirichlet-Laplacian
operator $-\Delta_{\Omega}$ under the radial assumption (see also [35, 36,
50]), to simplify the proof of the dispersive estimate and the global-in-time
(endpoint) Strichartz estimate (see Theorem 3.2 and Theorem 3.4) of linear
wave equation outside of the ball with radial data. As for the distorted
Fourier transform adapted to the potential scattering, we can refer to [1, 10,
13, 33, 34], and reference therein.

As an application of the global-in-time (endpoint) Strichartz estimae and the
energy method, we can show global well-posedness of equation (1.1) with radial
data in low regularity Sobolev space $\left(\dot{H}^{s}_{D}(\Omega)\cap
L^{4}(\Omega)\right)\times\dot{H}^{s-1}_{D}(\Omega)$ for some $s<1$, where the
solution maybe have infinite energy. Unlike the energy-critical
wave/Schrödinger problem, there is no a priori conservation law for the rough
solution of (1.1) to ensure that the low regularity Sobolev norm of solution
will remain bounded for the whole time of its existence.

######  Theorem 1.4.

Let $\frac{3}{4}<s<1$, then the equation (1.1) with radial data is global
well-posedness in $\left(\dot{H}^{s}_{D}(\Omega)\cap
L^{4}(\Omega)\right)\times\dot{H}^{s-1}_{D}(\Omega)$. More precisely, for
arbitrarily large time $T$, the solution obeys the following estimate

$\displaystyle\left\|u\right\|_{C\left([0,T);\;\dot{H}^{s}_{D}(\Omega)\right)}\lesssim
T^{\frac{3(1-s)(2s-1)}{4s-3}}.$

To the best of the author’s knowledge, it is first result about low regularity
of semilinear wave equation with zero Dirichlet boundary value on the exterior
domain. We will take the strategy from [4, 18, 29] to decompose the solution
of (1.1) into small global solution $w$ of (1.1) with high-frequency-localized
initial data and the remaider with low-frequency-localized initial data, whose
energy is almost conserved for arbitrarily large time $T$ if choosing the
frequency cutoff on initial data carefully, which is similar to the well-known
almost conservation law developed by Colliander, Keel, Staffilani, Takaoka and
Tao in [9].

### Notation

Throughout the paper, we use the notation $X\lesssim Y$, or $Y\gtrsim X$ to
denote the statement that $X\leq CY$ for some constant $C$, which may vary
from line to line. We use $X\approx Y$ to denote the statement $X\lesssim
Y\lesssim X$.

Lastly, this paper is organized as follows. In Section 2, we recall the
distorted Fourier transform adapted to the Dirichlet-Laplacian operator
$-\Delta_{\Omega}$ in the radial case and the related Littlewood-Paley theory.
In Section 3, we use the distorted Fourier transform to show the dispersive
estimate and the global-in-time (endpoint) Strichartz estimate to linear wave
equation outside of the ball with radial data, which gives a short proof of
the global-in-time Strichartz estimate when the initial data is radial. In
Section 4, we combine the Fourier truncation method, the energy method with
the (endpoint) Strichartz estimate to show that by use of small global
solution with high-frequency-localized initial data, the solution to the
difference equation with low-frequency-localized initial data has almost
conserved energy and can exist for arbitrarily large time $T$, which can be
used to complete the proof of the global well-posedness in the low regularity
Sobolev space in Theorem 1.4.

### Acknowledgements.

G. Xu was supported by National Key Research and Development Program of China
(No. 2020YFA0712900) and by NSFC (No. 112371240).

##  2\. Distorted Fourier Transform and Littlewood-Paley theory

In this section, we consider the 3D Dirichlet-Laplacian operator outside of
the ball $\Omega=\mathbb{R}^{3}\backslash\bar{B}(0,1)$, which we denote by
$-\Delta_{\Omega}$. we recall the distorted Fourier transform adapted to the
operator $-\Delta_{\Omega}$ in the radial case from [35, 36] and the Sobolev
space $\dot{H}^{s,p}_{D}(\Omega)$ in [30]. We can also refer to [1, 39, 50]
for spectral properties of Schrödinger operator and [2, 21, 38, 47] for
classical Fourier analysis and the Littlewood-Paley theory on the whole space.

The general spectral theory of the Dirichlet-Laplacian operator
$-\Delta_{\Omega}$ is standard. The essential spectrum of the operator
$-\Delta_{\Omega}$ is $[0,\infty)$; The operator $-\Delta_{\Omega}$ has no
positive eigenvalues embedded into $(0,\infty)$ and has no negative
eigenvalues; Moreover $0$ is not an eigenvalue or a resonance of the operator
$-\Delta_{\Omega}$.

The spectral resolution for radial functions on
$\Omega=\mathbb{R}^{3}\backslash\bar{B}(0,1)$ is expressed by the radial,
generalized eigenfunctions

$-\Delta\;e_{\lambda}=\lambda^{2}\;e_{\lambda}$

for all $\lambda>0$, which satisfy the Sommerfeld radiation condition, namely

$e_{\lambda}(r)=\frac{\sin\lambda(r-1)}{r},\quad r=|x|\geq 1,$ (2.1)

which palys the similar rule as the functions $(r\lambda)^{-1/2}\cdot
J_{1/2}(r\lambda)$ for the radial funtion on the whole space $\mathbb{R}^{3}$,
where $J_{1/2}(r\lambda)$ is Bessel function of order $1/2$ (see [21]).

For the radial, tempered distributions
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{3})$, supported on $\Omega$, we denote
the distorted Fourier transformation $\mathcal{F}_{D}$ for $\lambda>0$ by

$\displaystyle\mathcal{F}_{D}f(\lambda)=\frac{\sqrt{2}}{\sqrt{\pi}}\int^{\infty}_{1}e_{\lambda}(s)f(s)s^{2}\;ds=\frac{\sqrt{2}}{\sqrt{\pi}}\int^{\infty}_{1}\frac{\sin\lambda(s-1)}{s}f(s)s^{2}\;ds.$
(2.2)

Note that the following resolution of identity

$\displaystyle\frac{2}{\pi}\int^{\infty}_{0}e_{\lambda}(r)\,e_{\lambda}(s)\,d\lambda=$
$\displaystyle\frac{1}{\pi}\int^{\infty}_{-\infty}e_{\lambda}(r)\,e_{\lambda}(s)\,d\lambda=\frac{\delta(r-s)}{s^{2}},$

from which it follows that

$\mathcal{F}^{-1}_{D}\mathcal{F}_{D}f=f$

for the radial function $f\in\mathcal{S}(\mathbb{R}^{3})$ supported in
$\Omega$, where $\mathcal{F}^{-1}_{D}$ is the formal adjoint, defined on the
tempered distributions $g$ as the restriction to $r=|x|\geq 1$ of

$\displaystyle\mathcal{F}^{-1}_{D}g(r)=\frac{\sqrt{2}}{\sqrt{\pi}}\int^{\infty}_{0}e_{\lambda}(r)g(\lambda)\;d\lambda=\frac{\sqrt{2}}{\sqrt{\pi}}\int^{\infty}_{0}\frac{\sin\lambda(r-1)}{r}g(\lambda)\;d\lambda.$
(2.3)

Consequently, $f\longrightarrow\mathcal{F}_{D}f$ induces an isometric map

$\displaystyle\mathcal{F}_{D}:\;L^{2}\left([1,\infty),s^{2}\,ds\right)\longrightarrow
L^{2}\left([0,\infty),\,d\lambda\right)$

It is worth noting that unlike classical Fourier transform on the whole space,
the spectral supports are not additive under function multiplication for the
above distorted Fourier transform.

Given a bounded function $m(\lambda)$, which for convenience we assume to be
defined on all of $\mathbb{R}$ and even in $\lambda$, and radial function
$f\in C^{\infty}_{c}(\Omega)$, we define

$\displaystyle
m\left(\sqrt{-\Delta_{\Omega}}\right)f(r)=\mathcal{F}^{-1}_{D}\big{(}m(\cdot)\,\mathcal{F}_{D}f\big{)}(r).$

This defines a functional calculus on $L^{2}_{rad}(\Omega)$ and takes the
expression as

$\displaystyle
m(\sqrt{-\Delta_{\Omega}})f(r)=\int^{\infty}_{1}K_{m}(r,s)f(s)\,s^{2}\,ds$

with

$\displaystyle
K_{m}(r,s)=\frac{2}{\pi}\,\int^{\infty}_{0}e_{\lambda}(r)\,e_{\lambda}(s)\,m(\lambda)\;d\lambda.$

In general, we have the following Mikhlin Multiplier theorem.

