# Bilinear optimal control for weak solutions of the Keller-Segel logistic
model in $2D$ domains

P. Braz e Silva  Departamento de Matemática. Universidade Federal de
Pernambuco, CEP 50740-560, Recife - PE. Brazil<EMAIL_ADDRESS>,  F.
Guillén-González  Dpto. de Ecuaciones Diferenciales y Análisis Numérico and
IMUS, Universidad de Sevilla, Sevilla. Spain<EMAIL_ADDRESS>,  Cilon F.
Perusato  Departamento de Matemática. Universidade Federal de Pernambuco, CEP
50740-560, Recife - PE. Brazil<EMAIL_ADDRESS>and  M.A. Rodríguez-
Bellido  Dpto. de Ecuaciones Diferenciales y Análisis Numérico and IMUS,
Universidad de Sevilla, Sevilla. Spain<EMAIL_ADDRESS>
###### Abstract.

An optimal control problem associated to the Keller-Segel with logistic
reaction system will be studied in $2D$ domains. The control acts in a
bilinear form only in the chemical equation. The existence of optimal control
and a necessary optimality system are deduced. The main novelty is that
control can be rather singular and the state (cell density $u$ and the
chemical concentration $v$) remains only in a weak setting, which is not usual
in the literature to solve optimal control problems subject to chemotaxis
models (see e.g. [16]).

###### Key words and phrases:

Chemotaxis model, logistic reaction, weak solutions, bilinear optimal control,
optimality conditions

###### 2021 Mathematics Subject Classification:

35K51, 35Q92, 49J20, 49K20

P. Braz e Silva was partially supported by by CAPES–PRINT -
88881.311964/2018–01, CAPES-MATHAMSUD $\\#$88881.520205/2020-0, and CNPq,
Brazil, $\\#$308758/2018-8 and $\\#$432387/2018-8

F. Guillén-González and M.A. Rodríguez-Bellido acknowledges funding from Grant
PGC2018-098308-B-I00 (MCI/AEI/FEDER, UE), Grant US-1381261 (US/JUNTA/FEDER,
UE) and Grant P20-01120 (PAIDI/JUNTA/FEDER, UE)

C. Perusato was partially supported by CAPES–PRINT - 88881.311964/2018–01 and
Propesq-UFPE - 08-2019 (Qualis A). He is also grateful for the warm
hospitality and the wonderful academic atmosphere during his visit at the
Universidad de Sevilla, where this work was started.

##  1\. Introduction

###  1.1. The controlled model

In this work we study an optimal control problem for the (attractive or
repulsive) Keller-Segel model in a $2D$ domain $\Omega\subset\mathbb{R}^{2}$
with logistic source term and bilinear control acting on the chemical
equation:

(1.1) $\left\\{\begin{array}[]{rcll}\partial_{t}u-\Delta
u+\kappa\nabla\cdot(u\,\nabla v)&=&r\,u-\mu\,u^{2}&\mbox{in
$\Omega\times(0,T)$,}\\\ \partial_{t}v-\Delta
v+v&=&u+f\,v\,1_{\Omega_{c}}&\mbox{in $\Omega\times(0,T)$,}\\\
\partial_{\boldsymbol{n}}u=\partial_{\boldsymbol{n}}v&=&0&\mbox{on
$\partial\Omega\times(0,T)$,}\\\ u(0,\cdot)=u_{0}\geq 0,\quad
v(0,\cdot)&=&v_{0}\geq 0&\mbox{in $\Omega$.}\end{array}\right.$

Here, $f:Q_{c}:=(0,T)\times\Omega_{c}\to\mathbb{R}$ is the control with
$\Omega_{c}\subset\Omega\subset\mathbb{R}^{2}$ the control domain, and the
states $u,v:Q:=(0,T)\times\Omega\to\mathbb{R}_{+}^{2}$ are the cellular
density and chemical concentration, respectively. Moreover, $r\in\mathbb{R}$
and $\mu>0$ are coefficients of the logistic reaction, and
$\kappa\in\mathbb{R}$ is the chemotaxis coefficient ($\kappa>0$ models
attraction and $\kappa<0$ repulsion). We are interested in the study of a
control problem associated to the following weak solution concept of system
(1.1). Hereafter, $L^{2+}$ means $L^{2+\varepsilon}$ for $\varepsilon$ small
enough.

######  Definition 1.1.

Let $f\in L^{2+}(Q_{c}):=L^{2+}(0,T;L^{2+}(\Omega_{c}))$, $u_{0}\in
L^{2}(\Omega)$, $v_{0}\in W^{1+,2+}(\Omega)$ with $u_{0}\geq 0$ and $v_{0}\geq
0$ a.e. in $\Omega$. A pair $(u,v)$ is called a weak solution of problem (1.1)
in $(0,T)$, if

$\displaystyle u\geq 0,\quad v\geq 0\quad\hbox{a.e. in
$Q=(0,T)\times\Omega$,}$ $\displaystyle u\in W_{2}:=\\{u\in
C([0,T];L^{2}(\Omega))\cap L^{2}(0,T;H^{1}(\Omega)),\ \partial_{t}u\in
L^{2}(0,T;H^{1}(\Omega)^{\prime})\\},$ $\displaystyle v\in X_{2+}:=\\{v\in
C([0,T];W^{1+,2+}(\Omega))\cap L^{2+}(0,T;W^{2,2+}(\Omega)),\ \partial_{t}v\in
L^{2+}(Q)\\},$

the equation (1.1)1 jointly boundary condition for $u$ hold in a variational
sense, while equation (1.1)2 and the boundary condition for $v$ are satisfied
pointwisely, and initial conditions (1.1)3 and (1.1)4 are satisfied in
$L^{2}(\Omega)$ and $W^{1+,2+}(\Omega)$, respectively.

Notice that, since we are in $2D$ bounded domains, $v\in C([0,T];W^{1+,2+})$
implies $v\in L^{\infty}(0,T;L^{\infty})$, hence using that $f\in
L^{2+}(Q_{c})$ one has $fv\in L^{2+}(Q)$. That means that $v\in X_{2+}$ is the
maximal regularity which can be obtained. The previous weak regularity for
$u\in W_{2}$ will be enough to solve the following optimal control problem.

(1.2) $\left\\{\begin{array}[]{l}\mbox{Find }(u,v,f)\in W_{2}\times
X_{2+}\times\mathcal{F}\mbox{ minimizing the functional }\\\
\vspace{-1ex}\cr\\\
J(u,v,f):=\displaystyle\frac{\gamma_{u}}{2}\int_{0}^{T}\|u(t)-u_{d}(t)\|^{2}_{L^{2}{(\Omega)}}dt\\\
+\displaystyle\frac{\gamma_{v}}{2}\int_{0}^{T}\|v(t)-v_{d}(t)\|^{2}_{L^{2}{(\Omega)}}dt+\displaystyle\frac{\gamma_{f}}{{2+}}\int_{0}^{T}\|f(t)\|^{2+}_{L^{2+}(\Omega_{c})}dt\\\
\vspace{-1ex}\cr\\\ \mbox{ subject to $(u,v,f)$ be a weak solution of the PDE
system (\ref{KS}),}\end{array}\right.$

where $(u_{d},\,v_{d})\in L^{2}(Q)^{2}$ represent the target states and the
nonnegative numbers $\gamma_{u}$, $\gamma_{v}$ and $\gamma_{f}$ measure the
cost of the states and control, respectively. With respect to the control
constraint, we assume

$\mathcal{F}\subset L^{2+}(Q_{c})\quad\mbox{ be a nonempty, closed and convex
set.}$

The functional $J$ defined in (1.2) describes the deviation of the cell
density $u$ and the chemical concentration $v$ from a target cell density
$u_{d}$ and chemical concentration $v_{d}$, respectively; plus the cost of the
control $f$ measured in the $L^{2+}$-norm.

###  1.2. Previous results

Before continuing our discussion, let us give some motivation for the study of
(1.1). In the last decades, there has been a surge of activity on the study of
the chemotaxis model which describes the movement of the cells directed by the
concentration gradient of a chemical substance in their environment. Moreover,
it is important to consider the biological situation where the bacterial
population may proliferate according to a logistic law. An interesting feature
in chemotaxis corresponds to the movement of cells directed by the gradient of
the chemical signal which is produced by cells themselves. When one considers
also the interactions between cells and the chemical signal with liquid
environments, one gets the chemotaxis-fluid system, which is basically the
chemotaxis model coupled with the Navier-Stokes equations. For more details,
see the excellent review [4] and the references therein.

Let us recall some issues related to the uncontrolled equations (1.1), i.e.,
when $f\equiv 0$. Plenty of results have been obtained here. Amongst the many
articles related to the uncontrolled sysstem, let us mention those on
existence of weak and strong solutions in $\mathbb{R}^{2}$. In this case,
without considering logistic reaction (i.e. $r=\mu=0$), the existence of
global weak solutions was provided by Liu and Lorz [24]. In two-dimensional
bounded convex domains, the existence of (global) classical solutions was
obtained by M. Winkler [33]. In the presence of logistic source, the existence
of global weak solutions (and the long time behavior) has been analyzed in
[21] by J. Lankeit. In this case, the existence of global mild solutions was
examined in [12]. For 3D domains, we also refer [34] and the references
therein.

It is important to mention that remarkable progress has been made in
mathematical and numerical analysis of optimal control problems for viscous
flows described by the Navier-Stokes equations and other related models, see
e.g., [1, 7, 26]. However, the literature related to optimal control for
chemotaxis problems is still scarce. In [27], the authors study an optimal
(distributed) control problem where the state problem is given by a stationary
chemotaxis model coupled with the Navier-Stokes equations. We note that in [8,
9] the authors provide some results related to the controllability for the
nonstationary Keller-Segel system and the nonstationary chemotaxis-fluid model
with consumption of chemoattractant substance associated to a chemotaxis
system, based on Carleman-type estimates for the solutions of the adjoint
system. Recently, a bilinear optimal control problem associated to the
chemotaxis-Navier-Stokes model (without logistic source) in bounded three-
dimensional domains was examined in [25]. For the chemo-repulsion case, this
problem was studied in [16, 18] for $2D$ and $3D$ domains respectively, and in
[17] for $2D$ domains with a potential nonlinear production term, that is
changing the production term $u$ in the $v$ equation by $u^{p}$, with $p>1$.

