# A uniform bound on costs of controlling semilinear heat equations on a
sequence of increasing domains and its application

Lijuan Wang, Can Zhang School of Mathematics and Statistics, Wuhan University;
Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072,
China; e-mail<EMAIL_ADDRESS>author. School of
Mathematics and Statistics, Wuhan University; Computational Science Hubei Key
Laboratory, Wuhan University, Wuhan 430072, China; e-mail:
<EMAIL_ADDRESS>
###### Abstract

In this paper, we first prove a uniform upper bound on costs of null controls
for semilinear heat equations with globally Lipschitz nonlinearity on a
sequence of increasing domains, where the controls are acted on an
equidistributed set that spreads out in the whole Euclidean space
$\mathbb{R}^{N}$. As an application, we then show the exactly null
controllability for this semilinear heat equation in $\mathbb{R}^{N}$. The
main novelty here is that the upper bound on costs of null controls for such
kind of equations in large but bounded domains can be made uniformly with
respect to the sizes of domains under consideration. The latter is crucial
when one uses a suitable approximation argument to derive the global null
controllability for the semilinear heat equation in $\mathbb{R}^{N}$. This
allows us to overcome the well-known problem of the lack of compactness
embedding arising in the study of null controllability for nonlinear PDEs in
generally unbounded domains.

2010 Mathematics Subject Classifications. 35K05, 93B07, 93C20

Keywords. Semilinear heat equation, null controllability, uniform cost,
equidistributed set

##  1 Introduction and main results

This paper is concerned with the null control costs for semilinear heat
equations on a sequence of increasing bounded domains in $\mathbb{R}^{N}$
(with $N\in\mathbb{N}$), when the controls act on the interior subsets of
these domains. Generally speaking, the null control costs depend on the
geometry of both control regions and whole domains where the equations evolve.
Nevertheless, the goal of this paper is to investigate the uniform upper
estimate for null control costs with respect to the varying domains. As an
interesting application, we shall derive the null controllability of the
semilinear heat equation in $\mathbb{R}^{N}$.

The general formulation of the problem could be stated as follows. Let $T$ be
a positive time and let $E$ be a subset of positive Lebesgue measure in
$(0,T)$. For each $n\in\mathbb{N}$, let $\omega_{n}$ be a nonempty open subset
of bounded Lipschitz domain $\Omega_{n}$ in $\mathbb{R}^{N}$. Consider the
following semilinear heat equation:

${10301}\left\\{\begin{array}[]{lll}\partial_{t}z_{n}-\Delta
z_{n}+f(z_{n})=\chi_{\omega_{n}}\chi_{E}u_{n}&\mbox{in}&\Omega_{n}\times(0,T),\\\
z_{n}=0&\mbox{on}&\partial\Omega_{n}\times(0,T),\\\
z_{n}(0)=z_{0}|_{\Omega_{n}},\end{array}\right.$ (1.1)

where $z_{0}\in L^{2}(\mathbb{R}^{N})$, $u_{n}=u_{n}(x,t)\in
L^{2}(\Omega_{n}\times(0,T))$ is the control function, $\chi_{E}$ and
$\chi_{\omega_{n}}$ are characteristic functions of $E$ and control region
$\omega_{n}$, respectively. Throughout the paper, we suppose that $f$ is a
globally Lipschitz function. Then, there exists a unique solution $z_{n}\in
C([0,T];L^{2}(\Omega_{n}))\bigcap L^{2}(0,T;H_{0}^{1}(\Omega_{n}))$ for the
equation $\eqref{10301}$ (c.f., [10], for instance).

Recall that for each $n\in\mathbb{N}$, the control system $\eqref{10301}$ is
called exactly null controllable at the time $T>0$, if the following statement
is true: For any $z_{0}\in L^{2}(\mathbb{R}^{N})$, there is a control
$u_{n}\in L^{2}(0,T;L^{2}(\Omega_{n}))$ so that the corresponding solution
$z_{n}$ to $\eqref{10301}$ satisfies that $z_{n}(T)=0$ in $\Omega_{n}$.
Furthermore, the null control cost is the least constant $C_{n}$ so that
$\|u_{n}\|_{L^{2}(\Omega_{n}\times(0,T))}\leqslant
C_{n}\|z_{0}\|_{L^{2}(\mathbb{R}^{N})}$ holds for all $z_{0}\in
L^{2}(\mathbb{R}^{N})$. As mentioned at the beginning, the constant $C_{n}$
may depend on the geometry parameters of $\Omega_{n}$ and $\omega_{n}$.

The question whether or not a given control system is null controllable and
obtaining upper bounds for the associated null control costs, with respect to
the time interval or control regions, are important topics in control theory,
both for linear partial differential control equations and abstract linear
control systems. We refer the reader to [1, 3, 9, 15, 16, 23, 24, 26] and
references therein for a wider discussion on this activated research field.

Meanwhile, there are many fascinating works in the literature on the
approximate or exact null controllability for the semilinear heat equations in
bounded domains; see, for instance, [10, 12, 19]. Their proofs are usually
divided into two parts: (i) null controllability of the linearized system;
(ii) a fixed-point argument. The globally null controllability was also proved
for a class of nonlinearities for which blow-up phenomena may arise (see,
e.g., [2, 11, 14]).

To the best of our knowledge, there are very few works on establishing a
uniform upper bound for the null control cost for a partial differential
controlled equation in varying domains.

In this paper, we shall restrict the control on an equidistributed set. We say
a set as an equidistributed set in $\mathbb{R}^{N}$ if it contains a union of
suitably distributed balls of fixed radius. Recently, there are many beautiful
existing results on the quantitative unique continuation for general elliptic
operators on equidistributed sets (c.f., [15, 21] and references therein).
Meanwhile, we will consider the varying domains $\Omega_{n}$ as an
approximation of $\mathbb{R}^{N}$, for the simplicity.

In order to state our main results, we first introduce certain standard
notations. For each $r>0$ and $x_{0}\in\mathbb{R}^{N}$, $B_{r}(x_{0})$ stands
for the closed ball centered at $x_{0}$ and of radius $r$; $Q_{r}(x_{0})$
denotes the smallest closed cube centered at $x_{0}$ so that
$B_{r}(x_{0})\subset Q_{r}(x_{0})$; $\mathrm{int}(Q_{r}(x_{0}))$ is the
interior of $Q_{r}(x_{0})$.

Let $0<r_{1}<r_{2}<\infty$. The following three assumptions will be effective
throughout the paper:

$(H_{1})$. There is a sequence
$\\{x_{i}\\}_{i=1}^{\infty}\subset\mathbb{R}^{N}$ so that
$\mathbb{R}^{N}=\displaystyle{\bigcup_{i=1}^{\infty}}Q_{r_{2}}(x_{i})$,
$\mathrm{int}(Q_{r_{2}}(x_{i}))\bigcap\mathrm{int}(Q_{r_{2}}(x_{j}))=\emptyset$
for each $i\neq j$. Moreover,
$\omega\triangleq\displaystyle{\bigcup_{i=1}^{\infty}}\omega_{i}$, where
$\omega_{i}$ is an open set and $B_{r_{1}}(x_{i})\subset\omega_{i}\subset
B_{r_{2}}(x_{i})$ for each $i\in\mathbb{N}$.

$(H_{2})$. For each $n\in\mathbb{N}$,
$\Omega_{n}\triangleq\mbox{int}\Big{(}\displaystyle{\bigcup_{i\in
I_{n}}}Q_{r_{2}}(x_{i})\Big{)}$ is convex, with $\mbox{Card}\,(I_{n})<\infty$.
In addition, $I_{n}\subsetneq I_{m}$ when $n<m$, and
$\displaystyle{\bigcup_{n=1}^{\infty}}\Omega_{n}=\mathbb{R}^{N}$.

$(H_{3})$. The function $f:\mathbb{R}\rightarrow\mathbb{R}$ is globally
Lipschitz continuous, i.e., $|f(s)-f(\tau)|\leqslant L|s-\tau|$ for all
$s,\tau\in\mathbb{R}$ with some constant $L>0$, and satisfies that $f(0)=0$.

With regarding to the assumption $(H_{1})$, we may say that the set $\omega$
is an equidistributed set in $\mathbb{R}^{N}$. A particularly example is a
periodic arrangement of balls (see Figure LABEL:fig1 below).

Figure 1: Examplefig1

Very recently, we have proved in [8] the observability inequality and null
controllability on such kind of sets for the linear heat equation with time
and space dependent potentials in $\mathbb{R}^{N}$.

The first main result of this paper concerning the uniform upper bound on
costs of controlling a semilinear heat equation on increasing large domains
can be stated as follows.

######  Theorem 1.1.

null-3 Let $(H_{1}),(H_{2})$ and $(H_{3})$ hold. Let $T>0$ and $E$ be a subset
of positive measure in $(0,T)$. Then there is a positive constant
$\kappa=\kappa(r_{1},r_{2},E,T,L)$ so that for any $n\in\mathbb{N}$ and any
$z_{0}\in L^{2}(\mathbb{R}^{N})$, there is a control $u_{n}\in
L^{2}(\mathbb{R}^{N}\times(0,T))$, with the uniform bound

$\|u_{n}\|_{L^{2}(\mathbb{R}^{N}\times(0,T))}\leqslant\kappa\|z_{0}\|_{L^{2}(\mathbb{R}^{N})},$

so that the corresponding solution $z_{n}\in C([0,T];L^{2}(\Omega_{n}))$ to
the following semilinear heat equation

$\left\\{\begin{array}[]{lll}\partial_{t}z_{n}-\Delta
z_{n}+f(z_{n})=\chi_{\omega\cap\Omega_{n}}\chi_{E}u_{n}&\mbox{in}&\Omega_{n}\times(0,T),\\\
z_{n}=0&\mbox{on}&\partial\Omega_{n}\times(0,T),\\\
z_{n}(0)=z_{0}&\mbox{in}&\Omega_{n},\end{array}\right.$

satisfies that $z_{n}(T)=0$ over $\Omega_{n}$.

######  Remark 1.2.

Our argument does not allow us to establish a similar result for the global
null controllability of slightly superlinearities as studied in [2, 11, 14].

######  Remark 1.3.

We do not know how to extend this result from an equidistributed set $\omega$
to a more general thick set. We refer the reader to [8, Remark 1.7] for the
difficulty.

Another motivation of this paper is to establish the null controllability for
the semilinear heat equation in the whole space $\mathbb{R}^{N}$, when the
control is acted on an equidistributed set. As already remarked in Remark 3.5
of our recent work [8] on the null controllability for the linear heat
equation with bounded potentials, the linearized plus fixed-point approach in
[10, 12] cannot be directly applied in the framework of general unbounded
domains because of the lack of compactness of Sobolev’s embedding.

The authors of [7] studied the approximate controllability of a semilinear
heat equation in an unbounded domain $\mathcal{O}$ of $\mathbb{R}^{N}$, with
control only acted in an open and nonempty subset, by an approximation method.
More precisely, they first considered the approximate controllability problem
in bounded domains of the form $\mathcal{O}_{n}\triangleq\mathcal{O}\bigcap
B_{n}$, where $B_{n}$ denotes the ball centered at the origin and of radius
$n$; and then they showed that the controls proposed in [10] restricted to
$\mathcal{O}_{n}$ converge in some sense to a desired approximate control in
$\mathcal{O}$, as $n$ goes to infinity. One of main ingredients in their
proofs is a qualitative unique continuation property for the linear parabolic
equation. Nevertheless, the technical proof of [7] (see also [6]) is not valid
any more for the null controllability of a semilinear heat equation in
$\mathbb{R}^{N}$.