######  Theorem 2.1 ([30, 35]).

Suppose $m:[0,\infty)\rightarrow\mathbb{C}$ obeys

$\displaystyle\left|\partial^{k}_{\lambda}m(\lambda)\right|\lesssim\lambda^{-k}$

for all integer $k\in[0,2]$. Then $m(\sqrt{-\Delta_{\Omega}})$, which we
define via the $L^{2}$ functional calculus, extends uniquely from
$L^{2}(\Omega)\cap L^{p}(\Omega)$ to a bounded operator on $L^{p}(\Omega)$,
for all $1<p<\infty$.

###### Proof.

In the radial case, we can use the Schur’s Lemma and interpolation to obtain
the result, please refer to [35] for more details, and in general case, we
need to use the heat kernel estimate on the exterior domain $\Omega$ in [51]
and classical Calderón-Zygmund theory to show the boundedness result, please
see more details in [30]. ∎

More Mikhlin multiplier results adapted to potential scattering, we can refer
to [19, 40]. Based on the above Mikhlin’s multiplier theorem, we can now
describe basic ingredients of Littlewood-Paley theory adapted to the
Dirichlet-Laplacian operator $-\Delta_{\Omega}$. Fix
$\phi:[0,\infty)\rightarrow[0,1]$ a smooth non-negative function obeying

$\displaystyle\phi(\lambda)=1\quad\text{for}\quad 0\leq\lambda\leq
1\quad\text{and}\quad\phi(\lambda)=0\quad\text{for}\quad\lambda\geq 2.$

For each dyadic number $N\in 2^{\mathbb{Z}}$, we define

$\displaystyle\phi_{N}(\lambda):=\phi(\lambda/N)\quad\text{and}\quad\psi_{N}(\lambda):=\phi_{N}(\lambda)-\phi_{N/2}(\lambda).$

Notice that $\big{\\{}\psi_{N}(\lambda)\big{\\}}_{N\in 2^{\mathbb{Z}}}$ forms
a partition of unity for $(0,\infty)$. With these functions, we now define the
Littlewood-Paley projections:

$\displaystyle P^{\Omega}_{\leq N}f:=\phi_{N}(\sqrt{-\Delta_{\Omega}})f,\quad
P^{\Omega}_{N}f:=\psi_{N}(\sqrt{-\Delta_{\Omega}})f,\quad
P^{\Omega}_{>N}f:=I-P^{\Omega}_{\leq N}f,$

and

$\displaystyle\tilde{P}^{\Omega}_{N}f:=\tilde{\psi}_{N}\left(\sqrt{-\Delta_{\Omega}}\right)f=\psi_{N-1}\left(\sqrt{-\Delta_{\Omega}}\right)f+\psi_{N}\left(\sqrt{-\Delta_{\Omega}}\right)f+\psi_{N+1}\left(\sqrt{-\Delta_{\Omega}}\right)f.$

We introduce the homogeneous Besov space as the following.

######  Definition 2.2.

Let $s\in\mathbb{R}$ and $1\leq q,r\leq\infty$. The homogeneous Besov space
$\dot{B}^{s}_{D,q,r}(\Omega)$ consists of the distributions $f$ supported on
$\Omega$ such that

$\displaystyle\|f\|_{\dot{B}^{s}_{D,q,r}(\Omega)}:=\left(\sum_{N\in
2^{\mathbb{Z}}}N^{sr}\|P^{\Omega}_{N}f\|^{r}_{L^{q}(\Omega)}\right)^{1/r}<\infty.$

######  Lemma 2.3 ([30, 35]).

For any radial function $f\in C^{\infty}_{c}(\Omega)$, we have

$\displaystyle\left\|P^{\Omega}_{\leq N}f\right\|_{L^{p}(\Omega)}+$
$\displaystyle\left\|P^{\Omega}_{N}f\right\|_{L^{p}(\Omega)}\lesssim\left\|f\right\|_{L^{p}(\Omega)},$
(2.4) $\displaystyle N^{s}\left\|P^{\Omega}_{N}f\right\|_{L^{p}(\Omega)}$
$\displaystyle\approx\left\|\big{(}-\Delta_{\Omega}\big{)}^{s/2}P^{\Omega}_{N}f\right\|_{L^{p}(\Omega)}$
(2.5)

for any $1\leq p\leq\infty$ and $s\in\mathbb{R}$, Moreover, we have

$\displaystyle\left\|P^{\Omega}_{\leq
N}\right\|_{L^{q}(\Omega)}+\left\|P^{\Omega}_{N}\right\|_{L^{q}(\Omega)}\lesssim
N^{3\big{(}\frac{1}{p}-\frac{1}{q}\big{)}}\left\|f\right\|_{L^{p}(\Omega)}$

for all $1\leq p\leq q\leq\infty$. The implicit constants depend only on $p,q$
and $s$.

In general case, the estimates (2.4) and (2.5) only hold for $1<p<\infty$ by
Theorem 2.1. In the radial case, the correspoding integral kernels have good
prpperties by the distorted Fourier transform, see more details in [35].
Therefore, for any $1<p<\infty$ and any $f\in L^{p}(\Omega)$, we have the
following homogeneous decomposition

$\displaystyle f(x)=\sum_{N\in 2^{\mathbb{Z}}}P^{\Omega}_{N}f(x).$

In particular, the sums converge in $L^{p}(\Omega)$.

######  Definition 2.4.

For $s\geq 0$ and $1<p<\infty$, Let $\dot{H}^{s,p}_{D}(\Omega)$ and
$H^{s,p}_{D}(\Omega)$ denote the completions of $C^{\infty}_{c}(\Omega)$ under
the norms

$\displaystyle\|f\|_{\dot{H}^{s,p}_{D}}:=\left\|\big{(}-\Delta_{\Omega}\big{)}^{s/2}f\right\|_{L^{p}(\Omega)}\quad\text{and}\quad\|f\|_{H^{s,p}_{D}}:=\left\|\big{(}I-\Delta_{\Omega}\big{)}^{s/2}f\right\|_{L^{p}(\Omega)}.$

When $p=2$, we write $\dot{H}^{s}_{D}(\Omega)$ and $H^{s}_{D}(\Omega)$ for
$\dot{H}^{s,2}_{D}(\Omega)$ and $H^{s,2}_{D}(\Omega)$, respectively.

We will use the radial Sobolev spaces $\dot{H}^{s,p}_{D,rad}(\Omega)$ and
$H^{s,p}_{D,rad}(\Omega)$ in the context. If the radial function $f$ belongs
to $C^{\infty}_{c}(\Omega)$, then

$\displaystyle\mathcal{F}_{D}(-\Delta_{\Omega}f)(\lambda)=\lambda^{2}\mathcal{F}_{D}(f)(\lambda),$

and $\mathcal{F}_{D}$ induces an isometric map

$\displaystyle\mathcal{F}_{D}:\;\dot{H}^{1}_{D,rad}(\Omega)\longrightarrow
L^{2}\left([0,\infty),\lambda^{2}\,d\lambda\right).$

The Littlewood-Paley square function estimates and the dense result then
follow from the multiplier theorem by the usual argument.

######  Proposition 2.5 ([30]).

Fix $1<p<\infty$ and $s\geq 0$. Then for any $f\in C^{\infty}_{c}(\Omega)$, we
have

$\displaystyle\left\|\big{(}-\Delta_{\Omega}\big{)}^{s/2}f\right\|_{L^{p}(\Omega)}\sim\left\|\left(\sum_{N\in
2^{\mathbb{Z}}}N^{2s}\big{|}P^{\Omega}_{N}f(x)\big{|}^{2}\right)^{1/2}\right\|_{L^{p}(\Omega)}.$

######  Proposition 2.6 ([30]).

For $1<p<\infty$, and $s<1+\frac{1}{p}$, $C^{\infty}_{c}(\Omega)$ is dense in
$\dot{H}^{s,p}_{D}(\Omega)$.

######  Definition 2.7.

The space $\dot{H}^{s,p}_{00}(\Omega)$ and $H^{s,p}_{00}(\Omega)$ are defined
as the completion of $C^{\infty}_{c}(\Omega)$ in
$\dot{H}^{s,p}(\mathbb{R}^{3})$ and $H^{s,p}(\mathbb{R}^{3})$, respectively.