###  1.3. Main contributions of the paper

First of all, we will prove the existence and uniqueness of weak solutions of
(1.1) and the continuous dependence of the weak solution $(u,v)$ respect the
control $f$.

######  Theorem 1.2.

Let $u_{0}\in L^{2}(\Omega)$, $v_{0}\in W^{1+,2+}(\Omega)$ with $u_{0}\geq 0$
and $v_{0}\geq 0$ in $\Omega$, and $f\in L^{2+}(Q_{c})$. There exists a unique
weak solution $(u,v)$ of system (1.1) in sense of Definition 1.1. Moreover,
there exists a positive constant

$\mathcal{K}_{1}:=\mathcal{K}_{1}(r,\mu,\kappa,|\Omega|,T,\|u_{0}\|_{L^{2}},\|v_{0}\|_{W^{1+,2+}},\|f\|_{L^{2+}(Q_{c})}),$

such that

(1.3) $\|(u,v)\|_{W_{2}\times X_{2+}}\leq\mathcal{K}_{1}.$

where we denote

$\|(u,v)\|_{W_{2}\times
X_{2+}}:=\|(\partial_{t}u,\partial_{t}v)\|_{L^{2}(H^{1})^{\prime}\times
L^{2+}(L^{2+})}+\|(u,v)\|_{C(L^{2}\times W^{1+,2+})}$
$+\|(u,v)\|_{L^{2}(H^{1})\times L^{2+}(0,T;W^{2,2+})}$

Finally, for any $r,\mu,\kappa,\Omega,T,u_{0},v_{0}$, the constant
$\mathcal{K}_{1}$ is bounded if $f$ is bounded in $L^{2+}(Q_{c})$.

The second main result of this paper will be the existence of a global optimal
solution of (1.2):

######  Theorem 1.3.

Let $(u_{0},v_{0})\in L^{2}(\Omega)\times W^{1+,2+}(\Omega)$ with $u_{0}\geq
0$ and $v_{0}\geq 0$ in $\Omega$. Assuming that either $\gamma_{f}>0$ or
$\mathcal{F}$ is bounded in $L^{2+}(Q_{c})$, then the bilinear optimal control
problem (1.2) has at least one global optimal solution
$(\tilde{u},\tilde{v},\tilde{f})$.

Finally, we obtain the existence and uniqueness of Lagrange multipliers
associated to the optimal control (1.2):

######  Theorem 1.4.

Let $\tilde{s}=(\tilde{u},\tilde{v},\tilde{f})\in\mathcal{S}_{ad}$ be a local
optimal solution for the control problem (1.2). Then, there exists a unique
Lagrange multiplier $(\lambda,\eta)\in X_{2}\times W_{2}$ satisfying the
optimality system

(1.4)
$\left\\{\begin{array}[]{rcl}-\partial_{t}\lambda-\Delta\lambda+\kappa\,\nabla\lambda\cdot\nabla\tilde{v}-\eta&=&\gamma_{u}(\tilde{u}-u_{d})\
\ \mbox{ in Q,}\\\
-\partial_{t}\eta-\Delta\eta-\kappa\,\nabla\cdot(\tilde{u}\nabla\lambda)+\eta-\tilde{f}\,\eta\,1_{\Omega_{c}}&=&\gamma_{v}(\tilde{v}-v_{d})\
\ \mbox{ in Q,}\\\ \lambda(T)=0,\ \eta(T)&=&0\quad\mbox{ in }\Omega,\\\
\dfrac{\partial\lambda}{\partial{\boldsymbol{n}}}=0,\
\dfrac{\partial\eta}{\partial{\boldsymbol{n}}}&=&0\quad\mbox{ on
}(0,T)\times\partial\Omega,\\\
\displaystyle\int_{0}^{T}\int_{\Omega_{c}}(\gamma_{f}{\rm
sgn}\tilde{f}|\tilde{f}|^{1+}+\tilde{v}\,\eta)(f-\tilde{f})&\geq&0\ \ \
\forall f\in\mathcal{F}.\end{array}\right.$

######  Remark 1.5.

If $\gamma_{f}>0$ and there is no convexity constraint on the control, that
is, $\mathcal{F}\equiv L^{2+}(Q_{c})$, then optimality condition (1.4)5
becomes

$\gamma_{f}{\rm
sgn}\tilde{f}|\tilde{f}|^{1+}1_{\Omega_{c}}+\tilde{v}\,\eta\,1_{\Omega_{c}}=0.$

The rest of the paper is organized as follows. In Section 2 some Preliminairy
results which will be used later are introduced. The proofs of Theorems 1.2,
1.3 and 1.4 are given in Sections 3, 4 and 5 respectively.

##  2\. Preliminary results

Along this manuscript the following result on $L^{p}$ regularity will be
considered.

######  Theorem 2.1 ([13], page 344).

For $\Omega\in C^{2}$, let $1<p<3$, $u_{0}\in{W}^{2-2/p,p}(\Omega)$ and $g\in
L^{p}(Q)$. Then the problem

$\left\\{\begin{array}[]{rcl}\partial_{t}u-\Delta u&=&g\quad\mbox{ in }Q,\\\
u(0,\cdot)&=&u_{0}\quad\mbox{ in }\Omega,\\\ \dfrac{\partial u}{\partial{\bf
n}}&=&0\quad\mbox{ on }(0,T)\times\partial\Omega,\end{array}\right.$

admits a unique solution $u$ such that

$u\in C([0,T];{W}^{2-2/p,p})\cap L^{p}(W^{2,p}),\quad\partial_{t}u\in
L^{p}(Q).$

Moreover, there exists a positive constant $C:=C(p,\Omega,T)$ such that

$\|u\|_{C({W}^{2-2/p,p})}+\|\partial_{t}u\|_{L^{p}(Q)}+\|u\|_{L^{p}(W^{2,p})}\leq
C(\|g\|_{L^{p}(Q)}+\|u_{0}\|_{{W}^{2-2/p,p}}).$

The existence of solutions of some initial and boundary-value problems will be
proven by means of:

######  Theorem 2.2 (Leray-Schauder fixed-point Theorem).

Let $\mathcal{X}$ a Banach space and $T:\,\mathcal{X}\rightarrow\mathcal{X}$ a
continuous and compact operator. If the set

$\\{x\in\mathcal{X}:\,x=\alpha\,Tx\quad\mbox{for some $0\leq\alpha\leq 1$}\\}$

is bounded, then $T$ has (at least) a fixed point.

In this paper, the following two compactness results will be applied.

######  Theorem 2.3 (Aubin-Lions lemma).

(See [22, Théorème 5.1, p. 58].) Let $\mathcal{X}$, $B$ and $\mathcal{Y}$ be
reflexive Banach spaces such that $\mathcal{X}\subset B\subset\mathcal{Y}$,
with compact embedding $\mathcal{X}\mapsto B$ and continuous embedding
$B\hookrightarrow\mathcal{Y}$. It is defined

$W=\\{w:\,w\in L^{p_{0}}(0,T;\mathcal{X})),\,\partial_{t}w\in
L^{p_{1}}(0,T;\mathcal{Y})\\}$

for a finite $T>0$ and $p_{0}$, $p_{1}\in(1,+\infty)$. Then, the injection of
$W$ into $L^{p_{0}}(0,T;B)$ is compact.

######  Theorem 2.4 (Simon’s compactness result).

(See [30, Corollary 4].) Let $\mathcal{X}$, $B$ and $\mathcal{Y}$ be Banach
spaces such that $\mathcal{X}\subset B\subset\mathcal{Y}$, with compact
embedding $\mathcal{X}\mapsto B$ and continuous embedding
$B\hookrightarrow\mathcal{Y}$. Let $F$ be a bounded set in
$L^{\infty}(0,T;\mathcal{X})$ such that the set
$\partial_{t}F=\\{\frac{\partial f}{\partial t};\,f\in F\\}$ is bounded in
$L^{r}(0,T;\mathcal{Y})$ for some $r>1$. Then $F$ is relatively compact in
$C([0,T];B)$.

##  3\. Proof of Theorem 1.2

The proof of Theorem 1.2 will be made in the next two subsections. For the
existence, we use the Leray-Schauder fixed point theorem. The uniqueness is
get by a comparison argument.

###  3.1. Existence

Let us introduce the spaces

(3.1) $\mathcal{X}_{u}:=L^{4-}(Q)\quad\mbox{ and
}\quad\mathcal{X}_{v}:=L^{\infty}(Q),$

and the operator $R:\mathcal{X}_{u}\times\mathcal{X}_{v}\rightarrow
W_{2}\times X_{2+}\hookrightarrow\mathcal{X}_{u}\times\mathcal{X}_{v}$ defined
by $R(\bar{u},\bar{v})=(u,v)$ is the solution of the decoupled linear problem

(3.2)
$\left\\{\begin{array}[]{l}\displaystyle\int_{0}^{T}\langle\partial_{t}u,\varphi\rangle+\int_{0}^{T}\int_{\Omega}\nabla
u\cdot\nabla\varphi+\mu\,\bar{u}_{+}\,u\,\varphi\\\
\displaystyle\quad=\int_{0}^{T}\int_{\Omega}r\,\bar{u}_{+}\varphi-\kappa\int_{0}^{T}\int_{\Omega}\bar{u}_{+}\nabla
v\cdot\nabla\varphi,\quad\forall\,\varphi\in L^{2}(H^{1}),\\\ \\\
\partial_{t}v-\Delta
v+v=\bar{u}_{+}+f\,\bar{v}_{+}\,1_{\Omega_{c}}\quad\hbox{in $Q,$}\\\
u(0)=u_{0},\ v(0)=v_{0},\quad\hbox{in $\Omega,$}\\\ \dfrac{\partial
v}{\partial{\boldsymbol{n}}}=0,\quad\hbox{on
$(0,T)\times\partial\Omega,$}\end{array}\right.$

where $\bar{u}_{+}:=\max\\{\bar{u},0\\}\geq 0$,
$\bar{v}_{+}:=\max\\{\bar{v},0\\}\geq 0$. In fact, first we compute $v$ and
after $u$. In the following lemmas we will prove the hypotheses of Leray-
Schauder fixed point theorem.