It is also mentioning that the authors in [4] and [13] proved the null
controllability for some semilinear heat equations in an unbounded domain
$\mathcal{O}$ of $\mathbb{R}^{N}$, when the control is assumed to be
distributed along a subdomain $\omega$ so that the uncontrolled region
$\mathcal{O}\setminus\omega$ is bounded. The arguments therein are mainly
based on a Carleman estimate for the linear parabolic operator in bounded
domains.

Recently, the authors in [21] showed in a linear and abstract framework that
if the sequence of null controls associated to an exhaustion of an unbounded
domain converges, then the solutions converge in the same way to the limiting
problem on the unbounded domain. This result allows to infer the null
controllability on unbounded domain by studying the control problem on a
sequence of bounded domains. In particular, it recovers the null
controllability result for the linear heat equation in $\mathbb{R}^{N}$. The
latter has been already established well in [9] and [23] independently.

Inspired by these works, we could utilize Theorem LABEL:null-3 to prove the
null controllability for a semilinear heat equation in $\mathbb{R}^{N}$ with
the control acted on an equidistributed set $\omega$. In fact, since null
controls in Theorem LABEL:null-3 are uniformly bounded,
$\\{u_{n}\\}_{n\geqslant 1}$ has a weakly convergent subsequence with a limit
$u$ in $L^{2}(\mathbb{R}^{N}\times(0,T))$. Thus, one may expect that such
control maybe a null control for the corresponding semilinear heat controlled
equation in $\mathbb{R}^{N}$. Actually we have the following result:

######  Theorem 1.4.

null-2 Assume that $(H_{1})$ and $(H_{3})$ hold. Let $T>0$ and $E$ be a subset
of positive measure in $(0,T)$. Then, for each initial value $y_{0}\in
L^{2}(\mathbb{R}^{N})$, there is a control $u\in
L^{2}(0,T;L^{2}(\mathbb{R}^{N}))$ with an upper bound

$\|u\|_{L^{2}(\mathbb{R}^{N}\times(0,T))}\leqslant\kappa\|y_{0}\|_{L^{2}(\mathbb{R}^{N})},$

so that the corresponding solution

${null-wang}\left\\{\begin{array}[]{lll}\partial_{t}y-\Delta
y+f(y)=\chi_{\omega}\chi_{E}u&\mbox{in}&\mathbb{R}^{N}\times(0,T),\\\
y(0)=y_{0}&\mbox{in}&\mathbb{R}^{N},\end{array}\right.$ (1.2)

satisfies that $y(T)=0$ in $\mathbb{R}^{N}$. Here $\kappa$ is the same
constant as in Theorem LABEL:null-3.

######  Remark 1.5.

The well-posedness of such kind of semilinear heat equations in general
unbounded domains is stated without a proof in [4], for instance. We refer the
reader to Definition LABEL:weak_solution and Lemma LABEL:Existence in Section
LABEL:b5 for precise presentations.

######  Remark 1.6.

Instead of $\mathbb{R}^{N}$, our considerations are valid as well for every
set that can be approximated with cubes such as the half space
$\mathbb{R}^{N-1}\times\mathbb{R}^{+}$ and the infinite strip.

######  Remark 1.7.

Our method does not rely on any compactness argument in the whole space
$\mathbb{R}^{N}$.

The rest of this paper is organized as follows. Section LABEL:b1 is devoted to
prove the uniform bound of control costs for the linearized controlled system.
Section LABEL:b2 shows the proof of Theorem LABEL:null-3 by a fixed-point
argument. Finally, Section LABEL:b5 presents the proof of Theorem
LABEL:null-2.

##  2 The linear case

b1

For each $n\in\mathbb{N}$, let $\varphi_{n}$ be the solution to the following
linear heat equation with a bounded potential $a\in
L^{\infty}(\mathbb{R}^{N}\times(0,T))$:

${1.1}\left\\{\begin{array}[]{lll}\partial_{t}\varphi_{n}-\Delta\varphi_{n}+a\varphi_{n}=0&\mathrm{in}&\Omega_{n}\times(0,T),\\\
\varphi_{n}=0&\mathrm{on}&\partial\Omega_{n}\times(0,T),\\\
\varphi_{n}(0)=\varphi_{0}&\mathrm{in}&\Omega_{n},\\\ \end{array}\right.$
(2.1)

with $\varphi_{0}\in L^{2}(\mathbb{R}^{N})$.

We first state a uniform observability inequality for all solutions of
(LABEL:1.1) evolving in $\Omega_{n}$ for all $n\in\mathbb{N}$. Here and
throughout this paper, we simply write
$\|a\|_{\infty}=\|a\|_{L^{\infty}(\mathbb{R}^{N}\times(0,T))};$ and write
$C(\cdot)$ for a positive constant depending on what are enclosed in the
brackets.

######  Theorem 2.1.

Thm1 Assume that $(H_{1})$ and $(H_{2})$ hold. Let $T>0$ and $E$ be a subset
of positive measure in $(0,T)$. Then there are constants $C=C(r_{1},r_{2})$
and $\widetilde{C}=\widetilde{C}(r_{1},r_{2},E)$ so that the following
observability inequality

$\int_{\Omega_{n}}|\varphi_{n}(x,T)|^{2}\,\mathrm{d}x\leqslant
e^{\widetilde{C}}e^{C\left(T+T\|a\|_{\infty}+\|a\|_{\infty}^{2/3}\right)}\int_{E}\int_{\omega\bigcap\Omega_{n}}|\varphi_{n}(x,t)|^{2}\mathrm{d}x\mathrm{d}t,\;\;\;\;\forall\varphi_{0}\in
L^{2}(\mathbb{R}^{N}),$

holds uniformly for all $n\in\mathbb{N}$.

######  Remark 2.2.

Note that the two constants in the above theorem are independent of the sizes
of domains.

######  Remark 2.3.

Thm2 Under the same assumptions as in Theorem LABEL:Thm1, the following
refined estimate is also true: There are constants $C=C(r_{1},r_{2})$ and
$\widetilde{C}=\widetilde{C}(r_{1},r_{2},E)$ so that the $L^{1}$-type
observability inequality

$\|\varphi_{n}(T)\|_{L^{2}(\Omega_{n})}\leqslant
e^{\widetilde{C}}e^{C\left(T+T\|a\|_{\infty}+\|a\|_{\infty}^{2/3}\right)}\int_{E}\int_{\omega\bigcap\Omega_{n}}|\varphi_{n}(x,t)|\mathrm{d}x\mathrm{d}t,\;\;\;\;\forall\varphi_{0}\in
L^{2}(\mathbb{R}^{N}),$

holds uniformly for all $n\in\mathbb{N}$.

As a direct consequence of Theorem LABEL:Thm1, we present the linear version
of Theorem LABEL:null-3 as follows. Consider the following linear control
system:

${appli-1}\left\\{\begin{array}[]{lll}\partial_{t}z_{n}-\Delta
z_{n}+az_{n}=\chi_{\omega\bigcap\Omega_{n}}\chi_{E}u_{n}&\mbox{in}&\Omega_{n}\times(0,T),\\\
z_{n}=0&\mbox{on}&\partial\Omega_{n}\times(0,T),\\\
z_{n}(0)=z_{0}&\mbox{in}&\Omega_{n},\\\ \end{array}\right.$ (2.2)

where $z_{0}\in L^{2}(\mathbb{R}^{N})$ is an initial state and $u_{n}\in
L^{2}(\Omega_{n}\times(0,T))$ is a control. For each $n\in\mathbb{N}$, we
write $z_{n}(\cdot;z_{0},u_{n})$ for the solution to (LABEL:appli-1). By a
standard duality method (see, for instance, [5, Theorem 2.42]), we can easily
obtain the following null controllability result with a uniform upper bound on
control costs.

######  Corollary 2.4.

appli-2 Under the same assumptions of Theorem LABEL:Thm1, for each
$n\in\mathbb{N}$ and $z_{0}\in L^{2}(\mathbb{R}^{N})$, there is a control
$u_{n}\in L^{2}(0,T;L^{2}(\mathbb{R}^{N}))$, with a uniform cost

$\|u_{n}\|_{L^{2}(0,T;L^{2}(\mathbb{R}^{N}))}\leqslant
e^{\widetilde{C}}e^{C\left(T+T\|a\|_{\infty}+\|a\|_{\infty}^{2/3}\right)}\|z_{0}\|_{L^{2}(\mathbb{R}^{N})},$

where the constants $C$ and $\widetilde{C}$ are given by Theorem LABEL:Thm1,
so that $z_{n}(T;z_{0},u_{n})=0$ in $\Omega_{n}$.

###  2.1 Quantitative estimates of unique continuation

pre

Before giving the proof of Theorem LABEL:Thm1, we present the following
quantitative unique continuation property for all solutions of (LABEL:1.1).

######  Proposition 2.5.

lemma-2.2 Let $(H_{2})$ hold. Let $0<r<R<\infty$ and $\delta\in(0,1]$. Then
there are a universal constant $C>0$ and three positive constants
$C_{1}\triangleq C_{1}(R,\delta),C_{2}\triangleq C_{2}(R,\delta)$ and
$\gamma\triangleq\gamma(r,R,\delta)\in(0,1)$ so that for any $n\in\mathbb{N}$,
any $x_{0}\in\Omega_{n}$, and any $\varphi_{0}\in L^{2}(\Omega_{n})$, the
solution $\varphi_{n}$ of (LABEL:1.1) satisfies

$\displaystyle\int_{B_{R}(x_{0})\bigcap\Omega_{n}}\\!\\!\\!\\!|\varphi_{n}(x,T)|^{2}\mathrm{d}x$
$\displaystyle\\!\\!\leqslant$
$\displaystyle\\!\\!\\!\\!\left[C_{1}e^{[1+2C(1+\frac{1}{R^{2}})](1+\frac{4}{T}+\|a\|^{2/3}_{\infty})+\frac{C_{2}}{T}+2T\|a\|_{\infty}}\int_{T/2}^{T}\int_{Q_{2R_{0}}(x_{0})\bigcap\Omega_{n}}\\!\\!\varphi_{n}^{2}(x,t)\mathrm{d}x\mathrm{d}t\right]^{\gamma}$
$\displaystyle\times\left(2\int_{B_{r}(x_{0})\bigcap\Omega_{n}}|\varphi_{n}(x,T)|^{2}\mathrm{d}x\right)^{1-\gamma},$

where $R_{0}\triangleq(1+2\delta)R$.

######  Remark 2.6.

Note that the constants above are independent of the sizes of the domains
$\Omega_{n}$.

Here, we do not give the complete proof of Proposition LABEL:lemma-2.2 since
it is basically already done in [8]. We only point out the major difference
compared with the proof of [8, Lemma 3.2]. Indeed, the key ingredient in the
proof of [8, Lemma 3.2] is a monotonicity formula of parabolic frequency
function in a bounded ball (i.e., [8, Lemma 3.1]). Instead, the following
analogous monotonicity formula in a convex and bounded domain could be
utilized when one seeks for the detailed proof of Proposition LABEL:lemma-2.2.