At last, the equivalence between $\dot{H}^{s,p}_{D}(\Omega)$ and
$\dot{H}^{s,p}_{00}(\Omega)$ with proper exponents then follows from the
Hardy’s inequalities for the exterior domain $\Omega$ and the whole space
$\mathbb{R}^{3}$ in [30].

######  Theorem 2.8 ([30]).

Suppose $1<p<\infty$ and $0\leq s<\min\\{1+\frac{1}{p},\frac{3}{p}\\}$, then
for all $f\in C^{\infty}_{c}(\Omega)$

$\displaystyle\left\|\big{(}-\Delta_{\Omega}\big{)}^{s/2}f\right\|_{L^{p}(\Omega)}\approx_{p,s}\left\|\big{(}-\Delta_{\mathbb{R}^{3}}\big{)}^{s/2}f\right\|_{L^{p}(\mathbb{R}^{3})}$

Thus $\dot{H}^{s,p}_{D}(\Omega)=\dot{H}^{s,p}_{00}(\Omega)$ for these values
of the parameters.

######  Remark 2.9.

As the direct corollary of the above equivalence, the fractional product rule
directly follows from the classical Euclidean setting. More precisely, for all
$f,g\in C^{\infty}_{c}(\Omega)$, then

$\displaystyle\left\|\big{(}-\Delta_{\Omega}\big{)}^{s/2}\big{(}fg\big{)}\right\|_{L^{p}(\Omega)}\lesssim\left\|\big{(}-\Delta_{\Omega}\big{)}^{s/2}f\right\|_{L^{p_{1}}(\Omega)}\|g\|_{L^{q_{1}}(\Omega)}+\|f\|_{L^{q_{2}}(\Omega)}\left\|\big{(}-\Delta_{\Omega}\big{)}^{s/2}g\right\|_{L^{p_{2}}(\Omega)}$

with the exponents satisfying $1<p,p_{1},p_{2}<\infty$,
$1<q_{1},q_{2}\leq\infty$,

$\displaystyle\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac{1}{p_{2}}+\frac{1}{q_{2}},\quad
0<s<\min\left(1+\frac{1}{p_{1}},1+\frac{1}{p_{2}},\frac{3}{p_{1}},\frac{3}{p_{2}}\right).$

And the fractional chain rule holds in a similar way.

##  3\. Linear estimates

In this part, we will make use of the distorted Fourier transform adapted to
the Dirichlet-Laplacian operator $-\Delta_{\Omega}$ to show the dispersive
estimate and the global-in-time (endpoint) Strichartz estimate of the linear
wave equation with zero Dirichlet boundary value outside of the ball with
radial data, we can also refer to [35, 36] for the dispersive and Strichartz
estimates of the Schrödinger equation outside of the ball with radial data.

Now we consider the 3D Dirichlet boundary value problem of the linear wave
equation with radial data

$\begin{cases}\partial^{2}_{t}u-\Delta u=F,&(t,x)\in\mathbb{R}\times\Omega,\\\
u(0,x)=u_{0}(x),&x\in\Omega,\\\ \partial_{t}u(0,x)=u_{1}(x),&x\in\Omega,\\\
u(t,x)=0,&x\in\partial\Omega,\end{cases}$ (3.1)

where $\Omega=\mathbb{R}^{3}\backslash\bar{B}(0,1)$ and $u_{0},u_{1}$ and the
inhomogeneous term $F$ are radial in $x$. By the funcional calculus, we have

$u(t,x)=\cos(t\sqrt{-\Delta_{\Omega}})u_{0}+\frac{\sin(t\sqrt{-\Delta_{\Omega}})}{\sqrt{-\Delta_{\Omega}}}u_{1}+\int^{t}_{0}\frac{\sin((t-s)\sqrt{-\Delta_{\Omega}})}{\sqrt{-\Delta_{\Omega}}}F(s,x)\,ds.$

Let us denote the half-wave operator as

$U(t)=e^{it\sqrt{-\Delta_{\Omega}}},$

then

$\displaystyle\cos(t\sqrt{-\Delta_{\Omega}})u_{0}=\frac{U(t)+U(-t)}{2}u_{0},\quad\frac{\sin(t\sqrt{-\Delta_{\Omega}})}{\sqrt{-\Delta_{\Omega}}}u_{1}=\frac{U(t)-U(-t)}{2i\sqrt{-\Delta_{\Omega}}}u_{1}.$

By the distorted Fourier transform and the stationary phase estimate, we have
the following uniform dispersive estimate in the radial case.

######  Proposition 3.1.

Let $2\leq r\leq\infty$, and the radial function $f$ is supported on $\Omega$,
then

$\displaystyle\big{\|}U(t)f\big{\|}_{\dot{B}^{-\beta(r)}_{D,r,2}(\Omega)}\lesssim|t|^{-\gamma(r)}\big{\|}f\big{\|}_{\dot{B}^{\beta(r)}_{D,r^{\prime},2}(\Omega)}$

where $\beta(r)=\gamma(r)=1-\frac{2}{r}$.

###### Proof.

Taking the distorted Fourier transform, we have

$\displaystyle\big{\|}U(t)f\big{\|}_{L^{2}(\Omega)}=\big{\|}f\big{\|}_{L^{2}(\Omega)}.$

Therefore, by the interpolation theorem, it suffices to show the following
uniform dispersive estimate

$\displaystyle\big{\|}U(t)f\big{\|}_{\dot{B}^{-1}_{D,\infty,2}(\Omega)}\lesssim|t|^{-1}\big{\|}f\big{\|}_{\dot{B}^{1}_{D,1,2}(\Omega)}.$
(3.2)

Notice the fact that
$P^{\Omega}_{N}f(r)=\tilde{P}^{\Omega}_{N}P^{\Omega}_{N}f(r)$ for $N\in
2^{\mathbb{Z}}$, $r=|x|\geq 1$, we have

$\displaystyle U(t)P^{\Omega}_{N}f(r)=$
$\displaystyle\frac{2}{\pi}\int^{\infty}_{0}\int^{\infty}_{1}e_{\lambda}(r)e_{\lambda}(s)e^{it|\lambda|}\tilde{\psi}_{N}(\lambda)\big{(}P^{\Omega}_{N}f\big{)}(s)s^{2}\;ds\,d\lambda$
$\displaystyle=$
$\displaystyle\int^{\infty}_{1}K_{N}(t,r;s)\big{(}P^{\Omega}_{N}f\big{)}(s)s^{2}\;ds$
(3.3)

where the integral kernel $K_{N}(t,r;s)$ is

$\displaystyle K_{N}(t,r;s)=$
$\displaystyle\frac{2}{\pi}\int^{+\infty}_{0}e_{\lambda}(r)e_{\lambda}(s)e^{it|\lambda|}\tilde{\psi}_{N}(\lambda)\,d\lambda$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{+\infty}_{-\infty}\frac{\sin\lambda(r-1)}{r}\frac{\sin\lambda(s-1)}{s}e^{it|\lambda|}\tilde{\psi}_{N}(|\lambda|)\;d\lambda$
$\displaystyle=$ $\displaystyle\frac{N}{\pi}\int^{\infty}_{-\infty}\frac{\sin
N\lambda(r-1)}{r}\frac{\sin
N\lambda(s-1)}{s}e^{itN|\lambda|}\tilde{\psi}_{0}(|\lambda|)\;d\lambda$
$\displaystyle=$
$\displaystyle\frac{N^{3}}{\pi}\int_{-\infty}^{+\infty}\frac{s-1}{s}\frac{r-1}{r}\frac{\sin
N\lambda(s-1)}{N(s-1)}\frac{\sin
N\lambda(r-1)}{N(r-1)}e^{iNt|\lambda|}\psi_{0}(|\lambda|)d\lambda$
$\displaystyle=$
$\displaystyle\frac{N^{3}}{\pi}\frac{r-1}{r}\frac{s-1}{s}e^{it{\sqrt{-\Delta_{rad}}}}(N(s-1),N(r-1);Nt)\psi_{0}$

for the usual radial wave propagator

$e^{it{\sqrt{-\Delta_{rad}}}}(t,r;s)f:=\int_{-\infty}^{+\infty}\frac{\sin\lambda
s}{s}\frac{\sin\lambda r}{r}e^{it|\lambda|}f(\lambda)\,d\lambda,\quad r,s>0.$