######  Lemma 3.1.

The operator
$R:\mathcal{X}_{u}\times\mathcal{X}_{v}\rightarrow\mathcal{X}_{u}\times\mathcal{X}_{v}$
is well defined and compact.

###### Proof.

Since $f\in L^{2+}(Q_{c})$ and $\bar{v}\in L^{\infty}(Q)$ implies $f\bar{v}\in
L^{2+}(Q)$, hence there exists a unique $v\in X_{2+}$ solution of the
$v$-problem in (3.2). By considering the linear parabolic $u$-problem in
(3.2), one has $u\in W_{2}$ owing to $v\in X_{2+}$, hence $\nabla v\in
L^{4+}(Q)$ and then $\bar{u}_{+}\nabla v\in L^{2}(Q)$. Finally, since $R$ maps
bounded sets of $\mathcal{X}_{u}\times\mathcal{X}_{v}$ into bounded sets of
$W_{2}\times X_{2+}$, then $R$ is compact from
$\mathcal{X}_{u}\times\mathcal{X}_{v}$ to itself. ∎

######  Lemma 3.2.

The set

(3.3) $T_{\alpha}=\\{(u,v)\in W_{2}\times X_{2+}\,:\,(u,v)=\alpha R(u,v)\mbox{
for some }\alpha\in[0,1]\\}$

is bounded in $\mathcal{X}_{u}\times\mathcal{X}_{v}$ (independently of
$\alpha\in[0,1]$). In fact, $T_{\alpha}$ is also bounded in $W_{2}\times
X_{2+}$, because there exists

(3.4)
$M=M(r,\mu,\kappa,|\Omega|,T,\|u_{0}\|_{L^{2}},\|v_{0}\|_{W^{1^{+},2^{+}}},\|f\|_{L^{2+}(Q_{c})})>0,$

with $M$ independent of $\alpha$, such that all pairs of functions $(u,v)\in
T_{\alpha}$ for $\alpha\in[0,1]$ satisfy

(3.5) $\|(u,v)\|_{W_{2}\times X_{2+}}\leq M.$

###### Proof.

Let $(u,v)\in T_{\alpha}$ for $\alpha\in(0,1]$ (the case $\alpha=0$ is
trivial). Then, owing to Lemma 3.1, $(u,v)\in W_{2}\times X_{2+}$ and
satisfies the following problem:

(3.6)
$\left\\{\begin{array}[]{l}\displaystyle\int_{0}^{T}\langle\partial_{t}u,\varphi\rangle+\int_{0}^{T}\int_{\Omega}\nabla
u\cdot\nabla\varphi+\mu\,{u}_{+}\,u\,\varphi\\\
\displaystyle\quad=\alpha\int_{0}^{T}\int_{\Omega}r\,{u}_{+}\varphi-\kappa\int_{0}^{T}\int_{\Omega}{u}_{+}\nabla
v\cdot\nabla\varphi,\quad\forall\,\varphi\in L^{2}(H^{1}),\\\ \\\
\partial_{t}v-\Delta
v+v=\alpha\,{u}_{+}+\alpha\,f\,{v}_{+}\,1_{\Omega_{c}}\quad\hbox{a.e. in
$Q,$}\end{array}\right.$

endowed with the corresponding initial and boundary conditions. Therefore, it
suffices to look for a bound of $(u,v)$ in $W_{2}\times X_{2+}$ independent of
$\alpha$. This bound is carried out into six steps:

Step 1: Non-negativity: $u,v\geq 0$.

By taking in (3.6)1 $\varphi=u_{-}:=\min\\{u,0\\}\leq 0$ (that is possible
because $u\in L^{2}(H^{1})$), and considering that $u_{-}=0$ if $u\geq 0$,
$\nabla u_{-}=\nabla u$ if $u\leq 0$, and $\nabla u_{-}=0$ if $u>0$, we have

$\frac{1}{2}\frac{d}{dt}\|u_{-}\|^{2}+\|\nabla u_{-}\|^{2}=\kappa(u_{+}\nabla
v,\nabla u_{-})+\alpha\,r\,(u_{+},u_{-})-\mu\,((u_{+})^{2},u_{-})=0,$

thus $u_{-}\equiv 0$ and, consequently, $u\geq 0$. Similarly, testing (3.6)2
by $v_{-}:=\min\\{v,0\\}\leq 0$ we obtain

$\frac{1}{2}\frac{d}{dt}\|v_{-}\|^{2}+\|\nabla
v_{-}\|^{2}+\|v_{-}\|^{2}=\alpha(u_{+},v_{-})+\alpha(fv_{+},v_{-})_{\Omega_{c}}\leq
0,$

which implies $v_{-}\equiv 0$, then $v\geq 0$. Therefore
$(u_{+},v_{+})=(u,v)$. In particular, $(u,v,f)$ is also the solution of
problem (3.6) changing $u_{+}$ by $u$ and $v_{+}$ by $v$. Therefore, fixed-
point of $R$ are in particular weak solutions of problem (1.1).

Step 2: Boundedness of $\displaystyle\int_{\Omega}u(x,t)\,dx$ .

By taking $\varphi=1$ in (3.6)1, we obtain:

(3.7)
$\displaystyle\frac{d}{dt}\displaystyle\int_{\Omega}u(x,t)\,dx+\mu\,\displaystyle\int_{\Omega}u^{2}(x,t)\,dx=\alpha\,r\,\displaystyle\int_{\Omega}u(x,t)\,dx$

Using Cauchy-Schwartz inequality, we have:

$\displaystyle\int_{\Omega}u(x,t)\,dx\leq|\Omega|^{1/2}\,\left(\displaystyle\int_{\Omega}u^{2}(x,t)\,dx\right)^{1/2},$

which from (3.7) let us deduce that:

(3.8)
$\displaystyle\frac{d}{dt}\displaystyle\int_{\Omega}u(x,t)\,dx+\displaystyle\frac{\mu}{|\Omega|}\,\left(\displaystyle\int_{\Omega}u(x,t)\,dx\right)^{2}\leq\alpha\,r\,\displaystyle\int_{\Omega}u(x,t)\,dx.$

Using the change of variable $y(t)=\displaystyle\int_{\Omega}u(x,t)\,dx$,
(3.8) becomes:

(3.9)
$y^{\prime}(t)+\displaystyle\frac{\mu}{|\Omega|}\,y(t)^{2}\leq\alpha\,r\,y(t)\leq
r\,y(t)$

which is related to a Bernouilli ODE. Making use of $z(t)=y(t)^{-1}$, we can
deduce that:

$-z^{\prime}(t)+\displaystyle\frac{\mu}{|\Omega|}\leq r\,z(t),$

and thus

(3.10) $z^{\prime}(t)+r\,z(t)\geq\displaystyle\frac{\mu}{|\Omega|}.$

This inequality is equivalent to:

$\displaystyle\frac{d}{dt}\left(e^{rt}\,z(t)\right)\geq\displaystyle\frac{\mu}{|\Omega|}\,e^{rt}=\displaystyle\frac{\mu}{|\Omega|}\,\displaystyle\frac{d}{dt}\left(e^{rt}\right)$

and therefore

(3.11) $z(t)\geq
z(0)\,e^{-rt}+\displaystyle\frac{\mu}{r|\Omega|}\,\left(1-e^{-rt}\right)=\displaystyle\frac{\mu}{r|\Omega|}+\left(z(0)-\displaystyle\frac{\mu}{r|\Omega|}\right)e^{-rt}.$

Now, we consider two cases:

  * •

if $z(0)\geq\displaystyle\frac{\mu}{r|\Omega|}$ (i.e.,
$y(0)\leq\displaystyle\frac{r|\Omega|}{\mu}$), then:

$z(t)\geq\displaystyle\frac{\mu}{r|\Omega|},$

which implies that:

(3.12) $y(t)\leq\displaystyle\frac{r|\Omega|}{\mu},\qquad\forall t\geq
0\qquad\mbox{(independently of $\alpha$)}.$

  * •

if $z(0)\leq\displaystyle\frac{\mu}{r|\Omega|}$ (i.e.,
$y(0)\geq\displaystyle\frac{r|\Omega|}{\mu}$), then from (3.11) we can deduce
that:

$z(t)\geq
z(0)\,e^{-rt}+\displaystyle\frac{\mu}{r|\Omega|}\,\left(1-e^{-rt}\right)\geq
z(0)\,e^{-rt}+z(0)\,\left(1-e^{-rt}\right)=z(0)$

and therefore

(3.13) $y(t)\leq
y(0)=\displaystyle\int_{\Omega}u_{0}(x)\,dx=m_{0},\qquad\forall t\geq
0\qquad\mbox{(independently of $\alpha$)}.$

As a conclusion, from (3.12) and (3.13), we arrive at the bound

(3.14)
$y(t)=\displaystyle\int_{\Omega}u(x,t)\,dx\leq\max\left\\{m_{0},\displaystyle\frac{r|\Omega|}{\mu}\right\\}:=K_{1},\quad\forall
t\geq 0.$

Step 3: Bound of $u$ in $L^{2}(0,T;L^{2}(\Omega))$

Integrating directly in $(0,T)$ for a fixed $T$ in (3.7), and using (3.14), we
obtain that:

(3.15)
$\displaystyle\int_{0}^{T}\int_{\Omega}u^{2}(x,t)\,dx\,dt\leq\displaystyle\frac{m_{0}+\alpha\,r\,K_{1}\,T}{\mu}\leq\displaystyle\frac{m_{0}+r\,K_{1}\,T}{\mu}:=K_{2}(T),$

which implies that

(3.16) $\|u\|_{L^{2}(Q)}^{2}\leq\displaystyle K_{2}(T).$

Step 4: Bound of $v$ in $L^{\infty}(0,T;H^{1}(\Omega))\cap
L^{2}(0,T;H^{2}(\Omega))$