######  Proposition 2.7.

lemma-2.1([17] or [25]) Let $\Omega$ be a bounded and convex subset in
$\mathbb{R}^{N}$. Let $r>0$, $\lambda>0$, $T>0$ and $x_{0}\in\Omega$. Denote
by

$G_{\lambda}(x,t)\triangleq\frac{1}{(T-t+\lambda)^{N/2}}e^{-\frac{|x-x_{0}|^{2}}{4(T-t+\lambda)}},\;\;t\in[0,T].$

For $u\in H^{1}(0,T;L^{2}(B_{r}(x_{0})\bigcap\Omega))\bigcap
L^{2}(0,T;H^{2}(B_{r}(x_{0})\bigcap\Omega)\bigcap
H^{1}_{0}(B_{r}(x_{0})\bigcap\Omega))$ and $t\in(0,T],$ set

${3.111}N_{\lambda,r}(t)\triangleq\frac{\int_{B_{r}(x_{0})\bigcap\Omega}|\nabla
u(x,t)|^{2}G_{\lambda}(x,t)\mathrm{d}x}{\int_{B_{r}(x_{0})\bigcap\Omega}|u(x,t)|^{2}G_{\lambda}(x,t)\mathrm{d}x}\;\;\;\;\text{whenever}\;\;\int_{B_{r}(x_{0})\bigcap\Omega}|u(x,t)|^{2}\mathrm{d}x\neq
0.$

Then

$\frac{\mathrm{d}}{\mathrm{d}t}N_{\lambda,r}(t)\leqslant\frac{1}{T-t+\lambda}N_{\lambda,r}(t)+\frac{\int_{B_{r}(x_{0})\bigcap\Omega}|(\partial_{t}u-\Delta
u)(x,t)|^{2}G_{\lambda}(x,t)\mathrm{d}x}{\int_{B_{r}(x_{0})\bigcap\Omega}|u(x,t)|^{2}G_{\lambda}(x,t)\mathrm{d}x}.$

Based on Proposition LABEL:lemma-2.2, we then could obtain a global
interpolation inequality for solutions of (LABEL:1.1) at one point of time
variable.

######  Proposition 2.8.

2.3 Assume that $(H_{1})$ and $(H_{2})$ hold. Then there are two constants
$C_{3}\triangleq C_{3}(r_{2})>0$ and
$\theta\triangleq\theta(r_{1},r_{2})\in(0,1)$ so that for any $n\in\mathbb{N}$
and any $\varphi_{0}\in L^{2}(\Omega_{n})$, the solution $\varphi_{n}$ of
(LABEL:1.1) satisfies

${3.33333}\int_{\Omega_{n}}|\varphi_{n}(x,T)|^{2}\mathrm{d}x\leqslant
e^{C_{3}\left(T^{-1}+T+T\|a\|_{\infty}+\|a\|_{\infty}^{2/3}\right)}\left(\int_{\Omega_{n}}|\varphi_{0}(x)|^{2}\mathrm{d}x\right)^{\theta}\left(\int_{\omega\bigcap\Omega_{n}}|\varphi_{n}(x,T)|^{2}\mathrm{d}x\right)^{1-\theta}.$
(2.3)

###### Proof.

According to Proposition LABEL:lemma-2.2 (where $x_{0},r,R$ and $\delta$ are
replaced by $x_{i}\in\Omega_{n},r_{1},\sqrt{N}r_{2}$ and $1/2$, respectively),
we obtain

$\displaystyle\int_{\mbox{int}(Q_{r_{2}}(x_{i}))}|\varphi_{n}(x,T)|^{2}\mathrm{d}x\leqslant\int_{B_{\sqrt{N}r_{2}}(x_{i})\bigcap\Omega_{n}}|\varphi_{n}(x,T)|^{2}\mathrm{d}x$
$\displaystyle\leqslant$
$\displaystyle\left[\mathcal{\widehat{K}}_{1}e^{[1+2C(1+r_{2}^{-2})](1+4T^{-1}+\|a\|^{2/3}_{\infty})+\mathcal{\widehat{K}}_{2}T^{-1}+2T\|a\|_{\infty}}\int_{T/2}^{T}\int_{Q_{4\sqrt{N}r_{2}}(x_{i})\bigcap\Omega_{n}}\varphi_{n}^{2}\mathrm{d}x\mathrm{d}t\right]^{\theta}$
$\displaystyle\times\left[2\int_{B_{r_{1}}(x_{i})\bigcap\Omega_{n}}|\varphi_{n}(x,T)|^{2}\mathrm{d}x\right]^{1-\theta},$

where
$\mathcal{\widehat{K}}_{1}\triangleq\mathcal{\widehat{K}}_{1}(r_{2})>0,\mathcal{\widehat{K}}_{2}\triangleq\mathcal{\widehat{K}}_{2}(r_{2})>0$
and $\theta\triangleq\theta(r_{1},r_{2})\in(0,1).$ This, along with Young’s
inequality, implies that for each $\varepsilon>0,$

$\displaystyle\int_{\mbox{int}(Q_{r_{2}}(x_{i}))}|\varphi_{n}(x,T)|^{2}\mathrm{d}x$
$\displaystyle\leqslant$
$\displaystyle\varepsilon\theta\mathcal{\widehat{K}}_{1}e^{[1+2C(1+r_{2}^{-2})](1+4T^{-1}+\|a\|^{2/3}_{\infty})+\mathcal{\widehat{K}}_{2}T^{-1}+2T\|a\|_{\infty}}\int_{T/2}^{T}\int_{Q_{4\sqrt{N}r_{2}}(x_{i})\bigcap\Omega_{n}}\varphi_{n}^{2}\mathrm{d}x\mathrm{d}t$
$\displaystyle+2\varepsilon^{-\frac{\theta}{1-\theta}}(1-\theta)\int_{B_{r_{1}}(x_{i})\bigcap\Omega_{n}}|\varphi_{n}(x,T)|^{2}\mathrm{d}x.$

By $(H_{2})$, we have

${3.44444}\begin{array}[]{lll}&&\displaystyle{}\int_{\Omega_{n}}|\varphi_{n}(x,T)|^{2}\mathrm{d}x=\sum_{i\in
I_{n}}\int_{\mbox{int}(Q_{r_{2}}(x_{i}))}|\varphi_{n}(x,T)|^{2}\mathrm{d}x\\\
\\\
&\leqslant&\displaystyle{}\varepsilon\theta\mathcal{\widehat{K}}_{1}e^{[1+2C(1+r_{2}^{-2})](1+4T^{-1}+\|a\|^{2/3}_{\infty})+\mathcal{\widehat{K}}_{2}T^{-1}+2T\|a\|_{\infty}}\sum_{i\in
I_{n}}\int_{T/2}^{T}\int_{Q_{4\sqrt{N}r_{2}}(x_{i})\bigcap\Omega_{n}}\varphi_{n}^{2}\mathrm{d}x\mathrm{d}t\\\
\\\
&&\displaystyle{}+2\varepsilon^{-\frac{\theta}{1-\theta}}(1-\theta)\int_{\omega\bigcap\Omega_{n}}|\varphi_{n}(x,T)|^{2}\mathrm{d}x.\end{array}$
(2.4)

Denote by

${\widetilde{\varphi}}_{n}(x,t)\triangleq\left\\{\begin{array}[]{lll}\varphi_{n}(x,t)&\mbox{if}&(x,t)\in\Omega_{n}\times(0,T),\\\
0&\mbox{if}&(x,t)\in(\mathbb{R}^{N}\setminus\Omega_{n})\times(0,T).\end{array}\right.$

We can directly check that

$\displaystyle\sum_{i\in
I_{n}}\int_{T/2}^{T}\int_{Q_{4\sqrt{N}r_{2}}(x_{i})\bigcap\Omega_{n}}\varphi_{n}^{2}\mathrm{d}x\mathrm{d}t$
$\displaystyle\leqslant$ $\displaystyle\sum_{i\geqslant
1}\int_{T/2}^{T}\int_{Q_{4\sqrt{N}r_{2}}(x_{i})}(\chi_{\Omega_{n}}{\widetilde{\varphi}}_{n})^{2}\mathrm{d}x\mathrm{d}t$
$\displaystyle\leqslant$
$\displaystyle\mathcal{\widehat{K}}_{3}\int_{T/2}^{T}\int_{\mathbb{R}^{N}}(\chi_{\Omega_{n}}{\widetilde{\varphi}}_{n})^{2}\mathrm{d}x\mathrm{d}t=\mathcal{\widehat{K}}_{3}\int_{T/2}^{T}\int_{\Omega_{n}}\varphi_{n}^{2}\mathrm{d}x\mathrm{d}t,$

where $\mathcal{\widehat{K}}_{3}>0$. Then it follows from (LABEL:3.44444) that

$\displaystyle\int_{\Omega_{n}}|\varphi_{n}(x,T)|^{2}\mathrm{d}x$
$\displaystyle\leqslant$
$\displaystyle\varepsilon\theta\mathcal{\widehat{K}}_{1}\mathcal{\widehat{K}}_{3}e^{[1+2C(1+r_{2}^{-2})](1+4T^{-1}+\|a\|^{2/3}_{\infty})+\mathcal{\widehat{K}}_{2}T^{-1}+2T\|a\|_{\infty}}\int_{T/2}^{T}\int_{\Omega_{n}}\varphi_{n}^{2}\mathrm{d}x\mathrm{d}t$
$\displaystyle+2\varepsilon^{-\frac{\theta}{1-\theta}}(1-\theta)\int_{\omega\bigcap\Omega_{n}}|\varphi_{n}(x,T)|^{2}\mathrm{d}x\;\;\mathrm{for\
each}\ \varepsilon>0.$

This implies

${3.444440}\begin{array}[]{lll}&&\displaystyle{}\int_{\Omega_{n}}|\varphi_{n}(x,T)|^{2}\mathrm{d}x\\\
\\\
&\leqslant&\displaystyle{}\left[\mathcal{\widehat{K}}_{1}\mathcal{\widehat{K}}_{3}e^{[1+2C(1+r_{2}^{-2})](1+4T^{-1}+\|a\|^{2/3}_{\infty})+\mathcal{\widehat{K}}_{2}T^{-1}+2T\|a\|_{\infty}}\int_{T/2}^{T}\int_{\Omega_{n}}\varphi_{n}^{2}\mathrm{d}x\mathrm{d}t\right]^{\theta}\\\
\\\
&&\displaystyle{}\times\left[2\int_{\omega\bigcap\Omega_{n}}|\varphi_{n}(x,T)|^{2}\mathrm{d}x\right]^{1-\theta}.\end{array}$
(2.5)

Noting that

$\int_{\Omega_{n}}|\varphi_{n}(x,t)|^{2}\,\mathrm{d}x\leqslant
e^{2\|a\|_{\infty}t}\int_{\Omega_{n}}|\varphi_{0}(x)|^{2}\,\mathrm{d}x\ \ \
\mathrm{for\ each}\ t\in[0,T],$

by (LABEL:3.444440), we deduce

$\displaystyle\int_{\Omega_{n}}|\varphi_{n}(x,T)|^{2}\mathrm{d}x$
$\displaystyle\leqslant$
$\displaystyle\left[\mathcal{\widehat{K}}_{1}\mathcal{\widehat{K}}_{3}Te^{[1+2C(1+r_{2}^{-2})](1+4T^{-1}+\|a\|^{2/3}_{\infty})+\mathcal{\widehat{K}}_{2}T^{-1}+2T\|a\|_{\infty}}e^{2T\|a\|_{\infty}}\int_{\Omega_{n}}\varphi_{0}^{2}\mathrm{d}x\right]^{\theta}$
$\displaystyle\times\left[2\int_{\omega\bigcap\Omega_{n}}|\varphi_{n}(x,T)|^{2}\mathrm{d}x\right]^{1-\theta}.$

Hence, (LABEL:3.33333) follows from the latter inequality immediately. ∎

###  2.2 Proof of Theorem LABEL:Thm1

pro

Now, we are able to present the proof of Theorem LABEL:Thm1 by using the
telescoping series method. The proof is similar to that of [8, Theorem 1.1].
Here we only sketch the proof.