In fact, for all $t\in\mathbb{R}$ , we have

$e^{it\sqrt{-\Delta_{rad}}}f=e^{it\sqrt{-\Delta}}f=\mathcal{F}^{-1}(e^{it|\xi|}\mathcal{F}f(\xi)).$

where $\mathcal{F}^{\pm 1}$ is the usual Fourier transform for the operator
$-\Delta_{\mathbb{R}^{3}}$. By the properties of Bessel function $J_{n}(r)$
and van der Corput’s lemma, we obtain

$\displaystyle\|e^{it\sqrt{-\Delta_{rad}}}\|_{L^{1}(\mathbb{R}^{3})\to
L^{\infty}(\mathbb{R}^{3})}\lesssim|t|^{-1},$

which together with Young’s inequality implies that

$\|e^{it\sqrt{-\Delta_{rad}}}f\|_{L^{\infty}}\lesssim|t|^{-1}\|f\|_{L^{1}}.$

By the supported property of $\psi_{0}$, we have

$\displaystyle\sup_{r,s\geq 1}\big{|}K_{N}(t,r;s)\big{|}\lesssim
N^{3}\,|Nt|^{-1}\,\|\psi_{0}\|_{L^{1}}\lesssim|t|^{-1}N^{2}.$ (3.4)

By (3.3), (3.4) and the Minkowski inequality, we obtain

$\displaystyle\big{\|}U(t)P^{\Omega}_{N}f\big{\|}_{L^{\infty}(\Omega)}\lesssim|t|^{-1}N^{2}\big{\|}P^{\Omega}_{N}f\big{\|}_{L^{1}(\Omega)},$

which impies the result and completes the proof. ∎

By the dual argument in [20], and Christ-Kiselev lemma in [8], we have

######  Theorem 3.2 (Strichartz estimate with radial data).

Let $\rho_{1},\rho_{2},\mu\in\mathbb{R}$ and $2\leq
q_{1},q_{2},r_{1},r_{2}\leq\infty$ and let the following conditions be
satisfied

$\displaystyle 0\,\leq\,\frac{1}{q_{i}}+\frac{1}{r_{i}}\leq\,\frac{1}{2},$
$\displaystyle\quad r_{i}\not=\infty,\quad i=1,\,2,$
$\displaystyle\rho_{1}+3\left(\frac{1}{2}-\frac{1}{r_{1}}\right)-\frac{1}{q_{1}}=\mu,$
$\displaystyle\quad\rho_{2}+3\left(\frac{1}{2}-\frac{1}{r_{2}}\right)-\frac{1}{q_{2}}=1-\mu$

Let $u_{0},u_{1}$ and $F$ be radial in $x$, and
$u:I\times\Omega\longrightarrow\mathbb{R}$ be a solution to linear wave
equation (3.1) with $0\in I$. Then $u$ satisfies the estimates

$\displaystyle\left\|u\right\|_{L^{q_{1}}_{I}\dot{B}^{\rho_{1}}_{D,r_{1},2}(\Omega)\cap
C(I;\dot{H}^{\mu}(\Omega))}+\left\|\partial_{t}u\right\|$
${}_{L^{q_{1}}_{I}\dot{B}^{\rho_{1}-1}_{D,r_{1},2}(\Omega)\cap
C(I;\dot{H}^{\mu-1}(\Omega))}$
$\displaystyle\lesssim\;\|(u_{0},u_{1})\|_{\dot{H}^{\mu}_{D,rad}(\Omega)\times\dot{H}^{\mu-1}_{D,rad}(\Omega)}+\|F\|_{L^{q^{\prime}_{2}}_{I}\dot{B}^{-\rho_{2}}_{D,r^{\prime}_{2},2}(\Omega)}.$

######  Remark 3.3.

By the explicite distorted Fourier transform, we give a short proof of the
dispersive estimate and the global-in-time Strichartz estimate of the 3D
Dirichlet boundary value problem of linear wave equation outside of the ball
in the radial case. In general case, the related estimates are extremely
complicate, please refer to [5, 16, 22, 23, 43, 44] and [25, 26] for more
details.

In fact, we can also follow the argument in [48] and obtain improved endpoint
Strichartz estimates in the radial case, which depends on the non-uniform
dispersive estimate. As for the endpoint $L^{2}_{t}L^{\infty}_{x}$ estimate
for the radial case in $\mathbb{R}^{3}$, we can also refer to [32, 49].

######  Theorem 3.4 (Endpoint Strichartz estimates with radial data).

Let $u_{0},u_{1}$ and $F$ be radial in $x$ variable, and
$u:I\times\Omega\longrightarrow\mathbb{R}$ be a solution to linear wave
equation (3.1) with $0\in I$. If $q>4$ and $s=1-3/q,$ then we obtain

$\displaystyle\left\|u\right\|_{L^{2}_{t}L^{q}_{x}(I\times\Omega)}\lesssim\|(u_{0},u_{1})\|_{\dot{H}^{s}_{D,rad}(\Omega)\times\dot{H}^{s-1}_{D,rad}(\Omega)}+\|F\|_{L^{1}_{t}\left(I;\,\dot{H}^{s-1}_{D,rad}(\Omega)\right)}.$

###### Proof.

By the energy estimate, it suffices to show the following homogeneous estimate
for the half wave operator $U(t)$

$\displaystyle\big{\|}U(t)f\big{\|}_{L^{2}_{t}L^{q}_{x}(I\times\Omega)}\lesssim\|f\|_{\dot{H}^{s}_{D,rad}(\Omega)}.$

By the distorted Fourier transform once again, we have

$\displaystyle U(t)f(r)=$
$\displaystyle\frac{2}{\pi}\int^{\infty}_{0}\frac{\sin(\lambda(r-1))}{r}\,e^{it|\lambda|}\left(\mathcal{F}_{D}f\right)(\lambda)\,d\lambda=\frac{1}{i\pi}\int^{\infty}_{0}\frac{e^{i\lambda(r-1)}-e^{-i\lambda(r-1)}}{r}\,e^{it|\lambda|}\left(\mathcal{F}_{D}f\right)(\lambda)\,d\lambda.$

By the Littlewood-Paley decomposition, we have

$\displaystyle U(t)f(r)=$
$\displaystyle\sum\limits_{N}e^{it\sqrt{-\Delta_{\Omega}}}\tilde{P}^{\Omega}_{N}P^{\Omega}_{N}f(r)$
$\displaystyle=$ $\displaystyle\sum\limits_{N}\frac{1}{i\pi
r}\int^{\infty}_{0}\left(e^{i\lambda(r-1)}-e^{-i\lambda(r-1)}\right)\,e^{it|\lambda|}\tilde{\psi}_{N}(\lambda)\left(\mathcal{F}_{D}P^{\Omega}_{N}f\right)(\lambda)\,d\lambda$
$\displaystyle=$ $\displaystyle\sum\limits_{N}\frac{N}{i\pi
r}\int^{\infty}_{0}\left(e^{iN\lambda(r-1)}-e^{-iN\lambda(r-1)}\right)\,e^{itN|\lambda|}\tilde{\psi}_{0}(\lambda)\left(\mathcal{F}_{D}P^{\Omega}_{N}f\right)(N\lambda)\,d\lambda.$

Notice that $\mathcal{F}_{D}(P^{\Omega}_{N}f)$ is supported on $(N/2,2N)$, we
can make a decomposition by Fourier series as the following,

$\displaystyle\left(\mathcal{F}_{D}P^{\Omega}_{N}f\right)(N\lambda)=\sum_{k\in\mathbb{Z}}c_{k}^{N}e^{ik\frac{2\pi}{2N}N\lambda},$

then we have

$\displaystyle
e^{it\sqrt{-\Delta_{\Omega}}}\tilde{P}^{\Omega}_{N}P^{\Omega}_{N}f(r)=$
$\displaystyle\sum_{k\in\mathbb{Z}}\frac{\,Nc_{k}^{N}}{i\pi
r}\int^{\infty}_{0}\left(e^{iN\lambda(r-1)}-e^{-iN\lambda(r-1)}\right)\,e^{itN|\lambda|}\tilde{\psi}_{0}(\lambda)e^{i\pi
k\lambda}\,d\lambda$ $\displaystyle=$
$\displaystyle\sum_{k\in\mathbb{Z}}\frac{\,Nc_{k}^{N}}{i\pi
r}\Big{(}\psi^{+}_{k}(t,r)-\psi^{-}_{k}(t,r)\Big{)},$

where

$\displaystyle\psi^{\pm}_{k}(t,r)=\int^{\infty}_{0}e^{\pm
iN\lambda(r-1)}\,e^{itN|\lambda|}\tilde{\psi}_{0}(\lambda)e^{i\pi
k\lambda}\,d\lambda.$