Taking $v$ as test function in (3.6)2, we obtain:

(3.17)
$\begin{array}[]{rcl}\displaystyle\frac{1}{2}\frac{d}{dt}\|v\|_{L^{2}}^{2}+\|v\|_{H^{1}}^{2}&=&\alpha\,\displaystyle\int_{\Omega}u\,v\,dx+\alpha\,\displaystyle\int_{\Omega_{c}}f\,v^{2}\,dx\\\
\vspace{-1ex}\cr\\\ \mbox{because
$(\alpha\in(0,1])$}&\leq&\|u\|_{L^{2}}\|v\|_{L^{2}}+\|f\|_{L^{2}}\|v\|_{L^{4}}^{2}\\\
\vspace{-1ex}\cr\\\
&\leq&\|u\|_{L^{2}}\|v\|_{L^{2}}+\|f\|_{L^{2}}\|v\|_{L^{2}}\|v\|_{H^{1}}\\\
\vspace{-1ex}\cr\\\
&\leq&\delta\,\|v\|_{H^{1}}^{2}+C_{\delta}\,\left(\|u\|_{L^{2}}^{2}+\|f\|_{L^{2}}^{2}\|v\|_{L^{2}}^{2}\right)\end{array}$

where we used the following standard inequality in 2D domains

$\|u\|_{L^{4}}\leq C\|u\|_{L^{2}}^{1/2}\|u\|_{H^{1}}^{1/2},\quad\forall u\in
H^{1}(\Omega).$

Therefore, by taking $\delta$ small enough,

(3.18) $\displaystyle\frac{d}{dt}\|v\|_{L^{2}}^{2}+\|v\|_{H^{1}}^{2}\leq
C\,\left(\|u\|_{L^{2}}^{2}+\|f\|_{L^{2}}^{2}\|v\|_{L^{2}}^{2}\right)$

From Gronwall’s lemma, and thanks to the boundedness of $u$ and $f$ in
$L^{2}(Q)$, one has $v$ bounded in $L^{\infty}(0,T;L^{2}(\Omega))\cap
L^{2}(0,T;H^{1}(\Omega))$.

By taking $-\Delta v$ as test function in (3.6)2, we obtain:

(3.19) $\begin{array}[]{l}\displaystyle\frac{1}{2}\frac{d}{dt}\|\nabla
v\|_{L^{2}}^{2}+\|\Delta v\|_{L^{2}}^{2}+\|\nabla
v\|_{L^{2}}^{2}=-\alpha\,\displaystyle\int_{\Omega}u\,\Delta
v\,dx-\alpha\,\displaystyle\int_{\Omega}f\,v\,\Delta v\,dx\\\
\vspace{-1ex}\cr\\\ \leq\|u\|_{L^{2}}\|\Delta
v\|_{L^{2}}+\|f\|_{L^{2+}}\|v\|_{H^{1}}\|\Delta v\|_{L^{2}}\\\
\vspace{-1ex}\cr\\\ \leq\delta\,\left(\|\Delta
v\|_{L^{2}}^{2}+\|v\|_{H^{1}}^{2}\right)+C_{\delta}\,\left(\|u\|_{L^{2}}^{2}+\|f\|_{L^{2+}}^{2}\|v\|_{H^{1}}^{2}\right)\end{array}$

Adding (3.17) to (3.19), we obtain:

(3.20)
$\begin{array}[]{rcl}\displaystyle\frac{d}{dt}\|v\|_{H^{1}}^{2}+\|v\|_{H^{2}}^{2}&\leq&C\,\Big{(}\|u\|_{L^{2}}^{2}+\left(\|f\|_{L^{2}}^{2}+\|f\|_{L^{2+}}^{2}\right)\,\|v\|_{H^{1}}^{2}\Big{)}\end{array}$

From Gronwall’s lemma, one has $v$ bounded in
$L^{\infty}(0,T;H^{1}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega))$.

Step 5: $u$ is bounded in $L^{\infty}(0,T;L^{2}(\Omega))\cap
L^{2}(0,T;H^{1}(\Omega))$.

By testing (3.6)1 by $u$, after a few computations, we get,

$\begin{split}\frac{1}{2}\frac{d}{dt}\|u\|_{L^{2}}^{2}+\|\nabla
u\|_{L^{2}}^{2}+\mu\,\|u\|_{L^{3}}^{3}+\|u\|_{L^{2}}^{2}\\\
\leq\kappa\,\|u\|_{L^{4}}\|\nabla v\|_{L^{4}}\|\nabla
u\|_{L^{2}}+(r_{+}\alpha+1)\|u\|_{L^{2}}^{2}\\\ \leq
C\|u\|_{L^{2}}^{2}\|\nabla
v\|_{L^{4}}^{4}+\frac{1}{2}\|u\|_{H^{1}}^{2}+(r_{+}+1)\|u\|_{L^{2}}^{2},\\\
\end{split}$

Then, we arrive at

(3.21)
$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\|u\|_{L^{2}}^{2}+\|u\|_{H^{1}}^{2}\leq
C\|\nabla
v\|_{L^{4}}^{4}\|u\|_{L^{2}}^{2}+2\,(r_{+}+1)\|u\|_{L^{2}}^{2}\end{split}$

Therefore, applying the Gronwall lemma and using Step 4, we obtain that $u$ is
bounded in $L^{\infty}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{1}(\Omega))$.

Step 6: $v$ is bounded in $L^{\infty}(0,T;W^{1+,2+}(\Omega))\cap
L^{2+}(0,T;W^{2,2+}(\Omega))$. From Step 4 we can deduce that $v\in
L^{\infty-}(Q)$, and from Step 5, we can deduce that $u\in L^{4}(Q)$.
Therefore, $u+f\,v\in L^{2+}(Q)$. Then, heat regularity result in Theorem 2.1,
allows us to deduce that $v\in X_{2^{+}}$ and the corresponding bound on
$X_{2^{+}}$ depending on $\|v_{0}\|_{W^{1^{+},2^{+}}(\Omega)}$ and the bound
of $u+f\,v$ in $L^{2+}(Q)$.

This finishes Lemma 3.2. ∎

######  Lemma 3.3.

The operator
$R:\mathcal{X}_{u}\times\mathcal{X}_{v}\rightarrow\mathcal{X}_{u}\times\mathcal{X}_{v}$,
defined in (3.2), is continuous.

The proof is similar to Lemma 3.4 in [16].

Consequently, from Lemmas 3.1, 3.2 and 3.3, it follows that the operator $R$
satisfy the conditions of the Leray-Schauder fixed-point theorem. Thus, we
conclude that the map $R(\bar{u},\bar{v})$ has at least a fixed point,
$R(u,v)=(u,v)$, which is a weak solution to system (1.1) in $(0,T)$.

Finally, we observe that estimate (1.3) follows the same steps giving in the
proof of Lemma 3.2 (now for the case $\alpha=1$).

###  3.2. Uniqueness of solution

This proof follows the same argument than in [16], but it is included here for
reader convenience. Let $(u_{1},v_{1}),\,(u_{2},v_{2})\in W_{2}\times X_{2}$
two weak solutions of system (1.1). Substracting equations (1.1) for
$(u_{1},v_{1})$ and $(u_{2},v_{2})$, and denoting $u:=u_{1}-u_{2}$ and
$v:=v_{1}-v_{2}$, we obtain the following system

(3.22) $\left\\{\begin{array}[]{rcl}\partial_{t}u-\Delta
u+\kappa\,\nabla\cdot(u_{1}\nabla v+u\nabla
v_{2})&=&r\,u-\mu\,u\,(u_{1}+u_{2})\ \mbox{ in }Q,\\\ \partial_{t}v-\Delta
v+v&=&u+f\,v\,1_{\Omega_{c}}\ \mbox{ in }Q,\\\ u(0,\cdot)&=&0,\ v(0,\cdot)=0\
\mbox{ in }\Omega,\\\ \dfrac{\partial u}{\partial{\bf n}}&=&0,\
\dfrac{\partial v}{\partial{\bf n}}=0\ \mbox{ on
}(0,T)\times\partial\Omega.\end{array}\right.$

Testing (3.22)1 by $u\in L^{2}(H^{1})$ and (3.22)2 by $-\Delta v\in L^{2}(Q)$
we have

(3.23)
$\begin{array}[]{rcl}\displaystyle\frac{1}{2}\frac{d}{dt}\left(\|u\|^{2}+\|\nabla
v\|^{2}\right)&+&\|\nabla u\|^{2}+\|\Delta v\|^{2}+\|\nabla
v\|^{2}+\mu\,\displaystyle\int_{\Omega}u^{2}\,(u_{1}+u_{2})\,dx\\\
\vspace{-1ex}\cr\\\ &=&r\,\|u\|^{2}+\kappa\,(u_{1}\nabla v+u\nabla
v_{2},\nabla u)+(u+fv,-\Delta v).\end{array}$

The term $\mu\,\displaystyle\int_{\Omega}u^{2}\,(u_{1}+u_{2})\,dx$ has the
good sign.