Proof of Theorem LABEL:Thm1. Arbitrarily fix $n\in\mathbb{N}$. For any
$0\leqslant t_{1}<t_{2}\leqslant T$, by a translation in the time variable and
Proposition LABEL:2.3, we obtain from Young’s inequality that

${2019-7-9}\|\varphi_{n}(t_{2})\|^{2}_{L^{2}(\Omega_{n})}\leqslant\varepsilon\|\varphi_{n}(t_{1})\|^{2}_{L^{2}(\Omega_{n})}+\frac{\mathcal{\widetilde{K}}_{1}}{\varepsilon^{\alpha}}e^{\frac{\mathcal{\widetilde{K}}_{2}}{t_{2}-t_{1}}}\|\varphi_{n}(t_{2})\|^{2}_{L^{2}(\omega\bigcap\Omega_{n})}\
\ \ \mathrm{for\ each}\ \varepsilon>0,$ (2.6)

where $\mathcal{\widetilde{K}}_{1}\triangleq
e^{\frac{C_{3}}{1-\theta}\left(T+T\|a\|_{\infty}+\|a\|_{\infty}^{2/3}\right)},$
$\mathcal{\widetilde{K}}_{2}\triangleq C_{3}/(1-\theta)$ and
$\alpha\triangleq\theta/(1-\theta)$.

Let $l$ be a density point of $E$. According to Proposition 2.1 in [18], for
each $\kappa>1$, there exists $l_{1}\in(l,T)$, depending on $\kappa$ and $E$,
so that the sequence $\\{l_{m}\\}_{m\geqslant 1}$, given by

$l_{m+1}=l+\frac{1}{\kappa^{m}}(l_{1}-l),$

satisfies

${3.2525251}l_{m}-l_{m+1}\leqslant 3|E\bigcap(l_{m+1},l_{m})|.$ (2.7)

Next, let $0<l_{m+2}<l_{m+1}\leqslant t<l_{m}<l_{1}<T$. It follows from
(LABEL:2019-7-9) that

${3.2525252}\|\varphi_{n}(t)\|^{2}_{L^{2}(\Omega_{n})}\leqslant\varepsilon\|\varphi_{n}(l_{m+2})\|^{2}_{L^{2}(\Omega_{n})}+\frac{\mathcal{\widetilde{K}}_{1}}{\varepsilon^{\alpha}}e^{\frac{\mathcal{\widetilde{K}}_{2}}{t-l_{m+2}}}\|\varphi_{n}(t)\|^{2}_{L^{2}(\omega\bigcap\Omega_{n})}\
\mathrm{for\ each}\ \varepsilon>0.$ (2.8)

By a standard energy estimate, we have

$\|\varphi_{n}(l_{m})\|_{L^{2}(\Omega_{n})}\leqslant
e^{T\|a\|_{\infty}}\|\varphi_{n}(t)\|_{L^{2}(\Omega_{n})}.$

This, along with (LABEL:3.2525252), implies

$\|\varphi_{n}(l_{m})\|^{2}_{L^{2}(\Omega_{n})}\leqslant
e^{2T\|a\|_{\infty}}\left(\varepsilon\|\varphi_{n}(l_{m+2})\|^{2}_{L^{2}(\Omega_{n})}+\frac{\mathcal{\widetilde{K}}_{1}}{\varepsilon^{\alpha}}e^{\frac{\mathcal{\widetilde{K}}_{2}}{t-l_{m+2}}}\|\varphi_{n}(t)\|^{2}_{L^{2}(\omega\bigcap\Omega_{n})}\right)\
\mathrm{for\ each}\ \varepsilon>0,$

which indicates that

$\|\varphi_{n}(l_{m})\|^{2}_{L^{2}(\Omega_{n})}\leqslant\varepsilon\|\varphi_{n}(l_{m+2})\|^{2}_{L^{2}(\Omega_{n})}+\frac{\mathcal{\widetilde{K}}_{3}}{\varepsilon^{\alpha}}e^{\frac{\mathcal{\widetilde{K}}_{2}}{t-l_{m+2}}}\|\varphi_{n}(t)\|^{2}_{L^{2}(\omega\bigcap\Omega_{n})}\
\mathrm{for\ each}\ \varepsilon>0,$

where
$\mathcal{\widetilde{K}}_{3}=(e^{2T\|a\|_{\infty}})^{1+\alpha}\mathcal{\widetilde{K}}_{1}$.
Integrating the latter inequality over $E\bigcap(l_{m+1},l_{m})$ gives

${3.2525253}\begin{array}[]{lll}\displaystyle{}|E\bigcap(l_{m+1},l_{m})|\|\varphi_{n}(l_{m})\|^{2}_{L^{2}(\Omega_{n})}&\leqslant&\displaystyle{}\varepsilon|E\bigcap(l_{m+1},l_{m})|\|\varphi_{n}(l_{m+2})\|^{2}_{L^{2}(\Omega_{n})}\\\
&&\displaystyle{}+\frac{\mathcal{\widetilde{K}}_{3}}{\varepsilon^{\alpha}}e^{\frac{\mathcal{\widetilde{K}}_{2}}{l_{m+1}-l_{m+2}}}\int_{l_{m+1}}^{l_{m}}\chi_{E}\|\varphi_{n}(t)\|^{2}_{L^{2}(\omega\bigcap\Omega_{n})}\mathrm{d}t\end{array}$
(2.9)

for each $\varepsilon>0$.

Since $l_{m}-l_{m+1}=(\kappa-1)(l_{1}-l)/\kappa^{m},$ by (LABEL:3.2525253) and
(LABEL:3.2525251), we obtain

$\displaystyle\|\varphi_{n}(l_{m})\|^{2}_{L^{2}(\Omega_{n})}$
$\displaystyle\leqslant$
$\displaystyle\frac{1}{|E\bigcap(l_{m+1},l_{m})|}\frac{\mathcal{\widetilde{K}}_{3}}{\varepsilon^{\alpha}}e^{\frac{\mathcal{\widetilde{K}}_{2}}{l_{m+1}-l_{m+2}}}\int_{l_{m+1}}^{l_{m}}\chi_{E}\|\varphi_{n}(t)\|^{2}_{L^{2}(\omega\bigcap\Omega_{n})}\mathrm{d}t+\varepsilon\|\varphi_{n}(l_{m+2})\|^{2}_{L^{2}(\Omega_{n})}$
$\displaystyle\leqslant$
$\displaystyle\frac{3\kappa^{m}}{(l_{1}-l)(\kappa-1)}\frac{\mathcal{\widetilde{K}}_{3}}{\varepsilon^{\alpha}}e^{\mathcal{\widetilde{K}}_{2}\left(\frac{1}{l_{1}-l}\frac{\kappa^{m+1}}{\kappa-1}\right)}\int_{l_{m+1}}^{l_{m}}\chi_{E}\|\varphi_{n}(t)\|^{2}_{L^{2}(\omega\bigcap\Omega_{n})}\mathrm{d}t+\varepsilon\|\varphi_{n}(l_{m+2})\|^{2}_{L^{2}(\Omega_{n})}$

for each $\varepsilon>0$. This yields

$\begin{array}[]{lll}&&\|\varphi_{n}(l_{m})\|^{2}_{L^{2}(\Omega_{n})}\\\
&\leqslant&\displaystyle{}\frac{1}{\varepsilon^{\alpha}}\frac{3}{\kappa}\frac{\mathcal{\widetilde{K}}_{3}}{\mathcal{\widetilde{K}}_{2}}e^{2\mathcal{\widetilde{K}}_{2}\left(\frac{1}{l_{1}-l}\frac{\kappa^{m+1}}{\kappa-1}\right)}\int_{l_{m+1}}^{l_{m}}\chi_{E}\|\varphi_{n}(t)\|^{2}_{L^{2}(\omega\bigcap\Omega_{n})}\mathrm{d}t\displaystyle{}+\varepsilon\|\varphi_{n}(l_{m+2})\|^{2}_{L^{2}(\Omega_{n})}\end{array}{3.2525254}$
(2.10)

for each $\varepsilon>0$. Denote by $d\triangleq
2\mathcal{\widetilde{K}}_{2}/[\kappa(l_{1}-l)(\kappa-1)]$. It follows from
(LABEL:3.2525254) that

$\displaystyle\varepsilon^{\alpha}e^{-d\kappa^{m+2}}\|\varphi_{n}(l_{m})\|^{2}_{L^{2}(\Omega_{n})}-\varepsilon^{1+\alpha}e^{-d\kappa^{m+2}}\|\varphi_{n}(l_{m+2})\|^{2}_{L^{2}(\Omega_{n})}\leqslant\frac{3}{\kappa}\frac{\mathcal{\widetilde{K}}_{3}}{\mathcal{\widetilde{K}}_{2}}\int_{l_{m+1}}^{l_{m}}\chi_{E}\|\varphi_{n}(t)\|^{2}_{L^{2}(\omega\bigcap\Omega_{n})}\mathrm{d}t$

for each $\varepsilon>0$.

Choosing $\varepsilon=e^{-d\kappa^{m+2}}$ in the above inequality gives

${3.25252555}\begin{array}[]{lll}&&\displaystyle{}e^{-(1+\alpha)d\kappa^{m+2}}\|\varphi_{n}(l_{m})\|^{2}_{L^{2}(\Omega_{n})}-e^{-(2+\alpha)d\kappa^{m+2}}\|\varphi_{n}(l_{m+2})\|^{2}_{L^{2}(\Omega_{n})}\\\
&\leqslant&\displaystyle{}\frac{3}{\kappa}\frac{\mathcal{\widetilde{K}}_{3}}{\mathcal{\widetilde{K}}_{2}}\int_{l_{m+1}}^{l_{m}}\chi_{E}\|\varphi_{n}(t)\|^{2}_{L^{2}(\omega\bigcap\Omega_{n})}\mathrm{d}t.\end{array}$
(2.11)

Taking $\kappa=\sqrt{(\alpha+2)/(\alpha+1)}$ in (LABEL:3.25252555), we then
have

$\displaystyle
e^{-(2+\alpha)d\kappa^{m}}\|\varphi_{n}(l_{m})\|^{2}_{L^{2}(\Omega_{n})}-e^{-(2+\alpha)d\kappa^{m+2}}\|\varphi_{n}(l_{m+2})\|^{2}_{L^{2}(\Omega_{n})}\leqslant\frac{3}{\kappa}\frac{\mathcal{\widetilde{K}}_{3}}{\mathcal{\widetilde{K}}_{2}}\int_{l_{m+1}}^{l_{m}}\chi_{E}\|\varphi_{n}(t)\|^{2}_{L^{2}(\omega\bigcap\Omega_{n})}\mathrm{d}t.$

Changing $m$ to $2m^{\prime}$ and summing the above inequality from
$m^{\prime}=1$ to $\infty$ give the desired result. This finishes the proof of
Theorem LABEL:Thm1. ∎

##  3 Proof of Theorem LABEL:null-3

b2

By a density argument, we can assume that $f\in C^{1}$. We will use the
linearized result (i.e., Corollary LABEL:appli-2) and the Kakutani-Fan-
Glicksberg fixed point theorem (see, e.g., [22, Theorem 1.14]) to prove
Theorem LABEL:null-3.