By the integrating by parts, we have the local dispersive esitmate for any
$M\geqslant 1$ that

$\left|\psi^{\pm}_{k}(t,r)\right|\lesssim\frac{C_{M}}{\big{(}1+N|t+\frac{\pi
k}{N}\pm(r-1)|\big{)}^{M}}.$

Then we can make the simple calculation and obtain

$\displaystyle\left\|e^{it\sqrt{-\Delta_{D}}}P^{\Omega}_{N}f\right\|^{q}_{L^{q}_{x}(\Omega)}\lesssim$
$\displaystyle\int^{\infty}_{1}\left|\sum_{k\in\mathbb{Z}}c_{k}^{N}\,\psi^{\pm}_{k}(t,r)\right|^{q}\frac{1}{r^{q}}r^{2}\;dr$
$\displaystyle\lesssim$
$\displaystyle\int^{\infty}_{1}\left|\sum_{k\in\mathbb{Z}}\frac{|c_{k}^{N}|}{\big{(}1+N|t+\frac{\pi
k}{N}\pm(r-1)|\big{)}^{M}}\right|^{q}r^{2-q}\;dr$ $\displaystyle\lesssim$
$\displaystyle\sum_{k\in\mathbb{Z}}\int^{\infty}_{0}\frac{|c_{k}^{N}|^{q}}{\big{(}1+N||t+\frac{\pi
k}{N}|-r|\big{)}^{2}}(1+r)^{2-q}\;dr,$

where we choose $M$ with $(M-2/q)\cdot q^{\prime}>1$ and use the Young
inequality in the last inequality. We have for $s=1-\frac{3}{q}$ that

$\displaystyle\left\|e^{it\sqrt{-\Delta_{D}}}P^{\Omega}_{N}f\right\|_{L^{q}_{x}(\Omega)}\lesssim$
$\displaystyle\left(\sum_{k\in\mathbb{Z}}\int^{\infty}_{0}\frac{|c_{k}^{N}|^{q}}{\big{(}1+N||t+\frac{\pi
k}{N}|-r|\big{)}^{2}}(1+r)^{2-q}\;dr\right)^{\frac{1}{q}}$
$\displaystyle\lesssim$
$\displaystyle\left(\sum_{k\in\mathbb{Z}}N^{q-3}\int^{\infty}_{0}\frac{|c_{k}^{N}|^{q}}{\big{(}1+||Nt+\pi
k|-r|\big{)}^{2}}(1+r)^{2-q}\;dr\right)^{\frac{1}{q}}$ $\displaystyle\lesssim$
$\displaystyle
N^{s}\left(\sum_{k\in\mathbb{Z}}\frac{|c_{k}^{N}|^{q}}{\big{(}1+|Nt+\pi
k|\big{)}^{q-2}}\right)^{\frac{1}{q}}\lesssim
N^{s}\left(\sum_{k\in\mathbb{Z}}\frac{|c_{k}^{N}|^{2}}{\big{(}1+|Nt+\pi
k|\big{)}^{2-\frac{4}{q}}}\right)^{\frac{1}{2}}.$

where we use the embedding fact that $l^{2}\subset l^{q}$ in the last
inequality. By Bernstein’s inequality, we have for $q>4$ that

$\displaystyle\left\|e^{it\sqrt{-\Delta_{D}}}P^{\Omega}_{N}f\right\|^{2}_{L^{2}_{t}L^{q}_{x}(I\times\Omega)}\lesssim
N^{2s}\int_{I}\sum_{k\in\mathbb{Z}}\frac{|c_{k}^{N}|^{2}}{\big{(}1+|Nt+\pi
k|\big{)}^{2-\frac{4}{q}}}\,dt\lesssim
N^{2s}\sum_{k\in\mathbb{Z}}|c_{k}^{N}|^{2}\lesssim\|P_{N}^{\Omega}f\|^{2}_{\dot{H}^{s}_{D}(\Omega)},$

which together with Proposition 2.5 implies the result and completes the
proof. ∎

##  4\. Global well-posedness: Proof of Theorem 1.4

In this part, we combine the Fourier truncation method in [4, 29], and the
global-in-time (endpoint) Strichartz esitmate for the radial solution with the
energy method to prove the global well-posedness result in Theorem 1.4, which
is the similar as those in [18].

Let $\frac{1}{2}<s\leq 1$ ,
$(u_{0},u_{1})\in\left(\dot{H}^{s}_{D,rad}(\Omega)\cap
L^{4}(\Omega)\right)\times\dot{H}^{s-1}_{D,rad}(\Omega)$.

###  4.1. Global analysis for High frequency part

Let us consider the following Dirichlet problem of nonlinear wave equation
outside of the ball with high-frequency-localized radial data.

$\begin{cases}\partial^{2}_{t}w-\Delta
w+w^{3}=0,&(t,x)\in\mathbb{R}\times\Omega,\\\
w(0,x)=w_{0}(x)=P_{>2^{J}}u_{0}(x),&x\in\Omega,\\\
\partial_{t}w(0,x)=w_{1}(x)=P_{>2^{J}}u_{1}(x),&x\in\Omega,\\\
w(t,x)=0,&x\in\partial\Omega,\end{cases}$ (4.1)

Notice that the distorted Fourier transform preserve the zero Dirichlet
boundary value structure.

Let $\epsilon>0$ sufficiently small, and choose the dyadic number
$J=J(\epsilon)\gg 1$ such that

$\displaystyle\left\|(w_{0},w_{1})\right\|_{\dot{H}^{s}_{D,rad}(\Omega)\times\dot{H}^{s-1}_{D,rad}(\Omega)}=\left\|\big{(}P_{>2^{J}}u_{0},P_{>2^{J}}u_{1}\big{)}\right\|_{\dot{H}^{s}_{D,rad}(\Omega)\times\dot{H}^{s-1}_{D,rad}(\Omega)}\lesssim\epsilon.$
(4.2)

By the Strichartz estimates in Theorem 3.2 and Theorem 3.4, the standard well-
posedness theory in [37, 45] together with the regularity theory implies that

######  Proposition 4.1.

Let $\frac{1}{2}<s\leq 1$, There exists a large constant $J=J(\epsilon)>0$
such that if

$2^{J(\frac{1}{2}-s)}\lesssim\epsilon$

then (4.1) is global well-posedness in
$\dot{H}^{\frac{1}{2}}_{D}(\Omega)\cap\dot{H}^{s}_{D}(\Omega)$. Moreover, we
have the following estimates.

$\displaystyle\|w\|_{L^{4}_{t,x}(\mathbb{R}\times\Omega)}+\|w\|_{L^{\infty}_{t}L^{3}_{x}(\mathbb{R}\times\Omega)}+\|w\|_{L^{2}_{t}L^{6}_{x}(\mathbb{R}\times\Omega)}$
$\displaystyle\lesssim 2^{J(\frac{1}{2}-s)},$ (4.3)

and

$\displaystyle\|w\|_{L^{\infty}_{t}\left(\mathbb{R};\,\dot{H}^{s}_{D,rad}(\Omega)\right)}$
$\displaystyle\lesssim\epsilon.$ (4.4)

###### Proof.

The result follows from the standard contraction mapping argument. More
precisely, by the Strichartz estimates in Theorem 3.2, we will show the
following map $w\mapsto\mathcal{T}(w)$ defined by

$\displaystyle\mathcal{T}(w):=\cos(t\sqrt{-\Delta_{\Omega}})w_{0}+\frac{\sin(t\sqrt{-\Delta_{\Omega}})}{\sqrt{-\Delta_{\Omega}}}w_{1}-\int^{t}_{0}\frac{\sin((t-s)\sqrt{-\Delta_{\Omega}})}{\sqrt{-\Delta_{\Omega}}}w^{3}(s)\;ds$

is a contraction on the set $X\subset
C_{b}\left(\mathbb{R};\dot{H}^{\frac{1}{2}}_{D,rad}(\Omega)\right)$

$\displaystyle X:=\Big{\\{}w\in
C_{b}\left(\mathbb{R};\dot{H}^{\frac{1}{2}}_{D,rad}(\Omega)\right)$
$\displaystyle\cap L^{4}(\mathbb{R}\times\Omega):$
$\displaystyle\|w\|_{L^{\infty}_{t}\big{(}\mathbb{R};\dot{H}^{\frac{1}{2}}_{D,rad}(\Omega)\big{)}\cap
L^{4}_{t,x}(\mathbb{R}\times\Omega)}\leq
2\,C\left\|\big{(}w_{0},w_{1}\big{)}\right\|_{\dot{H}^{1/2}_{D,rad}(\Omega)\times\dot{H}^{-1/2}_{D,rad}(\Omega)}\Big{\\}}$

under the metric given by
$d(w_{1},w_{2}):=\|w_{1}-w_{2}\|_{L^{4}_{t,x}(\mathbb{R}\times\Omega)}.$ Here
$C$ denotes the constant from the Strichartz inequality.