Applying the Hölder and Young inequalities, we obtain

(3.24) $\displaystyle(u_{1}\nabla v,\nabla u)$ $\displaystyle\leq$
$\displaystyle\|u_{1}\|_{L^{4}}\|\nabla v\|_{L^{4}}\|\nabla u\|\leq
C\|u_{1}\|_{L^{4}}\|\nabla v\|^{1/2}\|\nabla v\|^{1/2}_{H^{1}}\|\nabla u\|$
$\displaystyle\leq$ $\displaystyle\delta(\|\nabla v\|^{2}_{H^{1}}+\|\nabla
u\|^{2})+C_{\delta}\|u_{1}\|^{4}_{L^{4}}\|\nabla v\|^{2},$ (3.25)
$\displaystyle(u\nabla v_{2},\nabla u)$ $\displaystyle\leq$
$\displaystyle\|u\|_{L^{4}}\|\nabla v_{2}\|_{L^{4}}\|\nabla u\|\leq
C\|u\|^{1/2}\|u\|_{H^{1}}^{1/2}\|\nabla v_{2}\|_{L^{4}}\|\nabla u\|$
$\displaystyle\leq$ $\displaystyle\delta\|u\|^{2}_{H^{1}}+C_{\delta}\|\nabla
v_{2}\|^{4}_{L^{4}}\|u\|^{2},$ (3.26) $\displaystyle(u,-\Delta v)$
$\displaystyle\leq$ $\displaystyle\delta\|\Delta v\|^{2}+C_{\delta}\|u\|^{2},$
(3.27) $\displaystyle(fv,-\Delta v)$ $\displaystyle\leq$
$\displaystyle\|f\|_{L^{2+}}\|v\|_{H^{1}}\|\Delta
v\|_{L^{2}}\leq\delta\|v\|^{2}_{H^{2}}+C_{\delta}\|f\|^{2}_{L^{2+}}\|v\|^{2}_{H^{1}}.$

Adding (3.17) to (3.23), and using (3.24)-(3.27), we obtain:

(3.28) $\begin{array}[]{l}\displaystyle\frac{d}{dt}\left(\|u\|^{2}+\|\nabla
v\|^{2}\right)+\|\nabla u\|^{2}+\|\nabla v\|_{H^{1}}^{2}\\\ \quad\leq
C\,\Big{(}\|u\|^{2}+\|u_{1}\|^{4}_{L^{4}}\|\nabla v\|^{2}+\|\nabla
v_{2}\|^{4}_{L^{4}}\|u\|^{2}+\|f\|^{2}_{L^{2+}}\|v\|^{2}_{H^{1}}\Big{)}\end{array}$

In order to consider the completed norm for $v$, we take $v$ as test function
in (3.22)2, we obtain:

$\displaystyle\frac{1}{2}\,\frac{d}{dt}\left(\|v\|^{2}\right)+\|v\|_{H^{1}}^{2}=\displaystyle\int_{\Omega}u\,v\,dx+\int_{\Omega_{c}}f\,v^{2}\,dx$

which implies:

(3.29) $\displaystyle\,\frac{d}{dt}\|v\|_{L^{2}}^{2}+\|v\|_{H^{1}}^{2}\leq
C\left(\|u\|_{L^{2}}^{2}+\|f\|^{2}_{L^{2}}\|v\|^{2}_{L^{2}}\right)$

Adding (3.28) to (3.29), we obtain:

(3.30)
$\begin{array}[]{l}\displaystyle\frac{d}{dt}\left(\|u\|^{2}+\|v\|_{H^{1}}^{2}\right)+\|u\|_{H^{1}}^{2}+\|v\|_{H^{2}}^{2}\\\
\quad\leq C\,\Big{(}\|u\|^{2}+\|u_{1}\|^{4}_{L^{4}}\|\nabla v\|^{2}+\|\nabla
v_{2}\|^{4}_{L^{4}}\|u\|^{2}+\|f\|^{2}_{L^{2+}}\|v\|^{2}_{H^{1}}\Big{)}\end{array}$

Since $\|u_{1}\|^{4}_{L^{4}}+\|\nabla
v_{2}\|^{4}_{L^{4}}+\|f\|^{2}_{L^{2+}}\in L^{1}(0,T)$ and $u_{0}=v_{0}=0$,
then Gronwall lemma implies uniqueness.

Thus, the proof of Theorem 1.2 is finished.

##  4\. Proof of Theorem 1.3

The admissible set for the optimal control problem (1.2) is defined by

$\mathcal{S}_{ad}=\\{s=(u,v,f)\in W_{2}\times
X_{2+}\times\mathcal{F}\,:\,s\mbox{ is a weak solution of (\ref{KS}) in
}(0,T)\\}.$

From Theorem 1.2 one has $\mathcal{S}_{ad}\neq\emptyset$. Let
$\\{s_{m}\\}_{m\in\mathbb{N}}:=\\{(u_{m},v_{m},f_{m})\\}_{m\in\mathbb{N}}\subset\mathcal{S}_{ad}$
be a minimizing sequence of $J$, that is,
$\displaystyle\lim_{m\rightarrow+\infty}J(s_{m})=\inf_{s\in\mathcal{S}_{ad}}J(s)$.
Then, by definition of $\mathcal{S}_{ad}$, for each $m\in\mathbb{N}$, $s_{m}$
satisfies system $(\ref{eq1})_{1}$ variationally in $L^{2}((H^{1})^{\prime})$
and $(\ref{eq1})_{2}$ a.e. $(t,x)\in Q$.

From the definition of $J$ and the assumption $\gamma_{f}>0$ or $\mathcal{F}$
is bounded in $L^{2+}(Q_{c})$, it follows that

(4.1) $\\{f_{m}\\}_{m\in\mathbb{N}}\mbox{ is bounded in }L^{{2+}}(Q_{c})$

From (3.4)-(3.5) there exists a positive constant $C$, independent of $m$,
such that

(4.2) $\|(u_{m},v_{m})\|_{W_{2}\times X_{2+}}\leq C.$

Therefore, from (4.1), (4.2), and taking into account that $\mathcal{F}$ is a
closed convex subset of $L^{2+}(Q_{c})$ (hence is weakly closed in
$L^{2+}(Q_{c})$), it is deduced that there exists
$\tilde{s}=(\tilde{u},\tilde{v},\tilde{f})\in W_{2}\times
X_{2+}\times\mathcal{F}$ such that, for some subsequence of
$\\{s_{m}\\}_{m\in\mathbb{N}}$, still denoted by
$\\{s_{m}\\}_{m\in\mathbb{N}}$, the following convergences hold, as
$m\rightarrow+\infty$:

(4.3) $\displaystyle u_{m}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{u}\quad\mbox{weakly in }L^{2}(H^{1})\mbox{ and weakly* in
}L^{\infty}(L^{2}),$ (4.4) $\displaystyle v_{m}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{v}\quad\mbox{weakly in }L^{2+}(W^{2,2+})\mbox{ and
weakly* in }L^{\infty}(W^{1+,2+}),$ (4.5) $\displaystyle\partial_{t}u_{m}$
$\displaystyle\rightarrow$
$\displaystyle\partial_{t}\tilde{u}\quad\mbox{weakly in
}L^{2}((H^{1})^{\prime}),$ (4.6) $\displaystyle\partial_{t}v_{m}$
$\displaystyle\rightarrow$
$\displaystyle\partial_{t}\tilde{v}\quad\mbox{weakly in }L^{2+}(Q),$ (4.7)
$\displaystyle f_{m}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{f}\quad\mbox{weakly in }L^{2+}(Q_{c}),\mbox{ and
}\tilde{f}\in\mathcal{F}.$

From (4.3)-(4.6), and using Sobolev embedding and Aubin-Lions compactness
results, one has

(4.8) $\displaystyle(u_{m},v_{m})$ $\displaystyle\rightarrow$
$\displaystyle(\tilde{u},\tilde{v})\quad\mbox{strongly in
}C^{0}([0,T];((H^{1}(\Omega))^{\prime}\times L^{2}(\Omega))$ (4.9)
$\displaystyle v_{m}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{v}\quad\mbox{strongly in }L^{\infty}(Q)).$ (4.10)
$\displaystyle(u_{m},\nabla v_{m})$ $\displaystyle\rightarrow$
$\displaystyle(\tilde{u},\nabla\tilde{v})\quad\mbox{strongly in
}L^{4-}(Q))\times L^{4+}(Q).$

In particular, using (4.7), (4.9) and (4.10) the limit of the nonlinear terms
of (3.7) can be controlled as follows

(4.11) $\displaystyle u_{m}\cdot\nabla v_{m}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{u}\cdot\nabla\tilde{v}\quad\hbox{ strongly in
$L^{2}(Q)$},$ (4.12) $\displaystyle f_{m}v_{m}1_{\Omega_{c}}$
$\displaystyle\rightarrow$
$\displaystyle\tilde{f}\,\tilde{v}\,1_{\Omega_{c}}\quad\hbox{ weakly in
$L^{2+}(Q)$}.$

Moreover, from (4.8), $(u_{m}(0),v_{m}(0))$ converges to
$(\tilde{u}(0),\tilde{v}(0))$ in $H^{1}(\Omega)^{\prime}\times L^{2}(\Omega)$,
and since $u_{m}(0)=u_{0}$, $v_{m}(0)=v_{0}$, it is deduced that
$\tilde{u}(0)=u_{0}$ and $\tilde{v}(0)=v_{0}$. Thus, $\tilde{s}$ satisfies the
initial conditions given in (1.1). Therefore, considering the convergences
(4.3)-(4.12), and taking the limit in equation (3.6) replacing $(u,v,f)$ by
$(u_{m},v_{m},f_{m})$, as $m$ goes to $+\infty$, it is possible to conclude
that $\tilde{s}=(\tilde{u},\tilde{v},\tilde{f})$ is a weak solution of the
system (1.1), that is, $\tilde{s}\in\mathcal{S}_{ad}$. Therefore,

(4.13) $\lim_{m\rightarrow+\infty}J(s_{m})=\inf_{s\in\mathcal{S}_{ad}}J(s)\leq
J(\tilde{s}).$

Additionally, since $J$ is lower semicontinuous on $\mathcal{S}_{ad}$, one has
$J(\tilde{s})\leq\displaystyle\liminf_{m\rightarrow+\infty}J(s_{m})$, which
jointly to (4.13), implies that $\tilde{s}$ is a global optimal control.