To this end, we first define

$a(r)\triangleq\left\\{\begin{array}[]{lll}\displaystyle{\frac{f(r)}{r}}&\mbox{if}&r\not=0,\\\
f^{\prime}(0)&\mbox{if}&r=0.\end{array}\right.$

By $(H_{3})$, we have that

${null-4}|a(r)|\leqslant L\;\;\mbox{for all}\;\;r\in\mathbb{R}.$ (3.1)

For each $n\geqslant 1$, we set

$\mathcal{K}_{n}\triangleq\\{\xi\in
L^{2}(0,T;L^{2}(\Omega_{n}))\bigm{|}\|\xi\|_{L^{2}(0,T;H_{0}^{1}(\Omega_{n}))}+\|\xi\|_{W^{1,2}(0,T;H^{-1}(\Omega_{n}))}\leqslant
c_{0}\\},$

where $c_{0}>0$ will be determined later. For each $\xi\in\mathcal{K}_{n}$, we
consider the following linear equation:

${null-5}\left\\{\begin{array}[]{lll}\partial_{t}z-\Delta
z+a(\xi(x,t))z=\chi_{\omega\bigcap\Omega_{n}}\chi_{E}u&\mbox{in}&\Omega_{n}\times(0,T),\\\
z=0&\mbox{on}&\partial\Omega_{n}\times(0,T),\\\
z(0)=z_{0}&\mbox{in}&\Omega_{n}.\end{array}\right.$ (3.2)

We simply write $z(\cdot)$ for the solution of (LABEL:null-5). According to
Corollary LABEL:appli-2, there is a positive constant
$\kappa\triangleq\kappa(r_{1},r_{2},E,T,L)$ (independent of $z_{0}$, $n$ and
$\xi$) and a control $u$ so that

${null-6}\|u\|_{L^{2}(0,T;L^{2}(\mathbb{R}^{N}))}\leqslant\kappa\|z_{0}\|_{L^{2}(\mathbb{R}^{N})}\;\;\mbox{and}\;\;z(T)=0.$
(3.3)

For each $n\geqslant 1$, we next define a set-valued mapping

$\Phi_{n}:\mathcal{K}_{n}\rightarrow 2^{L^{2}(0,T;L^{2}(\Omega_{n}))}$

by setting

$\displaystyle\Phi_{n}(\xi)\triangleq\Big{\\{}z\bigm{|}\mbox{there exists a
control}\;\;u\in L^{2}(0,T;L^{2}(\mathbb{R}^{N}))$ $\displaystyle\mbox{so that
(\ref{null-5}) and (\ref{null-6}) hold}\Big{\\}},\;\;\xi\in\mathcal{K}_{n}.$

One can easily check that $\Phi_{n}(\xi)\not=\emptyset$ for each
$\xi\in\mathcal{K}_{n}$.

The rest of the proof will be organized by several steps as follows.

Step 1. We show that $\mathcal{K}_{n}$ is compact and convex in
$L^{2}(0,T;L^{2}(\Omega_{n}))$, and that each $\Phi_{n}(\xi)$ is convex in
$L^{2}(0,T;L^{2}(\Omega_{n}))$.

These can be directly checked.

Step 2. We claim that $\Phi_{n}(\mathcal{K}_{n})\subset\mathcal{K}_{n}$.

Given $\xi\in\mathcal{K}_{n}$, there is a control $u$ satisfying
(LABEL:null-5) and (LABEL:null-6). By a standard energy estimate method and by
(LABEL:null-4)-(LABEL:null-6), we can easily check that

$\|z\|_{L^{2}(0,T;H_{0}^{1}(\Omega_{n}))}+\|z\|_{W^{1,2}(0,T;H^{-1}(\Omega_{n}))}\leqslant
c_{1}\|z_{0}\|_{L^{2}(\Omega_{n})},$

for a positive constant $c_{1}$ (independent of $z_{0}$, $n$ and $\xi$).
Hence,

$z\in\mathcal{K}_{n}\;\;\mbox{if}\;\;c_{0}=c_{1}\|z_{0}\|_{L^{2}(\Omega_{n})}.$

Step 3. We show that Graph($\Phi_{n}$) is closed.

It suffices to show that $z\in\Phi_{n}(\xi)$, provided that

$\xi_{\ell}\in\mathcal{K}_{n}\rightarrow\xi\;\;\mbox{ strongly
in}\;\;L^{2}(0,T;L^{2}(\Omega_{n}))$

and

$z_{\ell}\in\Phi_{n}(\xi_{\ell})\rightarrow z\;\;\mbox{ strongly
in}\;\;L^{2}(0,T;L^{2}(\Omega_{n})).$

To this end, we first observe that $\xi\in\mathcal{K}_{n}$, since
$\mathcal{K}_{n}$ is convex and closed. Next we claim that there exists a
subsequence of $\\{\ell\\}_{\ell\geqslant 1}$, denoted in the same manner, so
that

${null-7}a(\xi_{\ell})z_{\ell}\rightarrow a(\xi)z\;\;\mbox{strongly
in}\;\;L^{2}(0,T;L^{2}(\Omega_{n})).$ (3.4)

Indeed, since

$\xi_{\ell}\rightarrow\xi\;\;\mbox{strongly
in}\;\;L^{2}(0,T;L^{2}(\Omega_{n})),$

we have a subsequence of $\\{\ell\\}_{\ell\geqslant 1}$, still denoted by
itself, so that

$\xi_{\ell}(x,t)\rightarrow\xi(x,t)\;\;\mbox{for
a.e.}\;\;(x,t)\in\Omega_{n}\times(0,T).$

Then, by the definition of the function $a$, we conclude that

$a(\xi_{\ell}(x,t))\rightarrow a(\xi(x,t))\;\;\mbox{for
a.e.}\;\;(x,t)\in\Omega_{n}\times(0,T).$

By this and (LABEL:null-4), we can apply the Lebesgue Dominated Convergence
Theorem to obtain that

$\displaystyle\|a(\xi_{\ell})z_{\ell}-a(\xi)z\|_{L^{2}(0,T;L^{2}(\Omega_{n}))}^{2}$
$\displaystyle\leqslant$ $\displaystyle
2\|a(\xi_{\ell})(z_{\ell}-z)\|_{L^{2}(0,T;L^{2}(\Omega_{n}))}^{2}+2\|(a(\xi_{\ell})-a(\xi))z\|_{L^{2}(0,T;L^{2}(\Omega_{n}))}^{2}$
$\displaystyle\leqslant$ $\displaystyle
2L^{2}\|z_{\ell}-z\|_{L^{2}(0,T;L^{2}(\Omega_{n}))}^{2}+2\|(a(\xi_{\ell})-a(\xi))z\|_{L^{2}(0,T;L^{2}(\Omega_{n}))}^{2}$
$\displaystyle\rightarrow$ $\displaystyle 0.$

This leads to (LABEL:null-7).

Finally, for each $\ell\geqslant 1$, since
$z_{\ell}\in\Phi_{n}(\xi_{\ell})\subset\mathcal{K}_{n}$, there is $u_{\ell}\in
L^{2}(0,T;L^{2}(\mathbb{R}^{N}))$ with

${null-8}\|u_{\ell}\|_{L^{2}(0,T;L^{2}(\mathbb{R}^{N}))}\leqslant\kappa\|z_{0}\|_{L^{2}(\mathbb{R}^{N})},$
(3.5)

so that

${null-9}\left\\{\begin{array}[]{lll}\partial_{t}z_{\ell}-\Delta
z_{\ell}+a(\xi_{\ell}(x,t))z_{\ell}=\chi_{\omega\bigcap\Omega_{n}}\chi_{E}u_{\ell}&\mbox{in}&\Omega_{n}\times(0,T),\\\
z_{\ell}=0&\mbox{on}&\partial\Omega_{n}\times(0,T),\\\
z_{\ell}(0)=z_{0}&\mbox{in}&\Omega_{n},\\\
z_{\ell}(T)=0&\mbox{in}&\Omega_{n}\end{array}\right.$ (3.6)

and

${null-10}\|z_{\ell}\|_{L^{2}(0,T;H_{0}^{1}(\Omega_{n}))}+\|z_{\ell}\|_{W^{1,2}(0,T;H^{-1}(\Omega_{n}))}\leqslant
c_{0}.$ (3.7)

According to (LABEL:null-8) and (LABEL:null-10), there is a control $u$ and a
subsequence of $\\{\ell\\}_{\ell\geqslant 1}$, denoted in the same manner, so
that

${null-11}u_{\ell}\rightarrow u\;\;\mbox{weakly
in}\;\;L^{2}(0,T;L^{2}(\mathbb{R}^{N})),$ (3.8) ${null-12}z_{\ell}\rightarrow
z\;\;\mbox{weakly in}\;\;L^{2}(0,T;H_{0}^{1}(\Omega_{n}))\bigcap
W^{1,2}(0,T;H^{-1}(\Omega_{n})),\\\ $ (3.9)

and

${null-13}z_{\ell}(T)\rightarrow z(T)\;\;\mbox{strongly
in}\;\;L^{2}(\Omega_{n}).$ (3.10)

Passing to the limit for $\ell\rightarrow+\infty$ in (LABEL:null-8) and
(LABEL:null-9), making use of (LABEL:null-7) and
(LABEL:null-11)-(LABEL:null-13), we obtain that $z\in\Phi_{n}(\xi)$.

Step 4. We apply the Kakutani-Fan-Glicksberg Theorem to end the proof.

From the conclusions in the above three steps, we find that the map $\Phi_{n}$
satisfies conditions of the Kakutani-Fan-Glicksberg Theorem. Thus we can apply
this theorem to conclude that there exists $z\in\mathcal{K}_{n}$ so that
$z\in\Phi_{n}(z)$. Then, by the definition of $\Phi_{n}$ and using the fact
that

$a(z(x,t))z(x,t)=f(z(x,t)),$

one can finish the proof of Theorem LABEL:null-3. ∎

##  4 Proof of Theorem LABEL:null-2

b5 For the sake of completeness111Although we believe that the existence and
uniqueness of weak solutions for nonlinear heat equations in $\mathbb{R}^{N}$
have been well established in the literature, we did not yet find the exact
reference with precise proofs., we first consider the well-posedness of the
following non-homogeneous semilinear heat equation in $\mathbb{R}^{N}$:

${null-1-16}\left\\{\begin{array}[]{lll}\partial_{t}y-\Delta
y+f(y)=g&\mbox{in}&\mathbb{R}^{N}\times(0,T),\\\
y(0)=y_{0}&\mbox{in}&\mathbb{R}^{N},\end{array}\right.$ (4.1)

where $y_{0}\in L^{2}(\mathbb{R}^{N})$, $g\in
L^{2}(\mathbb{R}^{N}\times(0,T))$, and the nonlinearity $f$ satisfies the
assumption $(H_{3})$.

######  Definition 4.1.

weak solution We say that $y$ is a weak solution of (LABEL:null-1-16) if

($i$)

$y\in L^{2}(0,T;H^{1}(\mathbb{R}^{N}))\bigcap
W^{1,2}(0,T;H^{-1}(\mathbb{R}^{N}))\subset C([0,T];L^{2}(\mathbb{R}^{N}))$ and
$y(0)=y_{0}$;

($ii$)

For each $\psi\in H^{1}(\mathbb{R}^{N})$, the following equality holds:

$\langle\partial_{t}y(t),\psi\rangle_{H^{-1}(\mathbb{R}^{N}),H^{1}(\mathbb{R}^{N})}+\langle\nabla
y(t),\nabla\psi\rangle_{L^{2}(\mathbb{R}^{N})}+\langle
f(y(t)),\psi\rangle_{L^{2}(\mathbb{R}^{N})}=\langle
g(t),\psi\rangle_{L^{2}(\mathbb{R}^{N})}$

for a.e. $t\in(0,T)$.

Before presenting the proof of Theorem LABEL:null-2, we first state the well-
posedness of (LABEL:null-1-16) in the sense of the above definition.

######  Lemma 4.2.

Existence Under the assumption $(H_{3})$, for each $y_{0}\in
L^{2}(\mathbb{R}^{N})$ and each $g\in L^{2}(0,T;L^{2}(\mathbb{R}^{N}))$, the
equation (LABEL:null-1-16) has a unique weak solution.

######  Remark 4.3.

The argument below is also effective in our proof of Theorem LABEL:null-2.
Indeed, it is inspired by the approach in [20, Section 4.4] constructing weak
solutions for the Navier-Stokes equations on the whole space. The main idea
there is to use solutions of the Navier-Stokes equations on a sequence of
expanding bounded domains as a sequence of approximate solutions on the whole
space and show the convergence of such solutions.