Using the Strichartz estimate and Hölder inequality, we have

$\displaystyle\|\mathcal{T}(w)\|_{L^{\infty}_{t}\left(\mathbb{R};\dot{H}^{\frac{1}{2}}_{D,rad}(\Omega)\right)\cap
L^{4}_{t,x}(\mathbb{R}\times\Omega)}\lesssim$
$\displaystyle\left\|\big{(}w_{0},w_{1}\big{)}\right\|_{\dot{H}^{1/2}_{D,rad}(\Omega)\times\dot{H}^{-1/2}_{D,rad}(\Omega)}+\big{\|}w^{3}\big{\|}_{L^{4/3}_{t,x}(\mathbb{R}\times\Omega)}$
$\displaystyle\lesssim$
$\displaystyle\left\|\big{(}w_{0},w_{1}\big{)}\right\|_{\dot{H}^{1/2}_{D,rad}(\Omega)\times\dot{H}^{-1/2}_{D,rad}(\Omega)}+\big{\|}w\big{\|}^{3}_{L^{4}_{t,x}(\mathbb{R}\times\Omega)}$
$\displaystyle\leq$
$\displaystyle\;2\,C\left\|\big{(}w_{0},w_{1}\big{)}\right\|_{\dot{H}^{1/2}_{D,rad}(\Omega)\times\dot{H}^{-1/2}_{D,rad}(\Omega)}.$

Arguing as above, we obtain

$\displaystyle\|\mathcal{T}(w_{1})-\mathcal{T}(w_{2})\|_{L^{4}_{t,x}(\mathbb{R}\times\Omega)}\lesssim$
$\displaystyle\big{\|}w_{1}^{3}-w_{2}^{3}\big{\|}_{L^{4/3}_{t,x}(\mathbb{R}\times\Omega)}$
$\displaystyle\lesssim$
$\displaystyle\big{\|}w_{1}-w_{2}\big{\|}_{L^{4}_{t,x}(\mathbb{R}\times\Omega)}\left(\big{\|}w_{1}\big{\|}^{2}_{L^{4}_{t,x}(\mathbb{R}\times\Omega)}+\big{\|}w_{2}\big{\|}^{2}_{L^{4}_{t,x}(\mathbb{R}\times\Omega)}\right).$

Thus, choosing $J=J(\epsilon)$ even larger (if necessary), we can guarantee
that $\mathcal{T}$ maps the set $X$ back to itself and is a contraction on the
set $X$. By the contraction mapping theorem, it follows that $\mathcal{T}$ has
a fixed point $w$ in $X$. In addition, by the endpoint Strichartz estimate in
Theorem 3.4, the Sobolev embedding that
$L^{3/2}(\Omega)\subset\dot{H}^{-1/2}_{D}(\Omega)$ and the product rule in
Remark 2.9, we have

$\displaystyle\|w\|_{L^{2}_{t}L^{6}_{x}(\mathbb{R}\times\Omega)}\lesssim$
$\displaystyle\left\|\big{(}w_{0},w_{1}\big{)}\right\|_{\dot{H}^{1/2}_{D,rad}(\Omega)\times\dot{H}^{-1/2}_{D,rad}(\Omega)}+\big{\|}w^{3}\big{\|}_{L^{1}_{t}\left(\mathbb{R};\,\dot{H}^{-1/2}_{D,rad}(\Omega)\right)}$
$\displaystyle\lesssim$
$\displaystyle\left\|\big{(}w_{0},w_{1}\big{)}\right\|_{\dot{H}^{1/2}_{D,rad}(\Omega)\times\dot{H}^{-1/2}_{D,rad}(\Omega)}+\big{\|}w^{3}\big{\|}_{L^{1}_{t}\left(\mathbb{R};\,L^{3/2}_{x}(\Omega)\right)}$
$\displaystyle\lesssim$
$\displaystyle\left\|\big{(}w_{0},w_{1}\big{)}\right\|_{\dot{H}^{1/2}_{D,rad}(\Omega)\times\dot{H}^{-1/2}_{D,rad}(\Omega)}+\big{\|}w\big{\|}^{2}_{L^{4}_{t,x}(\mathbb{R}\times\Omega)}\big{\|}w\big{\|}_{L^{2}_{t}\left(\mathbb{R};\,L^{6}(\Omega)\right)}$
$\displaystyle\lesssim$
$\displaystyle\left\|\big{(}w_{0},w_{1}\big{)}\right\|_{\dot{H}^{1/2}_{D,rad}(\Omega)\times\dot{H}^{-1/2}_{D,rad}(\Omega)}\lesssim
2^{J(\frac{1}{2}-s)},$

and

$\displaystyle\|w\|_{L^{\infty}_{t}\left(\mathbb{R};\dot{H}^{s}_{D,rad}(\Omega)\right)\cap
L^{4}_{t}\dot{H}^{s-1/2,4}_{D,rad}(\Omega)}\lesssim$
$\displaystyle\,\left\|\big{(}w_{0},w_{1}\big{)}\right\|_{\dot{H}^{s}_{D,rad}(\Omega)\times\dot{H}^{s-1}_{D,rad}(\Omega)}+\big{\|}w^{3}\big{\|}_{L^{4/3}_{t}\left(\mathbb{R};\,\dot{H}^{s-1/2,4/3}_{D,rad}(\Omega)\right)}$
$\displaystyle\lesssim$
$\displaystyle\,\left\|\big{(}w_{0},w_{1}\big{)}\right\|_{\dot{H}^{s}_{D,rad}(\Omega)\times\dot{H}^{s-1}_{D,rad}(\Omega)}+\big{\|}w\big{\|}^{2}_{L^{4}_{t,x}(\mathbb{R}\times\Omega)}\big{\|}w\big{\|}_{L^{4}_{t}\left(\mathbb{R};\,\dot{H}^{s-1/2,4}_{D,rad}(\Omega)\right)}$
$\displaystyle\lesssim$
$\displaystyle\,\left\|\big{(}w_{0},w_{1}\big{)}\right\|_{\dot{H}^{s}_{D,rad}(\Omega)\times\dot{H}^{s-1}_{D,rad}(\Omega)}\lesssim\epsilon,$

then we completes the proof. ∎

###  4.2. Local-in-time energy analysis for low frequency part

Let $w$ be the small solution of (4.1) in Proposition 4.1, we now consider the
following difference equation in the energy space
$\left(\dot{H}^{1}_{D,rad}(\Omega)\cap L^{4}(\Omega)\right)\times
L^{2}_{rad}(\Omega)$.