##  5\. Proof of Theorem 1.4

###  5.1. A generic Lagrange multipliers theorem

We introduce a Lagrange multipliers theorem given by J.Zowe and S.Kurcyusz
[35] (see also [32, Chapter 6], for more details) that we will apply to get
first-order necessary optimality conditions for a local optimal solution
$(\tilde{u},\tilde{v},\tilde{f})$ of problem (1.2). First, we consider the
following (generic) optimization problem:

(5.1) $\min_{s\in\mathbb{M}}J(s)\ \mbox{ subject to }G(s)=0,$

where $J:\mathbb{X}\rightarrow\mathbb{R}$ is a functional,
$G:\mathbb{X}\rightarrow\mathbb{Y}$ is an operator, $\mathbb{X}$ and
$\mathbb{Y}$ are Banach spaces, and $\mathbb{M}$ is a nonempty closed and
convex subset of $\mathbb{X}$. The corresponding admissible set for problem
(5.1) is

$\mathcal{S}=\\{s\in\mathbb{M}\,:\,G(s)=0\\}.$

######  Definition 5.1.

(Lagrangian) The functional
$\mathcal{L}:\mathbb{X}\times{\mathbb{Y}}^{\prime}\rightarrow\mathbb{R}$,
given by

(5.2) $\mathcal{L}(s,\xi)=J(s)-\langle\xi,G(s)\rangle_{\mathbb{Y}^{\prime}}$

is called the Lagrangian functional related to problem (5.1).

######  Definition 5.2.

(Lagrange multiplier) Let $\tilde{s}\in\mathcal{S}$ be a local optimal
solution for problem (5.1). Suppose that $J$ and $G$ are Fréchet
differentiable in $\tilde{s}$. Then, any $\xi\in\mathbb{Y}^{\prime}$ is called
a Lagrange multiplier for (5.1) at the point $\tilde{s}$ if

(5.3)
$\mathcal{L}_{s}^{\prime}(\tilde{s},\xi)[r]=J^{\prime}(\tilde{s})[r]-\langle\xi,G^{\prime}(\tilde{s})[r]\rangle_{\mathbb{Y}^{\prime}}\geq
0\quad\forall r\in\mathcal{C}(\tilde{s}),$

where
$\mathcal{C}(\tilde{s})=\\{\theta(s-\tilde{s})\,:\,s\in\mathbb{M},\,\theta\geq
0\\}$ is the conical hull of $\tilde{s}$ in $\mathbb{M}$.

######  Definition 5.3.

Let $\tilde{s}\in\mathcal{S}$ be a local optimal solution for problem (5.1).
It will be said that $\tilde{s}$ is a regular point if

$G^{\prime}(\tilde{s})[\mathcal{C}(\tilde{s})]=\mathbb{Y}.$

######  Theorem 5.4.

([32, Theorem 6.3, p.330], [35, Theorem 3.1]) Let $\tilde{s}\in\mathcal{S}$ be
a local optimal solution for problem (5.1). Suppose that $J$ is Fréchet
differentiable in $\tilde{s}$, and $G$ is continuous Fréchet-differentiable in
$\tilde{s}$ . If $\tilde{s}$ is a regular point, then there exists Lagrange
multipliers for (5.1) at $\tilde{s}$.

###  5.2. Application of the Lagrange multiplier theory

Now, in order to reformulate the optimal control problem (1.2) in the abstract
setting (5.1), we introduce the Banach spaces

$\mathbb{X}:=\widehat{W}_{2}\times\widehat{X}_{2+}\times L^{2+}(Q_{c}),\
\mathbb{Y}:=L^{2}((H^{1})^{\prime})\times L^{2+}(Q),$

where

$\widehat{W}_{2}=\\{u\in W_{2}\,:\,u(0)=0\\},\quad\widehat{X}_{2+}=\\{v\in
X_{2+}\,:\,v(0)=0,\ \partial_{n}v|_{\partial\Omega}=0\\}$

and the operator $G=(G_{1},G_{2}):\mathbb{X}\rightarrow\mathbb{Y}$, where

$G_{1}:\mathbb{X}\rightarrow L^{2}((H^{1})^{\prime}),\quad
G_{2}:\mathbb{X}\rightarrow L^{2+}(Q)$

are defined at each point $s=(u,v,f)\in\mathbb{X}$ by

$\left\\{\begin{array}[]{l}\langle
G_{1}(s),\varphi\rangle=\langle\partial_{t}u,\varphi\rangle_{L^{2}(H^{1}),L^{2}((H^{1})^{\prime})}+(\nabla
u-\kappa\,u\nabla v,\nabla\varphi)_{L^{2}}\\\ \vspace{-2ex}\cr\\\
\qquad\qquad+(-r\,u+\mu\,u^{2},\varphi)_{L^{2}}\quad\forall\,\varphi\in
L^{2}(H^{1})\vspace{0.1cm}\\\ G_{2}(s)=\partial_{t}v-\Delta
v+v-u-f\,v\,1_{\Omega_{c}}\quad\hbox{in $L^{2+}(Q).$}\end{array}\right.$

Thus, the optimal control problem (1.2) is reformulated as follows

(5.4) $\min_{s\in{\mathbb{M}}}J(s)\quad\mbox{ subject to }\quad{G}(s)={\bf
0},$

where

(5.5)
$\mathbb{M}:=(\hat{u},\hat{v},0)+\widehat{W}_{2}\times\widehat{X}_{2+}\times\mathcal{F},$

with $(\hat{u},\hat{v})$ the global weak solution of (1.1) without control
$(\hat{f}=0$) and $\mathcal{F}$ is defined in (1.1).

######  Remark 5.5.

From Definition 5.1, the Lagragian associated to optimal control problem (5.4)
is the functional $\mathcal{L}:\mathbb{X}\times L^{2}(H^{1})\times
L^{2-}(Q)\rightarrow\mathbb{R}$ given by

$\mathcal{L}(s,\lambda,\eta)=J(s)-\langle\lambda,G_{1}(s)\rangle_{L^{2}(H^{1}),L^{2}((H^{1})^{\prime})}-(\eta,G_{2}(s))_{L^{2-},L^{2+}}.$

The set ${\mathbb{M}}$ defined in (5.5) is a closed convex subset of
$\mathbb{X}$ and the admissible set of control problem (5.4) is

(5.6) $\mathcal{S}_{ad}=\\{s=(u,v,f)\in{\mathbb{M}}\,:\,{G}(s)={\bf 0}\\}.$

Concerning to the differentiability of the functional $J$ and the constraint
operator ${G}$, one has the following results.

######  Lemma 5.6.

The functional $J:\mathbb{X}\rightarrow\mathbb{R}$ is Fréchet differentiable
and the Fréchet derivative of $J$ in
$\tilde{s}=(\tilde{u},\tilde{v},\tilde{f})\in\mathbb{X}$ in the direction
$r=(U,V,F)\in\mathbb{X}$ is

(5.7)
$\begin{split}J^{\prime}(\tilde{s})[r]=\gamma_{u}\int_{0}^{T}\int_{\Omega}(\tilde{u}-u_{d})U+\gamma_{v}\int_{0}^{T}\int_{\Omega}(\tilde{v}-v_{d})V\\\
+\gamma_{f}\int_{0}^{T}\int_{\Omega_{c}}{\rm
sgn}(\tilde{f})|\tilde{f}|^{1+}F.\end{split}$

######  Lemma 5.7.

The operator ${G}:\mathbb{X}\rightarrow\mathbb{Y}$ is continuous-Fréchet
differentiable and the Fréchet derivative of ${G}$ in
$\tilde{s}=(\tilde{u},\tilde{v},\tilde{f})\in\mathbb{X}$ in the direction
$r=(U,V,F)\in\mathbb{X}$ is the linear operator
${G}^{\prime}(\tilde{s})[r]=(G_{1}^{\prime}(\tilde{s})[r],G_{2}^{\prime}(\tilde{s})[r])$
defined by

(5.8) $\left\\{\begin{array}[]{l}\langle
G_{1}^{\prime}(\tilde{s})[r],\varphi\rangle=\langle\partial_{t}U,\varphi\rangle+(\nabla
U-\kappa\,U\nabla\tilde{v}-\kappa\,\tilde{u}\nabla V,\nabla\varphi)\\\
\qquad\qquad\qquad+(-r\,U+2\mu\,\tilde{u}\,U,\varphi),\quad\forall\,\varphi\in
L^{2}(H^{1})\vspace{0.1cm}\\\
G_{2}^{\prime}(\tilde{s})[r]=\partial_{t}V-\Delta
V+V-U-\tilde{f}\,V\,1_{\Omega_{c}}-F\tilde{v}\,1_{\Omega_{c}}.\end{array}\right.$

######  Remark 5.8.

From Definition 5.3 one has that
$\tilde{s}=(\tilde{u},\tilde{v},\tilde{f})\in\mathcal{S}_{ad}$ is a regular
point if for any $(g_{u},g_{v})\in\mathbb{Y}$ there exists
$r=(U,V,F)\in\widehat{W}_{2}\times\widehat{X}_{2+}\times\mathcal{C}(\tilde{f})$
such that

${G}^{\prime}(\tilde{s})[r]=(g_{u},g_{v}),$

where $\mathcal{C}(\tilde{f}):=\\{\theta(f-\tilde{f})\,:\,\theta\geq
0,\,f\in\mathcal{F}\\}$ is the conical hull of $\tilde{f}$ in $\mathcal{F}$.

###  5.3. The linearized problem (5.8) is surjective

######  Lemma 5.9.

Let $\tilde{s}=(\tilde{u},\tilde{v},\tilde{f})\in\mathcal{S}_{ad}$
($\mathcal{S}_{ad}$ defined in (5.6)), then $\tilde{s}$ is a regular point.