###### Proof.

For each $n\geqslant 1$, we shall consider the equation

${null-14}\left\\{\begin{array}[]{lll}\partial_{t}y_{n}-\Delta
y_{n}+f(y_{n})=g&\mbox{in}&\Omega_{n}\times(0,T),\\\
y_{n}=0&\mbox{on}&\partial\Omega_{n}\times(0,T),\\\
y_{n}(0)=y_{0}&\mbox{in}&\Omega_{n},\end{array}\right.$ (4.2)

where $\Omega_{n}$ is constructed as in $(H_{2})$. By $(H_{3})$,
(LABEL:null-14), and a standard energy estimate method, we can easily check
that

${null-17}\|y_{n}\|_{L^{2}(0,T;H_{0}^{1}(\Omega_{n}))}+\|y_{n}\|_{W^{1,2}(0,T;H^{-1}(\Omega_{n}))}\leqslant
C.$ (4.3)

Here and throughout the proof of this lemma, $C$ denotes a positive constant
independent of $n$.

We extend $y_{n}$ to $\mathbb{R}^{N}\times(0,T)$ by zero and still denote this
extension by $y_{n}$. On one hand, by (LABEL:null-17), there is a subsequence
of $\\{n\\}_{n\geqslant 1}$, still denoted by itself, and $y^{*}\in
L^{2}(0,T;H^{1}(\mathbb{R}^{N}))$, so that

${null-18}y_{n}\rightarrow y^{*}\;\;\mbox{weakly
in}\;\;L^{2}(0,T;H^{1}(\mathbb{R}^{N})).$ (4.4)

On the other hand, for each $M>0$, there exists a positive integer $n_{0}(M)$
so that

$H^{-1}(\Omega_{n})\subset
H^{-1}(B_{M}(0))\;\;\mbox{and}\;\;\|\cdot\|_{H^{-1}(B_{M}(0))}\leqslant\|\cdot\|_{H^{-1}(\Omega_{n})}\;\;\mbox{for
all}\;\;n\geqslant n_{0}(M).$

These, along with (LABEL:null-17) and (LABEL:null-18), imply that $y^{*}\in
W^{1,2}(0,T;H^{-1}(B_{M}(0)))$ and there is a subsequence of
$\\{n\\}_{n\geqslant 1}$, denoted in the same manner, so that

${null-20}y_{n}\rightarrow y^{*}\;\;\mbox{weakly
in}\;\;L^{2}(0,T;H^{1}(\mathbb{R}^{N}))\bigcap W^{1,2}(0,T;H^{-1}(B_{M}(0))),$
(4.5)

and

${null-20-1}y_{n}\rightarrow y^{*}\;\;\mbox{strongly
in}\;\;L^{2}(0,T;L^{2}(B_{M}(0))).$ (4.6)

It follows from $(H_{3})$ and (LABEL:null-20-1) that

${null-21}f(y_{n})\rightarrow f(y^{*})\;\;\mbox{strongly
in}\;\;L^{2}(0,T;L^{2}(B_{M}(0))).$ (4.7)

Arbitrarily fix $\varphi\in C_{0}^{\infty}(\mathbb{R}^{N}\times(0,T))$. Let
$M>0$ be large enough so that the support of $\varphi$ is contained in
$B_{M}(0)\times(0,T)$. Then for all $n\geqslant n_{0}(M)$, we have that

${null-22}\begin{array}[]{lll}&&-\displaystyle{\int_{0}^{T}}\langle
y_{n},\partial_{t}\varphi\rangle_{L^{2}(B_{M}(0))}\mathrm{d}t+\displaystyle{\int_{0}^{T}}\langle\nabla
y_{n},\nabla\varphi\rangle_{L^{2}(B_{M}(0))}\mathrm{d}t+\displaystyle{\int_{0}^{T}}\langle
f(y_{n}),\varphi\rangle_{L^{2}(B_{M}(0))}\mathrm{d}t\\\
&=&\displaystyle{\int_{0}^{T}}\langle
g,\varphi\rangle_{L^{2}(B_{M}(0))}\mathrm{d}t.\end{array}$ (4.8)

Passing to the limit for $n\rightarrow\infty$ in (LABEL:null-22), by
(LABEL:null-20) and (LABEL:null-21), we obtain that

$\begin{array}[]{lll}&&-\displaystyle{\int_{0}^{T}}\int_{\mathbb{R}^{N}}y^{*}\partial_{t}\varphi\;\mathrm{d}x\mathrm{d}t+\displaystyle{\int_{0}^{T}}\int_{\mathbb{R}^{N}}\nabla
y^{*}\cdot\nabla\varphi\;\mathrm{d}x\mathrm{d}t+\displaystyle{\int_{0}^{T}}\int_{\mathbb{R}^{N}}f(y^{*})\varphi\;\mathrm{d}x\mathrm{d}t\\\
&=&\displaystyle{\int_{0}^{T}}\int_{\mathbb{R}^{N}}g\varphi\;\mathrm{d}x\mathrm{d}t,\end{array}$

which indicates that

${null-23}\partial_{t}y^{*}=\Delta y^{*}-f(y^{*})+g\;\;\mbox{in the sense of
distribution}.$ (4.9)

Since $\Delta y^{*}-f(y^{*})+g\in L^{2}(0,T;H^{-1}(\mathbb{R}^{N}))$, it
follows from (LABEL:null-23) that $\partial_{t}y^{*}\in
L^{2}(0,T;H^{-1}(\mathbb{R}^{N}))$. Hence,

${null-23-1}y^{*}\in L^{2}(0,T;H^{1}(\mathbb{R}^{N}))\bigcap
W^{1,2}(0,T;H^{-1}(\mathbb{R}^{N}))\subset C([0,T];L^{2}(\mathbb{R}^{N})).$
(4.10)

This, along with (LABEL:null-23) and a density argument, implies that for each
$\psi\in H^{1}(\mathbb{R}^{N})$,

${null-23-2}\langle\partial_{t}y^{*}(t),\psi\rangle_{H^{-1}(\mathbb{R}^{N}),H^{1}(\mathbb{R}^{N})}+\langle\nabla
y^{*}(t),\nabla\psi\rangle_{L^{2}(\mathbb{R}^{N})}+\langle
f(y^{*}(t)),\psi\rangle_{L^{2}(\mathbb{R}^{N})}=\langle
g(t),\psi\rangle_{L^{2}(\mathbb{R}^{N})}$ (4.11)

for a.e. $t\in(0,T)$.

We next show that

${null-23-3}y^{*}(0)=y_{0}.$ (4.12)

To this end, we arbitrarily fix $\psi\in C_{0}^{\infty}(\mathbb{R}^{N})$. Let
$M>0$ be large enough so that the support of $\psi$ is contained in
$B_{M}(0)$. Set $\varphi(x,t)\triangleq(T-t)\psi(x)/T$. Then for all
$n\geqslant n_{0}(M)$, multiplying both sides of (LABEL:null-14) by $\varphi$
and integrating it over $\mathbb{R}^{N}\times(0,T)$, we obtain that

${null-24}\begin{array}[]{lll}&&-\displaystyle{\int_{0}^{T}}\langle
y_{n},\partial_{t}\varphi\rangle_{L^{2}(B_{M}(0))}\mathrm{d}t+\displaystyle{\int_{0}^{T}}\langle\nabla
y_{n},\nabla\varphi\rangle_{L^{2}(B_{M}(0))}\mathrm{d}t+\displaystyle{\int_{0}^{T}}\langle
f(y_{n}),\varphi\rangle_{L^{2}(B_{M}(0))}\mathrm{d}t\\\
&=&\displaystyle{\int_{0}^{T}}\langle
g,\varphi\rangle_{L^{2}(B_{M}(0))}\mathrm{d}t+\langle
y_{0},\psi\rangle_{L^{2}(B_{M}(0))}.\end{array}$ (4.13)

Passing to the limit for $n\rightarrow\infty$ in (LABEL:null-24), by
(LABEL:null-20) and (LABEL:null-21), we obtain that

${null-25}\begin{array}[]{lll}&&-\displaystyle{\int_{0}^{T}}\langle
y^{*},\partial_{t}\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t+\displaystyle{\int_{0}^{T}}\langle\nabla
y^{*},\nabla\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t+\displaystyle{\int_{0}^{T}}\langle
f(y^{*}),\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t\\\
&=&\displaystyle{\int_{0}^{T}}\langle
g,\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t+\langle
y_{0},\psi\rangle_{L^{2}(\mathbb{R}^{N})}.\end{array}$ (4.14)

Moreover, multiplying both sides of (LABEL:null-23-2) by $(T-t)/T$ and
integrating it over $(0,T)$, we have that

${null-26}\begin{array}[]{lll}&&-\displaystyle{\int_{0}^{T}}\langle
y^{*},\partial_{t}\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t+\displaystyle{\int_{0}^{T}}\langle\nabla
y^{*},\nabla\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t+\displaystyle{\int_{0}^{T}}\langle
f(y^{*}),\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t\\\
&=&\displaystyle{\int_{0}^{T}}\langle
g,\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t+\langle
y^{*}(0),\psi\rangle_{L^{2}(\mathbb{R}^{N})}.\end{array}$ (4.15)

It follows from (LABEL:null-25) and (LABEL:null-26) that

$\langle y^{*}(0)-y_{0},\psi\rangle_{L^{2}(\mathbb{R}^{N})}=0\;\;\mbox{for
each}\;\;\psi\in C_{0}^{\infty}(\mathbb{R}^{N}),$

which indicates (LABEL:null-23-3).

By (LABEL:null-23-1)–(LABEL:null-23-3) and Definition LABEL:weak_solution, we
see that $y^{*}$ is a weak solution of (LABEL:null-1-16). Finally, we show the
uniqueness of the weak solution. WLOG, we assume that $\widetilde{y}$ is also
a weak solution of (LABEL:null-1-16). According to Definition
LABEL:weak_solution, it holds that

${null-
wang-20}\begin{array}[]{l}\langle\partial_{t}(y^{*}-\widetilde{y})(t),(y^{*}-\widetilde{y})(t)\rangle_{H^{-1}(\mathbb{R}^{N}),H^{1}(\mathbb{R}^{N})}\\\
+\langle\nabla(y^{*}-\widetilde{y})(t),\nabla(y^{*}-\widetilde{y})(t)\rangle_{L^{2}(\mathbb{R}^{N})}+\langle
f(y^{*}(t))-f(\widetilde{y}(t)),(y^{*}-\widetilde{y})(t)\rangle_{L^{2}(\mathbb{R}^{N})}=0\end{array}$
(4.16)

for a.e. $t\in(0,T)$, and $y^{*}(0)=\widetilde{y}(0)=y_{0}$. Integrating
(LABEL:null-wang-20) over $(0,t)$, we obtain that

$\|(y^{*}-\widetilde{y})(t)\|_{L^{2}(\mathbb{R}^{N})}^{2}\leqslant
2L\int_{0}^{t}\|(y^{*}-\widetilde{y})(s)\|_{L^{2}(\mathbb{R}^{N})}^{2}\mathrm{d}s.$

By Gronwall’s inequality, we obtain from the latter inequality that
$y^{*}=\widetilde{y}$.

In summary, we finish the proof of Lemma LABEL:Existence. ∎

Now, we are able to present the proof of Theorem LABEL:null-2.