$\begin{cases}\partial^{2}_{t}v-\Delta
v+v^{3}=F(v,w),&(t,x)\in\mathbb{R}\times\Omega,\\\ v(0,x)=v_{0}(x)=P_{\leq
2^{J}}u_{0}(x),&x\in\Omega,\\\ \partial_{t}v(0,x)=v_{1}(x)=P_{\leq
2^{J}}u_{1}(x),&x\in\Omega,\\\ v(t,x)=0,&x\in\partial\Omega,\end{cases}$ (4.5)

where $F(v,w)=-3v^{2}w-3vw^{2}.$

######  Proposition 4.2.

Let $w$ be the solution in Proposition 4.1 and
$T=T\big{(}\|v_{0}\|_{\dot{H}^{1}_{D,rad}(\Omega)},\|v_{1}\|_{L^{2}(\Omega)}\big{)}$
such that

$\displaystyle
T\cdot\left\|\big{(}v_{0},v_{1}\big{)}\right\|^{2}_{\dot{H}^{1}_{D,rad}(\Omega)\times
L^{2}(\Omega)}\lesssim 1,$

then there exists a unique solution $v\in
C\left([0,T);\dot{H}^{1}_{D,rad}(\Omega)\right)$ to (4.5). Moreover, we have

$\displaystyle\|v\|_{L^{\infty}\big{(}[0,T);\dot{H}^{1}_{D,rad}(\Omega)\big{)}}\lesssim\left\|\big{(}v_{0},v_{1}\big{)}\right\|_{\dot{H}^{1}_{D,rad}(\Omega)\times
L^{2}(\Omega)}.$

###### Proof.

The result also follows from the standard contraction mapping argument. More
precisely, using the Strichartz estimates from Theorem 3.2, we will show the
following map $v\mapsto\mathcal{T}(v)$ defined by

$\displaystyle\mathcal{T}(v):=\cos(t\sqrt{-\Delta_{\Omega}})v_{0}$
$\displaystyle+\frac{\sin(t\sqrt{-\Delta_{\Omega}})}{\sqrt{-\Delta_{\Omega}}}v_{1}$
$\displaystyle-\int^{t}_{0}\frac{\sin((t-s)\sqrt{-\Delta_{\Omega}})}{\sqrt{-\Delta_{\Omega}}}\left(v^{3}+3v^{2}w+3vw^{2}\right)(s)\;ds$

is a contraction on the set $X\subset
C\left([0,T);\dot{H}^{1}_{D}(\Omega)\right)$:

$\displaystyle\left\\{v\in
C\left([0,T);\dot{H}^{1}_{D,rad}(\Omega)\right):\|v\|_{L^{\infty}\big{(}[0,T);\dot{H}^{1}_{D,rad}(\Omega)\big{)}}\leq
2\,C\left\|\big{(}v_{0},v_{1}\big{)}\right\|_{\dot{H}^{1}_{D,rad}(\Omega)\times
L^{2}(\Omega)}\right\\},$

where $T$ is determined later and the metric on $X$ is given by

$\displaystyle
d(u,v):=\|u-v\|_{L^{\infty}\big{(}[0,T);\dot{H}^{1}_{D,rad}(\Omega)\big{)}}.$

Here $C$ denotes the constant from the Strichartz inequality.

Using the Strichartz estimate and Hölder inequality, we have

$\displaystyle\;\|\mathcal{T}(v)\|_{L^{\infty}_{t}\left([0,T);\dot{H}^{1}_{D,rad}(\Omega)\right)}$
$\displaystyle\lesssim$
$\displaystyle\left\|\big{(}v_{0},v_{1}\big{)}\right\|_{\dot{H}^{1}_{D,rad}(\Omega)\times
L^{2}(\Omega)}+\big{\|}v^{3}+3v^{2}w+3vw^{2}\big{\|}_{L^{1}_{t}L^{2}_{x}([0,T)\times\Omega)}$
$\displaystyle\lesssim$
$\displaystyle\left\|\big{(}v_{0},v_{1}\big{)}\right\|_{\dot{H}^{1}_{D,rad}(\Omega)\times
L^{2}(\Omega)}+T\,\big{\|}v\big{\|}^{3}_{L^{\infty}_{t}L^{6}_{x}([0,T)\times\Omega)}+\big{\|}w\|^{2}_{L^{2}_{t}L^{6}_{x}(\mathbb{R}\times\Omega)}\big{\|}v\big{\|}_{L^{\infty}_{t}L^{6}_{x}([0,T)\times\Omega)}$
$\displaystyle\lesssim$
$\displaystyle\left\|\big{(}v_{0},v_{1}\big{)}\right\|_{\dot{H}^{1}_{D,rad}(\Omega)\times
L^{2}(\Omega)}+T\,\big{\|}v\big{\|}^{3}_{L^{\infty}_{t}\left([0,T);\dot{H}^{1}_{D}(\Omega)\right)}+\big{\|}w\|^{2}_{L^{2}_{t}L^{6}_{x}(\mathbb{R}\times\Omega)}\big{\|}v\big{\|}_{L^{\infty}_{t}\left([0,T);\dot{H}^{1}_{D,rad}(\Omega)\right)}.$

Arguing as above, we obtain

$\displaystyle\|\mathcal{T}(v_{1})-\mathcal{T}(v_{2})\|_{L^{\infty}_{t}\left([0,T);\dot{H}^{1}_{D,rad}(\Omega)\right)}$
$\displaystyle\lesssim$
$\displaystyle\;\big{\|}v_{1}^{3}-v_{2}^{3}+3v_{1}^{2}w-3v_{2}^{2}w+3v_{1}w^{2}-3v_{2}w^{2}\big{\|}_{L^{1}_{t}L^{2}_{x}([0,T)\times\Omega)}$
$\displaystyle\lesssim$
$\displaystyle\;T\big{\|}v_{1}-v_{2}\big{\|}_{L^{\infty}_{t}\left([0,T);\,L^{6}_{x}(\Omega)\right)}\left(\big{\|}v_{1}\big{\|}^{2}_{L^{\infty}_{t}\left([0,T);\,L^{6}_{x}(\Omega)\right)}+\big{\|}v_{2}\big{\|}^{2}_{L^{\infty}_{t}\left([0,T);\,L^{6}_{x}(\Omega)\right)}\right)+\big{\|}w\|^{2}_{L^{2}_{t}L^{6}_{x}(\mathbb{R}\times\Omega)}\big{\|}v_{1}-v_{2}\big{\|}_{L^{\infty}_{t}\left([0,T);\,L^{6}_{x}(\Omega)\right)}$
$\displaystyle\lesssim$
$\displaystyle\;T\big{\|}v_{1}-v_{2}\big{\|}_{L^{\infty}_{t}\dot{H}^{1}_{D,rad}(\Omega)}\left(\big{\|}v_{1}\big{\|}^{2}_{L^{\infty}_{t}\dot{H}^{1}_{D,rad}(\Omega)}+\big{\|}v_{2}\big{\|}^{2}_{L^{\infty}_{t}\dot{H}^{1}_{D,rad}(\Omega)}\right)+\big{\|}w\|^{2}_{L^{2}_{t}L^{6}_{x}(\mathbb{R}\times\Omega)}\big{\|}v_{1}-v_{2}\big{\|}_{L^{\infty}_{t}\dot{H}^{1}_{D,rad}(\Omega)}.$

Thus, by Proposition 4.1 and choosing $T$ such that

$2\;T\cdot\left(2\;C\left\|\big{(}v_{0},v_{1}\big{)}\right\|_{\dot{H}^{1}_{D,rad}(\Omega)\times
L^{2}(\Omega)}\right)^{2}\leq\frac{1}{2},$

we can guarantee that $\mathcal{T}$ maps the set $X$ back to itself and is a
contraction on the set $X$. By the contraction mapping theorem, it follows
that $\mathcal{T}$ has a fixed point $v$ in $X$. ∎

###  4.3. Global-in-time energy analysis for low frequency part

In this part, we extend local solution $v$ of (4.5) in Proposition 4.2 to
global ones in the energy space $\dot{H}^{1}_{D,rad}(\Omega)\times
L^{2}_{rad}(\Omega)$. By Proposition 4.2, It suffices to control the growth of
the energy of the solution $v$ of (4.5), which is not conserved because of the
perturbation term $F(v,w)$.

Let us denote the energy of $v$ by

$\displaystyle
E(v)(t)=\int_{\Omega}\frac{1}{2}\left(\partial_{t}v(t)\right)^{2}+\frac{1}{2}\left|\nabla
v(t)\right|^{2}+\frac{1}{4}\left(v(t)\right)^{4}\;dx.$

and take $\displaystyle E_{T}=\sup_{0\leq t<T}E(v)(t),$ where $[0,T)$ is the
maximal lifespan interval in Proposition 4.2.

We now turn to control the energy of $v$.