###### Proof.

Fixed $(\tilde{u},\tilde{v},\tilde{f})\in\mathcal{S}_{ad}$, let
$(g_{u},g_{v})\in L^{2}((H^{1})^{\prime})\times L^{2+}(Q)$. Since
$0\in\mathcal{C}(\tilde{f})=\\{\theta(f-\tilde{f})\,:\,\theta\geq
0,\,f\in\mathcal{F}\\}$, it suffices to show the existence of
$(U,V)\in\widehat{W}_{2}\times\widehat{X}_{2+}$ solving the linear problem

(5.9) $\left\\{\begin{array}[]{l}\langle\partial_{t}U,\varphi\rangle+(\nabla
U-\kappa\,U\nabla\tilde{v}-\kappa\,\tilde{u}\nabla V,\nabla\varphi)\\\
\qquad+(-r\,U+2\mu\,\tilde{u}\,U,\varphi)=\langle
g_{u},\varphi\rangle\quad\forall\,\varphi\in L^{2}(H^{1}),\vspace{0.1cm}\\\
\partial_{t}V-\Delta V+V-U-\tilde{f}\,V\,1_{\Omega_{c}}=g_{v}\quad\mbox{ in
}L^{2+}(Q).\end{array}\right.$

For this, we will use the Leray-Schauder fixed-point Theorem, for the operator

(5.10) $S:(\overline{U},\overline{V})\in L^{4-}(Q)\times
L^{\infty}(Q)\mapsto(U,V)\in\widehat{W}_{2}\times\widehat{X}_{2+}$

where $(U,V)$ is the solution of the decoupled problem (first computing $V$
and after $U$)

(5.11) $\left\\{\begin{array}[]{l}\langle\partial_{t}U,\varphi\rangle+(\nabla
U-\kappa\,U\nabla\tilde{v}-\kappa\,\tilde{u}\nabla V,\nabla\varphi)\\\
\qquad=(r\,\overline{U}-2\mu\,\tilde{u}\,\overline{U},\varphi)+\langle
g_{u},\varphi\rangle\quad\forall\,\varphi\in L^{2}(H^{1}),\vspace{0.1cm}\\\
\partial_{t}V-\Delta
V+V=\overline{U}+\tilde{f}\overline{V}1_{\Omega_{c}}+g_{v}\quad\mbox{ in
}Q.\end{array}\right.$

Step 1 ($S$ is well-defined and bounded): Prove that operator $S$ defined in
(5.10) maps bounded sets in $L^{4-}(Q)\times L^{\infty}(Q)$ in bounded sets in
$(U,V)\in{W}_{2}\times{X}_{2+}$.

For this, one first bound $V$ and later bound $U$. Indeed, since
$(\overline{U},\overline{V})\in L^{4-}(Q)\times L^{\infty}(Q)$ implies
$f\,\overline{V}\in L^{2+}(Q)$, then by applying $L^{2+}$-regularity to the
heat equation (5.11)2 (Theorem 2.1), it is deduced that $V\in X_{2+}$ and

(5.12)
$\begin{array}[]{rcl}\|V\|_{X_{2+}}&\leq&C\,\left(\|\overline{U}+\tilde{f}\overline{V}1_{\Omega_{c}}\|_{L^{2+}(Q)}+\|g_{v}\|_{L^{2+}(Q)}\right)\vspace{0.1cm}\\\
&\leq&C\,\left(\|\overline{U}\|_{L^{2+}}+\|\tilde{f}\|_{L^{2+}(Q_{c})}\|\overline{V}\|_{L^{\infty}}+\|g_{v}\|_{L^{2+}(Q)}\right)\end{array}$

By taking $\varphi=U$ in (5.11)1, we arrive at

(5.13) $\begin{array}[]{l}\frac{d}{dt}\|U\|^{2}_{L^{2}}+\|U\|^{2}_{H^{1}}\leq
C_{1}(1+\|\nabla\tilde{v}\|_{L^{4}}^{4})\|U\|^{2}_{L^{2}}\vspace{0.1cm}\\\
\qquad\qquad+C_{2}\Big{(}\|\tilde{u}\|^{2}_{L^{4}}\|\nabla
V\|^{2}_{L^{4}}+(1+\|\tilde{u}\|_{L^{4}}^{2})\|\overline{U}\|_{L^{2}}^{2}+\|g_{u}\|_{(H^{1})^{\prime}}^{2}\Big{)}\end{array}$

Finally, using $2D$ interpolation estimates,

$\|\tilde{u}\|_{L^{4}(Q)}^{2}\|\nabla
V\|_{L^{4}(Q)}^{2}\leq\|\tilde{u}\|_{W_{2}}^{2}\|V\|_{X_{2}}^{2}$

Then, using (5.12), the Gronwall Lemma applied to (5.13) guarantees the bound
for $U$ in $W_{2}$.

Step 2 (compactness): By using that ${W}_{2}\times{X}_{2+}$ is compactly
embedded in $L^{4-}(Q)\times L^{\infty}(Q)$, then operator $S$ is compact.

Step 3 (continuity): In particular, using Steps 1 and 2, it is not difficult
to prove the continuity of $S$ from $L^{4-}(Q)\times L^{\infty}(Q)$ to itself.

Step 4 (boundedness of possible fixed-points): Now, the aim is to show that
the set of the possible fixed-points of $\alpha S$ with $\alpha\in[0,1]$
defined as
$S_{\alpha}:=\\{(U,V)\in\widehat{W}_{2}\times\widehat{X}_{2+}\,:\,(U,V)=\alpha
S(U,V)\mbox{ for some }\alpha\in[0,1]\\}$ is bounded in $L^{4-}(Q)\times
L^{\infty}(Q)$ (with respect to $\alpha$). Indeed, if $(U,V)\in S_{\alpha},$
then $(U,V)\in\widehat{W}_{2}\times\widehat{X}_{2+}$ and solves the coupled
linear problem

(5.14) $\left\\{\begin{array}[]{l}\langle\partial_{t}U,\varphi\rangle+(\nabla
U-\kappa\,U\nabla\tilde{v}-\kappa\,\tilde{u}\nabla V,\nabla\varphi)\\\
\qquad=\alpha(r\,U-2\mu\,\tilde{u}\,U,\varphi)+\alpha\langle
g_{u},\varphi\rangle\quad\forall\,\varphi\in L^{2}(H^{1}),\vspace{0.1cm}\\\
\partial_{t}V-\Delta V+V=\alpha U+\alpha\tilde{f}V1_{\Omega_{c}}+\alpha g_{v}\
\mbox{ in }Q.\end{array}\right.$

Then, taking $\varphi=U$ in (5.14)1, one obtains (see (5.13)):

(5.15) $\begin{array}[]{l}\displaystyle\frac{d}{dt}\|U\|^{2}+\|\nabla
U\|^{2}+2\alpha\,\mu\,\int_{\Omega}\tilde{u}\,U^{2}\\\ \qquad\leq
C\left(\alpha+\|\nabla\tilde{v}\|_{L^{4}}^{4}\right)\,\|U\|^{2}+C\,\|\tilde{u}\|_{L^{4}}^{4}\|\nabla
V\|^{2}+\alpha^{2}\,\|g_{u}\|_{(H^{1})^{\prime}}^{2}.\end{array}$

Now, testing (5.14)2 by $V-\Delta V\in L^{2+}(Q)$, it holds:

(5.16) $\frac{d}{dt}\|V\|_{H^{1}}^{2}+\|V\|_{H^{2}}^{2}\leq
C\,\alpha^{2}\|f\|_{L^{2+}}^{2}\,\|V\|_{H^{1}}^{2}+\alpha^{2}\,\left(\|g_{v}\|^{2}+\|U\|^{2}\right).$

From (5.15) and (5.16) and using that $\alpha\leq 1$:

$\begin{array}[]{l}\displaystyle\frac{d}{dt}\left(\|U\|^{2}+\|V\|_{H^{1}}^{2}\right)+\|U\|_{H^{1}}^{2}+\|V\|_{H^{2}}^{2}\vspace{0.1cm}\\\
\leq
C\,\left(1+\|\nabla\tilde{v}\|_{L^{4}}^{4}\right)\,\|U\|^{2}+C\,\Big{(}\|f\|_{L^{2+}}^{2}+\|\tilde{u}\|_{L^{4}}^{4}\Big{)}\|V\|_{H^{1}}^{2}\vspace{0.1cm}\\\
+C\,\Big{(}\|g_{u}\|_{(H^{1})^{\prime}}^{2}+\|g_{v}\|^{2}\Big{)}.\end{array}$

Using that $U(0)=V(0)=0$ and $\|g_{u}\|_{L^{2}((H^{1})^{\prime})}$,
$\|g_{v}\|_{L^{2}(L^{2})}$, $\|f\|_{L^{2+}}$, $\|\tilde{u}\|_{L^{4}(Q)}$ and
$\|\nabla\tilde{v}\|_{L^{4}(Q)}$ are constant finite values, then the Gronwall
Lemma implies that

(5.17) $\|(U,V)\|_{W_{2}\times X_{2}}\leq C.$

Finally, by applying $L^{2+}$-regularity provided by Theorem 2.1 related to
the parabolic-Neumann problem, one has

$\|V\|_{X_{2+}}\leq C.$

Step 5: Conclusion: By applying Leray-Schauder fixed-point theorem (Theorem
2.2), one has the existence of $(U,V)\in W_{2}\times X_{2+}$ a solution of
problem (5.9). The uniqueness of solution is directly deduced from the
linearity of problem (5.9). ∎

###  5.4. Existence of Lagrange multipliers

Now, the existence of Lagrange multiplier for problem (1.2) associated to any
local optimal solution
$\tilde{s}=(\tilde{u},\tilde{v},\tilde{f})\in\mathcal{S}_{ad}$ will be shown.