Proof of Theorem LABEL:null-2. For each $n\geqslant 1$, according to Theorem
LABEL:null-3, there is a control $u_{n}\in L^{2}(0,T;L^{2}(\mathbb{R}^{N}))$
so that

${null-27}\left\\{\begin{array}[]{lll}\partial_{t}y_{n}-\Delta
y_{n}+f(y_{n})=\chi_{\omega}\chi_{E}u_{n}&\mbox{in}&\Omega_{n}\times(0,T),\\\
y_{n}=0&\mbox{on}&\partial\Omega_{n}\times(0,T),\\\
y_{n}(0)=y_{0}&\mbox{in}&\Omega_{n},\end{array}\right.$ (4.17)
${null-28}\|u_{n}\|_{L^{2}(0,T;L^{2}(\mathbb{R}^{N}))}\leqslant\kappa\|y_{0}\|_{L^{2}(\mathbb{R}^{N})}$
(4.18)

and

${null-29}y_{n}(T)=0\;\;\mbox{in}\;\;\Omega_{n}.$ (4.19)

By a standard energy estimate method, $(H_{3})$, (LABEL:null-27) and
(LABEL:null-28), we can easily check that

${null-30}\|y_{n}\|_{L^{2}(0,T;H_{0}^{1}(\Omega_{n}))}+\|y_{n}\|_{W^{1,2}(0,T;H^{-1}(\Omega_{n}))}\leqslant
C,$ (4.20)

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

We extend $y_{n}$ to $\mathbb{R}^{N}\times(0,T)$ by $0$ and still denote this
extension by $y_{n}$. By (LABEL:null-30) and (LABEL:null-28), there is a
subsequence of $\\{n\\}_{n\geqslant 1}$, still denoted by itself, and
$(u^{*},y^{*})\in L^{2}(0,T;L^{2}(\mathbb{R}^{N}))\times
L^{2}(0,T;H^{1}(\mathbb{R}^{N}))$, so that

${null-31}y_{n}\rightarrow y^{*}\;\;\mbox{weakly
in}\;\;L^{2}(0,T;H^{1}(\mathbb{R}^{N}))$ (4.21)

and

${null-32}u_{n}\rightarrow u^{*}\;\;\mbox{weakly
in}\;\;L^{2}(0,T;L^{2}(\mathbb{R}^{N})).$ (4.22)

On one hand, by similar arguments as those in Lemma LABEL:Existence, we
observe that $y^{*}$ is the unique weak solution of (LABEL:null-wang) (where
$u$ is replaced by $u^{*}$), and for each $M>0$, there is a subsequence of
$\\{n\\}_{n\geqslant 1}$, denoted in the same manner, so that

${null-33}f(y_{n})\rightarrow f(y^{*})\;\;\mbox{strongly
in}\;\;L^{2}(0,T;L^{2}(B_{M}(0))).$ (4.23)

On the other hand, we arbitrarily fix $\psi\in
C_{0}^{\infty}(\mathbb{R}^{N})$. Let $M>0$ be large enough so that the support
of $\psi$ is contained in $B_{M}(0)$. Set $\varphi(x,t)\triangleq t\psi(x)/T$.
Then for all $n\geqslant n_{0}(M)$ (where $n_{0}(M)$ is the same integer as
that in Lemma LABEL:Existence), multiplying both sides of (LABEL:null-27) by
$\varphi$ and integrating it over $\mathbb{R}^{N}\times(0,T)$, we obtain that

${null-34}\begin{array}[]{lll}&&-\displaystyle{\int_{0}^{T}}\langle
y_{n},\partial_{t}\varphi\rangle_{L^{2}(B_{M}(0))}\mathrm{d}t+\displaystyle{\int_{0}^{T}}\langle\nabla
y_{n},\nabla\varphi\rangle_{L^{2}(B_{M}(0))}\mathrm{d}t+\displaystyle{\int_{0}^{T}}\langle
f(y_{n}),\varphi\rangle_{L^{2}(B_{M}(0))}\mathrm{d}t\\\
&=&\displaystyle{\int_{0}^{T}}\langle\chi_{\omega}\chi_{E}u_{n},\varphi\rangle_{L^{2}(B_{M}(0))}\mathrm{d}t-\langle
y_{n}(T),\psi\rangle_{L^{2}(B_{M}(0))}.\end{array}$ (4.24)

Passing to the limit for $n\rightarrow\infty$ in (LABEL:null-34), by
(LABEL:null-29) and (LABEL:null-31)-(LABEL:null-33), we obtain that

${null-35}\begin{array}[]{lll}&&-\displaystyle{\int_{0}^{T}}\langle
y^{*},\partial_{t}\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t+\displaystyle{\int_{0}^{T}}\langle\nabla
y^{*},\nabla\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t+\displaystyle{\int_{0}^{T}}\langle
f(y^{*}),\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t\\\
&=&\displaystyle{\int_{0}^{T}}\langle\chi_{\omega}\chi_{E}u^{*},\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t.\end{array}$
(4.25)

Moreover, since $y^{*}$ is the weak solution of (LABEL:null-wang) (where $u$
is replaced by $u^{*}$), we have that

${null-36}\begin{array}[]{lll}&&-\displaystyle{\int_{0}^{T}}\langle
y^{*},\partial_{t}\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t+\displaystyle{\int_{0}^{T}}\langle\nabla
y^{*},\nabla\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t+\displaystyle{\int_{0}^{T}}\langle
f(y^{*}),\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t\\\
&=&\displaystyle{\int_{0}^{T}}\langle\chi_{\omega}\chi_{E}u^{*},\varphi\rangle_{L^{2}(\mathbb{R}^{N})}\mathrm{d}t-\langle
y^{*}(T),\psi\rangle_{L^{2}(\mathbb{R}^{N})}.\end{array}$ (4.26)

It follows from (LABEL:null-35) and (LABEL:null-36) that $\langle
y^{*}(T),\psi\rangle_{L^{2}(\mathbb{R}^{N})}=0$ for each $\psi\in
C_{0}^{\infty}(\mathbb{R}^{N})$. This implies that $y^{*}(T)=0$.

Hence, we finish the proof of Theorem LABEL:null-2. ∎

Acknowledgments. This work was partially supported by the National Natural
Science Foundation of China under grants 11771344 and 11971363. The second
author is also partially supported by the Academic Team Building Plan for
Young Scholars from Wuhan University under grant 413100085.

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# Deep Clustering for General-Purpose Audio Representations

###### Abstract

We introduce DECAR, a self-supervised pre-training approach for learning
general-purpose audio representations. Our system is based on clustering: it
utilizes an offline clustering step to provide target labels that act as
pseudo-labels for solving a prediction task. We develop on top of recent
advances in self-supervised learning for computer vision and design a
lightweight, easy-to-use self-supervised pre-training scheme. We pre-train
DECAR embeddings on a balanced subset of the large-scale Audioset dataset and
transfer those representations to 9 downstream classification tasks, including
speech, music, animal sounds, and acoustic scenes. Furthermore, we conduct
ablation studies identifying key design choices and also make all our code and
pre-trained models publicly available 111https://github.com/Speech-Lab-
IITM/DECAR.

Index Terms—  Audio, Speech, Self-Supervision, Clustering

##  1 Introduction

Self-supervised representation learning aims at obtaining features without
using manual annotations, and is rapidly closing the performance gap with
supervised pre-training in a wide range of tasks across Computer Vision (CV)
[1, 2], Natural Language Processing (NLP) [3], and Speech Processing,
especially Automatic Speech Recognition (ASR) [4, 5]. These systems leverage
self-supervised learning algorithms to learn representations by solving
various tasks on large, unlabeled datasets, and the weights learned are then
either transferred for solving other downstream tasks or used as a feature
extractor for input into a system, for learning a different set of weights for
another task.

The goal of speech and audio representation learning is to learn a
transformation from the acoustic signal that makes high-level information more
accessible to downstream tasks. In the acoustics domain, the recent success of
unsupervised speech representation learning has shown to outperform low-level
features like mel-spectrograms, waveform, or filter-banks, in various Spoken
Language Processing (SLP) downstream tasks, the primary reason being that the
former tends to learn more detailed information from speech. Despite recent
progress, most work on self-supervised learning for speech and audio
processing focus mostly on ASR and ignores other speech and audio tasks such
as acoustic scene detection or animal vocalizations, except [6, 7, 8, 9].
Triplet-based objectives used in [6, 9], heavily rely on the mining of
negative samples, and the quality of learned features varies significantly
with the sample generation scheme [7]. The methodology used in [8] learns
general-purpose audio representations by solving a contrastive task.
Contrastive learning systems typically work online and rely on a large number
of explicit pairwise feature comparisons, which is computationally
challenging, given the fact they require large batches for mining negative
samples.

In this paper, we introduce DECAR: DEep Clustering for General-Purpose Audio
Representations, a simple self-supervised pre-training framework to learn
general-purpose audio representations of sounds beyond and including speech.
Our system requires no prior knowledge, minimal additional steps, alleviates
dependency on large batches, and closely resembles a general supervised
training procedure. The DECAR framework employs an offline clustering step to
generate noisy labels from discrete audio samples and learns by predicting
cluster assignments. We show that it is possible to obtain useful general-
purpose audio features with a clustering framework.

Our DECAR pre-training approach consists of two forward passes of the entire
dataset for each epoch. At each epoch, first, we cluster the output of our
ConvNet feature extractor for all audio samples, and then use the subsequent
cluster assignments as “pseudo-labels” for an augmented versions of the same
audio samples to optimize the network. Formally, our approach iteratively
learns features and groups them.

We demonstrate the effectiveness of DECAR over a range of challenging and
diverse downstream tasks, including speech, music, acoustic scenes, and animal
sounds, and intent to highlight its performance on low resource downstream
tasks. After pre-training on a relatively smaller subset of the large-scale
AudioSet database [10], compared to prior work, we show that only a linear
classifier trained over DECAR embeddings is competitive in performance to
fully-supervised in-domain convolutional network’s performance and exceeds it
when fine-tuned end-to-end, with significant performance boosts in low-
resource downstream tasks. These experiments demonstrate that DECAR offers a
simple, easy-to-implement unsupervised audio pre-training methodology for
large-scale end-to-end training and learns audio representations that can be
generalized across a variety of audio tasks, including speech, songs, musical
instruments, etc.

##  2 Related Work

The past decade has seen impressive success in self-supervised learning in
vision, speech, and text, pushing boundaries in low resource settings and
improving performance in various downstream tasks. Methodologies to learn
representations from discrete or continuous input sequences, such as in Speech
or Natural Language Processing applications, use either masked modeling, by
solving different tasks like contrastive learning [4], reconstruction [11] and
class prediction [3, 5], or auto-regressive generation of input sequences by
encoding temporal information of past acoustic sequence [12].

Fig. 1: Pretraining and Finetuning of DECAR

In recent times, contrastive learning has shown great success in various
settings like masked modeling in speech [4], or in vision where the algorithm
enforces similarity between pairs of different augmentations for the same
image [2]. However, the bottleneck of requiring large batches to mine
distractors has led to the recent proposal of clustering-based methodologies,
which learns representations by predicting pseudo-labels assigned to instances
in the form of cluster assignments, both in speech, [5, 13] and vision domain
[1, 14]. While HuBERT [5] exploits clustering to solve a masked prediction
problem on speech frames in an End-to-End transformer framework, for self-
supervised _speech_ representation learning, [13] uses it for solving a
different task in a multi-task learning framework. To the best of our
knowledge, there is no prior work in this domain that exploits a clustering
framework for self-supervised audio representation learning.

##  3 Methodology

We learn general-purpose audio representations from unlabeled data by pre-
training a neural network with a clustering approach. Our approach, inspired
from the DeepCluster framework [1], is based on pre-training a convolutional
feature extractor on unlabeled audio data by predicting pseudo-labels which
are obtained by clustering vector embedding representations _e_ of audio
samples from a large-scale dataset and later combining our feature extractor
with an additional classification layer for solving various audio
understanding tasks across several downstream tasks.