######  Proposition 4.3.

Let $w$ be the global solution of (4.1) in Proposition 4.1, $v$ be the local
energy solution of (4.5) in Proposition 4.2, and $[0,T)$ be the maximal
lifespan interval of $v$, then for any $t\in[0,T)$, we have

$\displaystyle E(v)(t)\leq E_{T}\lesssim
E(v)(0)+E_{T}^{3/2}T^{1/2}\|w\|_{L^{2}_{t}L^{6}_{x}(\mathbb{R}\times\Omega)}+E_{T}\|w\|^{2}_{L^{2}_{t}L^{6}_{x}(\mathbb{R}\times\Omega)}.$

###### Proof.

We now take the derivative in time and obtain that

$\displaystyle\frac{d}{dt}E(v)(t)=\int_{\Omega}\partial_{t}v\big{(}-3v^{2}w-3vw^{2}\big{)}\;dx.$
(4.6)

By the Hölder inequality, we obtain that

$\displaystyle\left|\int^{T}_{0}\int_{\Omega}\partial_{t}v\,v^{2}\,w\;dxdt\right|\lesssim$
$\displaystyle\int^{T}_{0}\|\partial_{t}v\|_{L^{2}_{x}(\Omega)}\|v\|^{2}_{L^{6}_{x}(\Omega)}\|w\|_{L^{6}_{x}(\Omega)}\,dt$
$\displaystyle\lesssim$
$\displaystyle\int^{T}_{0}\|\partial_{t}v\|_{L^{2}_{x}(\Omega)}\|v\|^{2}_{\dot{H}^{1}_{D}(\Omega)}\|w\|_{L^{6}_{x}(\Omega)}\;dt\lesssim
E_{T}^{3/2}\,T^{1/2}\,\|w\|_{L^{2}_{t}L^{6}_{x}(\mathbb{R}\times\Omega)},$
(4.7)

and

$\displaystyle\left|\int^{T}_{0}\int_{\Omega}\partial_{t}v\,v\,w^{2}\;dxdt\right|\lesssim$
$\displaystyle\int^{T}_{0}\|\partial_{t}v\|_{L^{2}_{x}(\Omega)}\|v\|_{L^{6}_{x}(\Omega)}\|w\|^{2}_{L^{6}_{x}(\Omega)}\,dt$
$\displaystyle\lesssim$
$\displaystyle\int^{T}_{0}\|\partial_{t}v\|_{L^{2}_{x}(\Omega)}\|v\|_{\dot{H}^{1}_{D}(\Omega)}\|w\|^{2}_{L^{6}_{x}(\Omega)}\;dt\lesssim
E_{T}\,\|w\|^{2}_{L^{2}_{t}L^{6}_{x}(\mathbb{R}\times\Omega)}.$ (4.8)

Taking (4.7) and (4.8) into (4.6), we obtain the result. ∎

By the fact that $u(0)\in L^{4}(\Omega)$ and the Bernstein inequality in Lemma
2.3, we have

$\displaystyle E(v)(0)\lesssim 2^{2J(1-s)}.$

By Proposition 4.1 and Proposition 4.3, we have

$\displaystyle E(v)(t)\leq E_{T}\lesssim
2^{2J(1-s)}+E_{T}^{3/2}\,T^{1/2}\,2^{J(1/2-s)}+E_{T}\,2^{2J(1/2-s)}.$

If choosing $T$ such that

$\displaystyle 2^{3J(1-s)}\,T^{1/2}\,2^{J(1/2-s)}\approx
2^{2J(1-s)}\Longleftrightarrow T\approx 2^{2J(2s-3/2)},$ (4.9)

we obtain

$\displaystyle E_{T}\lesssim 2^{2J(1-s)}\approx T^{\frac{1-s}{2s-3/2}}.$

which gives control of the energy of $v$ for arbitrarily large $T$ as long as
$s>\frac{3}{4}$.

###  4.4. Growth estimate of the solution $u$ of (1.1) in
$\dot{H}^{s}_{D,rad}(\Omega)$

From Subsection 4.1 and Subsection 4.3, we know that the solution $u$ of (1.1)
exists in $[0,T)$ for arbitrarily large $T$ in
$\dot{H}^{\frac{1}{2}}_{D,rad}(\Omega)\cap\dot{H}^{s}_{D,rad}(\Omega)+\dot{H}^{1}_{D,rad}(\Omega)$
with $s>\frac{3}{4}$. In this part, we show the estimate of the solution $u$
in $\dot{H}^{s}_{D,rad}(\Omega)$, and complete the proof of Theorem 1.4.

By Proposition 4.1, it suffices to show the estimate of $v$ in
$\dot{H}^{s}_{D,rad}(\Omega)$, in addition, the homogeneous part

$\cos(t\sqrt{-\Delta_{\Omega}})v_{0}+\frac{\sin(t\sqrt{-\Delta_{\Omega}})}{\sqrt{-\Delta_{\Omega}}}v_{1}$

of $v$ is bounded in $\dot{H}^{s}_{D}(\Omega)$ by the energy estimate. By
Proposition 4.3 and the interpolation argument, it reduces to estimate the
inhomogeneous part of $v$ in $L^{2}(\Omega)$. By the distorted Fourier
transform, Proposition 4.1 and Proposition 4.3, we have for any $0<t<T$

$\displaystyle\left\|\int^{t}_{0}\frac{\sin\big{(}(t-s)\sqrt{-\Delta_{\Omega}}\big{)}}{\sqrt{-\Delta_{\Omega}}}\big{(}v^{3}+3v^{2}w+3vw^{2}\big{)}\;ds\right\|_{L^{2}(\Omega)}$
$\displaystyle\lesssim$
$\displaystyle\int^{t}_{0}\left\|\frac{\sin\big{(}(t-s)\lambda\big{)}}{\lambda}\mathcal{F}_{D}\big{(}v^{3}+3v^{2}w+3vw^{2}\big{)}(\lambda)\right\|_{L^{2}_{\lambda}(\mathbb{R}^{+})}\;ds$
$\displaystyle\lesssim$
$\displaystyle\int^{t}_{0}(t-s)\left\|v^{3}+3v^{2}w+3vw^{2}\right\|_{L^{2}_{x}(\Omega)}\;ds$
$\displaystyle\lesssim$
$\displaystyle\int^{t}_{0}(t-s)\left(\|v\|^{3}_{\dot{H}^{1}_{D,rad}(\Omega)}+\|v\|^{2}_{\dot{H}^{1}_{D,rad}(\Omega)}\|w\|_{L^{6}_{x}(\Omega)}+\|v\|_{\dot{H}^{1}_{D,rad}(\Omega)}\|w\|^{2}_{L^{6}_{x}(\Omega)}\right)\;ds$
$\displaystyle\lesssim$
$\displaystyle\;E^{3/2}_{T}\,\int^{t}_{0}(t-s)^{2}\;ds+E_{T}\,\|w\|_{L^{2}_{t}L^{6}_{x}(\mathbb{R}\times\Omega)}\left(\int^{t}_{0}(t-s)\;ds\right)^{1/2}+E^{1/2}_{T}\,T\,\|w\|^{2}_{L^{2}_{t}L^{6}_{x}(\mathbb{R}\times\Omega)}$
$\displaystyle\lesssim$
$\displaystyle\;E^{3/2}_{T}\,T^{2}+E_{T}\,T^{3/2}\,\|w\|_{L^{2}_{t}L^{6}_{x}(\mathbb{R}\times\Omega)}+E^{1/2}_{T}\,T\,\|w\|^{2}_{L^{2}_{t}L^{6}_{x}(\mathbb{R}\times\Omega)}$
$\displaystyle\lesssim$ $\displaystyle\;T^{2+\frac{3(1-s)}{4s-3}},$

where $T$ is determined by (4.9). Therefore, by the interpolation between
$\dot{H}^{1}_{D}(\Omega)$ and $L^{2}(\Omega)$, we obtain

$\displaystyle\left\|\int^{t}_{0}\frac{\sin\big{(}(t-s)\sqrt{-\Delta_{\Omega}}\big{)}}{\sqrt{-\Delta_{\Omega}}}\big{(}v^{3}+3v^{2}w+3vw^{2}\big{)}\;ds\right\|_{\dot{H}^{s}_{D}(\Omega)}\lesssim\left(T^{2+\frac{3(1-s)}{4s-3}}\right)^{1-s}\,\left(T^{\frac{1-s}{4s-3}}\right)^{s}\lesssim
T^{\frac{3(1-s)(2s-1)}{4s-3}}.$

This completes the proof of Theorem 1.4.

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