As discussed earlier, our pre-training step makes two passes through our
entire dataset at each epoch. At the first forward pass for each epoch, for
each audio sample $x$, we extract the log-compressed mel-filterbanks and pass
it through our ConvNet feature extractor $f(.)$ to obtain a single embedding
$f(x)=h_{x}\in\mathbb{R}^{n}$. Then we pass these embeddings through a
feedforward layer $g(.)$, to obtain $g(h_{x})=z_{x}\in\mathbb{R}^{k}$. Post
this step, the outputs $z_{x}$ are clustered using our clustering algorithm.
Before clustering, $z_{x}$ is PCA-reduced to 128 dimensions, whitened, and
l2-normalized. We experiment with both K-Means and Power Iteration Clustering
(PIC) [15] as choices for our clustering algorithm.

$\min_{C\in\mathbb{R}^{d\times
k}}\frac{1}{N}\sum_{x=1}^{N}\min_{y_{x}\in\\{0,1\\}^{k}}\left\|z_{x}-Cy_{x}\right\|_{2}^{2}\text{
}\ni\text{ }y_{n}^{\top}1_{k}=1$ (1)

At the second pass, we predict the cluster assignment for each audio sample
assigned during the first step with an augmented version of the same sample.
To achieve this, we add a _prediction-head_ which is a linear layer and learns
a parameterized function $p(.)$ where
$p(z_{x})=y^{\prime}_{x}\in\mathbb{R}^{c}$ and _c_ is the number of cluster
centroids. Finally, we pass $y^{\prime}_{x}$ through a softmax activation
function to get the probability distribution of each cluster class for each
audio sample and train all the parameters of the network by minimizing Cross-
Entropy loss over the real and predicted class distribution (2).

$\text{C.E.}=-\frac{1}{N}\sum_{x=1}^{N}\sum_{j=1}^{M}y_{x_{j}}\log\left(\operatorname{softmax}(y^{\prime}_{x})_{{}_{j}}\right)$
(2)

Formally, DECAR alternates between clustering the features to produce pseudo-
labels by solving (1), where it jointly learns a ${d\times k}$ centroid matrix
$C$ and cluster assignments $y_{x}$, and updating the parameters of the
ConvNet by predicting these pseudo-labels using (2). Additionally, during the
second step, while assigning the pseudo-labels to the audio samples post
clustering, we perform SpecAugment augmentation [16] on the extracted log-
compressed Mel-filterbanks before passing it on to the ConvNet for the
prediction task when training the network. We hypothesize that this improves
the robustness of our learned weights and also improves performance by
enforcing invariance to data augmentation.

We handle the problem of trivial parameterization during the cluster label
prediction step, where a vast majority of all the audio samples in a batch
might belong to the same cluster, and minimizing (2) might lead to a trivial
solution, by using a custom sampler, which samples batches of audios based on
a uniform distribution over all the pseudo-labels.

##  4 Datasets

We pre-train DECAR embeddings on a balanced subset of the diverse, large-scale
Audioset [10]. The complete dataset contains 2 million excerpts of 10 seconds
of audio from YouTube videos that are annotated in a multi-label fashion with
527 classes. Our subset consists of 10% of the total number of audio samples
from the original dataset sampled. Before our work, this dataset has also been
used [7, 8, 9] for self-supervised pre-training. Since our pre-training
strategy is self-supervised, we don’t utilize the labels in the train set.

We perform the downstream evaluation on a variety of tasks, including both
speech and non-speech tasks. For the selection of datasets for these tasks, we
are inspired by previous work in the self-supervised audio pre-training domain
[7, 8, 9], with some of our additions to take into account more low-resource
tasks. For speaker identification, we use two datasets, namely, LibriSpeech
[17] and Voxceleb [18]. For keyword spotting, we use Google Speech Commands
V1, and V2 datasets [19]. For acoustic scene classification, we use TUT Urban
Acoustic Scenes 2018 (TUT), consisting of labeled audio segments from 10
different acoustic scenes. For animal vocalizations, we use the Bird Song
Detection (BSD) dataset [20] to solve a binary classification problem 111The
original test labels are no more available, and thus we took a random split of
80:20 for train and test from the combined development sets. For Speech
Emotion Recognition (SER) we use IEMOCAP dataset [21]. For music recognition,
we use the NSynth dataset [22] of musical notes from different instruments.
Finally, for language identification, we use the Voxforge dataset categorize
audio clips of different classes of spoken language.

More details about the downstream datasets can be found in Table 1.

Table 1: Dataset statistics for downstream benchmark tasks. Dataset |  Target |  No of classes |  No of samples |  Avg duration (sec)
---|---|---|---|---
Libri Speech |  Speaker Identification |  585 |  28538 |  12.69
Voxceleb 1 |  Speaker Identification |  1211 |  153397 |  8.20
IEMOCAP |  Emotion recognition |  4 |  4490 |  4.49
TUT Urban 2018 (TUT) |  Acuostic scene classification |  10 |  8640 |  10.00
Speech Commands V1 |  Key word recognition |  12 |  64721 |  0.98
Speech Commands V2 |  Key word recognition |  12 |  105829 |  0.98
Bird Song Detection(BSD) |  Song detection |  2 |  15690 |  10.08
Voxforge |  Language Identification |  6 |  8803 |  6.68
N-synth |  Music Identification |  11 |  301883 |  4.00

##  5 Experimental Setup

Given an audio input sequence, we extract log-compressed Mel-filterbanks with
a window size of 25 ms, a hop size of 10 ms, and N = 64 Mel-spaced frequency
bins in the range 60–7800 Hz. Our ConvNet feature encoder $h$ is based on the
lightweight and highly scalable EfficientNet-B0. We apply global max-pooling
to the last layer of the encoder to get an embedding $h_{x}$ of size 1280.
During pre-training, we pass $h_{x}$ through a linear layer $g$, which
contains a fully connected layer with 512 units followed by the Relu
activation function and a prediction head $p$ (for the 2nd pass) with units
equal to the number of cluster centroids. We pre-train for a total of 100
epochs with a patience of 10 epochs with the criterion of no improvement in
the Normalized Mutual Information (NMI) between the cluster assignments of the
current epoch $t$ and the previous epoch $t-1$, which are denoted as $A$ and
$B$ in (3) respectively. We learn the parameters of our model with Stochastic
Gradient Descent (SGD) and a learning rate of 0.05 in batched-mode with a
batch size of 64.

$\operatorname{NMI}(A;B)=\frac{\mathrm{I}(A;B)}{\sqrt{\mathrm{H}(A)\mathrm{H}(B)}}$
(3)

Post-self-supervised pre-training, we discard the linear layer $g(.)$, and
prediction-head $p(.)$ and evaluate in downstream tasks in 2 different
settings similar to [8] where 1) We use DECAR embeddings as a feature
extractor and only train a linear classifier with the frozen embeddings 2) We
fine-tune the entire End-to-End network on the downstream task. Importantly,
we assess the performance on several diverse datasets to determine the
transferability of learned representations across audio domains and recording
conditions. We show our results in 6.

We train all our downstream tasks with a similar hyper-parameter setting,
where we use Adam optimizer, a learning rate of $10^{-3}$, batch size of 64,
and train it for a maximum of 50 epochs.

##  6 Results

Table 2 reports accuracy of DECAR embeddings over 9 downstream classification
tasks, with PIC as our clustering algorithm in self-supervised pre-training.
As discussed earlier, we report results both on frozen and fine-tuned DECAR
embeddings for each downstream task and repeat the same for randomly
initialized weights, which act as our baseline. As we can see, a linear
classifier trained over frozen DECAR embeddings outperforms drastically the
same setting when trained with randomly initialized embeddings. This makes it
evident that we learn powerful embeddings from our pre-training step. Next, we
also fine-tune our pre-trained DECAR embeddings on each downstream task and
show significant gains compared to training with randomly initialized weights.
We show a boost in performance in both speech and non-speech tasks showing
that our DECAR pre-training learns representations that generalize well across
a wide range of audio tasks. Our results show better improvements in low-
resource downstream settings like IEMOCAP, Voxceleb, and Acoustic Scenes with
the limited number of samples per class when compared to tasks like Speech
Commands and BSD. Additionally, we observe better improvements in speech tasks
where the prosodic features in speech aid the model to learn to distinguish
among different classes, like SER, versus the ones which require sequence
information like Speech Commands.

Our results are not directly comparable with the prior art in this domain due
to 2 main reasons:

  1. 1.

There is no standard benchmark for comparing general-purpose audio
representations and the presence of several nuances around dataset settings in
certain tasks.

  2. 2.

We pre-train DECAR in a low-resource setting compared to most other work in
self-supervised audio and speech representation learning. We have pre-trained
using only 0.2 million (10$\%$) data samples compared to 2 million used in
[8]. We acknowledge the fact that more data can further boost the performance
in all tasks, including high-resource downstream tasks.

Table 2: Test accuracy (%) on downstream tasks | Random Init. | DECAR
---|---|---
Task | Frozen | Fine-tuned | Frozen | Fine-tuned
Speaker Id. (LBS) | 4.2 | 94.6 | 62.5 | 97.0
Speaker Id. (Voxceleb) | 0.2 | 57.6 | 2.5 | 57.5
Speech Commands (V1) | 62.0 | 97.4 | 63.9 | 97.6
Speech Commands (V2) | 62.9 | 97.7 | 65.7 | 97.6
Acoustic Scenes | 32.6 | 61.6 | 51.4 | 66.9
Speech Emotion | 52.3 | 61.8 | 60.5 | 62.4
Birdsong Detection | 67.6 | 90.0 | 76.4 | 90.3
Language ID | 26.3 | 73.4 | 46.0 | 76.5
Music Instrument | 23.6 | 78.6 | 59.9 | 78.4

##  7 Ablation Study

Clustering being an optimal part of our self-supervised learning algorithm, we
wanted to study how variations in the clustering algorithm affected
performance on downstream tasks. For this, we compare the performance of our
downstream tasks across PIC and K-Means clustering algorithms. Table 3 shows
that though PIC dominates in most cases, there is a negligible difference in
performance when it comes to K-Means. This shows that our system is robust to
the clustering algorithm used. Additionally, we also study the effect of the
hyperparameter $k$, or the number of centroids, on our downstream performance
for K-Means. We notice no particular trend here and would want to address this
as part of our future work with the entire Audioset.

Table 3: Impact of cluster variations on test accuracy(%) | K-Means | PIC
---|---|---
Task | 256 | 512 | 1024 |
Speaker Id. (LBS) | 96.6 | 96.2 | 97.3 | 97.0
Speaker Id. (Voxceleb) | 53.4 | 49.7 | 57.0 | 57.5
Speech Commands (V1) | 97.3 | 97.0 | 97.4 | 97.6
Speech Commands (V2) | 97.4 | 97.3 | 97.4 | 97.6
Acoustic Scenes | 68.2 | 68.7 | 66.5 | 66.9
Speech Emotion | 62.1 | 64.0 | 63.1 | 62.4
Birdsong Detection | 89.6 | 89.5 | 90.1 | 90.3
Language ID | 75.4 | 75.0 | 73.9 | 76.5
Music Instrument | 77.3 | 77.5 | 78.1 | 78.4

##  8 Conclusion and Future Work

In this paper, we present DECAR, a simple and easy-to-implement self-
supervised audio representation learning algorithm that alleviates the need
for large batches or any additional steps and prior domain knowledge. Our
system shows significant performance boosts in low resource downstream
settings even when pre-trained on a fraction of the total amount of data used
in prior work. Future work would include mitigating the need for an additional
forward pass by learning cluster assignments online, thus allowing it to scale
to practically unlimited amounts of data, applying better augmentation schemes
during the cluster assignment prediction step, and also increase the diversity
of downstream tasks.

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