######  Theorem 5.10.

Let $\tilde{s}=(\tilde{u},\tilde{v},\tilde{f})\in\mathcal{S}_{ad}$ be a local
optimal solution for the control problem (1.2). Then, there exists a Lagrange
multiplier $\xi=(\lambda,\eta)\in L^{2}(H^{1})\times L^{2-}(Q)$ such that for
all
$(U,V,F)\in\widehat{W}_{2}\times\widehat{X}_{2+}\times\mathcal{C}(\tilde{f})$

$\displaystyle\gamma_{u}\int_{0}^{T}\int_{\Omega}(\tilde{u}-u_{d})U+\gamma_{v}\int_{0}^{T}\int_{\Omega}(\tilde{v}-v_{d})V+\gamma_{f}\int_{0}^{T}\int_{\Omega_{c}}{\rm
sgn}(\tilde{f})|\tilde{f}|^{1+}F$
$\displaystyle-\int_{0}^{T}\langle\partial_{t}U,\lambda\rangle-\int_{0}^{T}\int_{\Omega}(\nabla
U-\kappa\,U\nabla\tilde{v}-\kappa\,\tilde{u}\nabla
V,\nabla\lambda)+(-rU+2\mu\tilde{u}U,\lambda)$ (5.18)
$\displaystyle-\int_{0}^{T}\int_{\Omega}\bigg{(}\partial_{t}V-\Delta
V+V-U-\tilde{f}V1_{\Omega_{c}}\bigg{)}\eta+\int_{0}^{T}\int_{\Omega_{c}}F\tilde{v}1_{\Omega_{c}}\eta\geq
0.$

###### Proof.

From Lemma 5.9, $\tilde{s}\in\mathcal{S}_{ad}$ is a regular point, then from
Theorem 5.4 there exists a Lagrange multiplier $\xi=(\lambda,\eta)\in
L^{2}(H^{1})\times L^{2-}(Q)$ such that by (5.3)2 and Remark 5.5 one must
satisfy

(5.19)
$\mathcal{L}^{\prime}_{s}(s,\lambda,\eta)[r]=J^{\prime}(\tilde{s})[r]-\langle\lambda,G_{1}^{\prime}(\tilde{s})[r]\rangle_{L^{2}(H^{1}),L^{2}((H^{1})^{\prime}}-(\eta,G_{2}^{\prime}(\tilde{s})[r])_{L^{2}}\geq
0,$

for all
$r=(U,V,F)\in\widehat{W}_{2}\times\widehat{X}_{2+}\times\mathcal{C}(\tilde{f}).$
The proof follows from (5.7), (5.8), and (5.19). ∎

From Theorem 5.10, an optimality system for problem (1.2) can be derived.

######  Corollary 5.11.

Let $\tilde{s}=(\tilde{u},\tilde{v},\tilde{f})\in\mathcal{S}_{ad}$ be a local
optimal solution for the control problem (1.2). Then any Lagrange multiplier
$(\lambda,\eta)\in L^{2}(H^{1})\times L^{2-}(Q)$, provided by Theorem 5.10,
satisfies the system

$\displaystyle\int_{0}^{T}\langle\partial_{t}U,\lambda\rangle+\int_{0}^{T}\int_{\Omega}(\nabla
U-\kappa\,U\nabla\tilde{v})\cdot\nabla\lambda+(-rU+2\mu\tilde{u}U,\lambda)-\int_{0}^{T}\int_{\Omega}U\eta$
(5.20) $\displaystyle\hskip
19.91684pt=\gamma_{u}\int_{0}^{T}\int_{\Omega}(\tilde{u}-u_{d})U,\qquad\forall\,U\in\widehat{W}_{2},$
$\displaystyle\int_{0}^{T}\int_{\Omega}\bigg{(}\partial_{t}V-\Delta
V+V\bigg{)}\eta-\int_{0}^{T}\int_{\Omega_{c}}\tilde{f}V\eta+\kappa\int_{0}^{T}\int_{\Omega}\tilde{u}\nabla
V\cdot\nabla\lambda$ (5.21) $\displaystyle\hskip
19.91684pt=\gamma_{v}\int_{0}^{T}\int_{\Omega}(\tilde{v}-v_{d})V,\qquad\forall\,V\in\widehat{X}_{2+},$

and the optimality condition

(5.22) $\int_{0}^{T}\int_{\Omega_{c}}(\gamma_{f}{\rm
sgn}(\tilde{f})|\tilde{f}|^{1+}+\tilde{v}\eta)(f-\tilde{f})\geq
0,\qquad\forall f\in\mathcal{F}.$

###### Proof.

From (5.18), taking $(V,F)=(0,0)$, and using that $\widehat{W}_{2}$ is a
vectorial space, (5.20) holds. Similarly, taking $(U,F)=(0,0)$ in (5.18), and
taking into account that $\widehat{X}_{2+}$ is a vectorial space, (5.21) is
deduced. Finally, taking $(U,V)=(0,0)$ in (5.18) it holds

$\gamma_{f}\int_{0}^{T}\int_{\Omega_{c}}{\rm
sgn}(\tilde{f})|\tilde{f}|^{1+}F+\int_{0}^{T}\int_{\Omega_{c}}\tilde{v}\,\eta\,F\geq
0,\quad\forall\,F\in\mathcal{C}(\tilde{f}).$

Thus, choosing $F=\theta(f-\tilde{f})\in\mathcal{C}(\tilde{f})$ for all
$f\in\mathcal{F}$ and $\theta\geq 0$, (5.22) is deduced. ∎

######  Remark 5.12.

A pair $(\lambda,\eta)\in L^{2}(H^{1})\times L^{2-}(Q)$ satisfying
(5.20)-(5.21) corresponds to the concept of very weak solution (at least for
the $\eta$-variable) of the linear problem

(5.23)
$\left\\{\begin{array}[]{rcl}-\partial_{t}\lambda-\Delta\lambda-\kappa\,\nabla\lambda\cdot\nabla\tilde{v}-\eta-r\lambda+2\mu\tilde{u}\lambda&=&\gamma_{u}(\tilde{u}-u_{d})\quad\mbox{
in }Q,\\\
-\partial_{t}\eta-\Delta\eta+\eta-\kappa\,\nabla\cdot(\tilde{u}\nabla\lambda)-\tilde{f}\,\eta\,1_{\Omega_{c}}&=&\gamma_{v}(\tilde{v}-v_{d})\quad\mbox{
in }Q,\\\ \lambda(T)=0,\ \ \eta(T)&=&0\quad\mbox{ in }\Omega,\\\
\dfrac{\partial\lambda}{\partial{\boldsymbol{n}}}=0,\
\dfrac{\partial\eta}{\partial{\boldsymbol{n}}}&=&0\quad\mbox{ on
}(0,T)\times\partial\Omega.\end{array}\right.$

###  5.5. Regularity of Lagrange multipliers

######  Theorem 5.13.

Let $\tilde{s}=(\tilde{u},\tilde{v},\tilde{f})\in\mathcal{S}_{ad}$ be a local
optimal solution for the problem (1.2). Then the problem (5.23) has a unique
solution $(\lambda,\eta)$ such that

(5.24) $(\lambda,\eta)\in X_{2}\times W_{2}$

###### Proof.

Let $s=T-t$, with $t\in(0,T)$ and $\tilde{\lambda}(s)=\lambda(t)$,
$\tilde{\eta}(s)=\eta(t)$. Then, system (5.23) is equivalent to

(5.25)
$\left\\{\begin{array}[]{rcl}\partial_{s}\tilde{\lambda}-\Delta\tilde{\lambda}-\kappa\,\nabla\tilde{\lambda}\cdot\nabla\tilde{v}-\tilde{\eta}-r\tilde{\lambda}+2\mu\tilde{u}\tilde{\lambda}&=&\gamma_{u}(\tilde{u}-u_{d})\quad\mbox{
in }Q,\\\
\partial_{s}\tilde{\eta}-\Delta\tilde{\eta}+\tilde{\eta}-\kappa\,\nabla\cdot(\tilde{u}\nabla\tilde{\lambda})-\tilde{f}\,\tilde{\eta}\,1_{\Omega_{c}}&=&\gamma_{v}(\tilde{v}-v_{d})\quad\mbox{
in }Q,\\\ \tilde{\lambda}(0)=0,\ \tilde{\eta}(0)&=&0\quad\mbox{ in }\Omega,\\\
\dfrac{\partial\tilde{\lambda}}{\partial{\boldsymbol{n}}}=0,\
\dfrac{\partial\tilde{\eta}}{\partial{\boldsymbol{n}}}&=&0\quad\mbox{ on
}(0,T)\times\partial\Omega.\end{array}\right.$

In order to prove the existence of a solution for (5.25), the Leray-Schauder
fixed-point Theorem can be applied as before, now over the operator

(5.26) $\widehat{T}:(\bar{\lambda},\bar{\eta})\in L^{\infty-}\times
L^{4-}\mapsto(\lambda,\eta)\in X_{2}\times W_{2}$

where $(\lambda,\eta)=\widehat{T}(\bar{\lambda},\bar{\eta})$ solves the
decoupled problem (first computing $\lambda$ and after $\mu$):

(5.27)
$\left\\{\begin{array}[]{rcl}\partial_{s}{\lambda}-\Delta{\lambda}-\kappa\,\nabla{\lambda}\cdot\nabla\tilde{v}&=&\bar{\eta}+r\bar{\lambda}-2\mu\tilde{u}\bar{\lambda}+\gamma_{u}(\tilde{u}-u_{d})\quad\mbox{
in }Q,\\\
\partial_{s}{\eta}-\Delta\eta+{\eta}-\kappa\,\nabla\cdot(\tilde{u}\nabla{\lambda})&=&\tilde{f}\,\bar{\eta}\,1_{\Omega_{c}}+\gamma_{v}(\tilde{v}-v_{d})\quad\mbox{
in }Q,\\\ {\lambda}(0)=0,\ {\eta}(0)&=&0\quad\mbox{ in }\Omega,\\\
\dfrac{\partial{\lambda}}{\partial{\boldsymbol{n}}}=0,\
\dfrac{\partial{\eta}}{\partial{\boldsymbol{n}}}&=&0\quad\mbox{ on
}(0,T)\times\partial\Omega,\end{array}\right.$

The proof follows the same lines and it will be omitted. Indeed, the key point
is to show that the set of possible fixed-points

$\widehat{T}_{\alpha}:=\\{(\lambda,\eta)\in X_{2}\times
W_{2}\,:\,(\lambda,\eta)=\alpha\widehat{T}(\lambda,\eta)\mbox{ for some
}{\alpha}\in[0,1]\\}$

is bounded in $X_{2}\times W_{2}$ (with respect to ${\alpha}$). In fact, if
$(\lambda,\eta)\in\widehat{T}_{\alpha},$ then
$(\lambda,\eta)\in{X}_{2}\times{W}_{2}$ and solves the coupled linear problem:

(5.28)
$\left\\{\begin{array}[]{rcl}\partial_{s}{\lambda}-\Delta{\lambda}+{\kappa}\,\nabla{\lambda}\cdot\nabla\tilde{v}-\alpha\,r\lambda+2\alpha\,\mu\tilde{u}\lambda-{\alpha}\,{\eta}&=&{\alpha}\,\gamma_{u}(\tilde{u}-u_{d})\quad\mbox{
in }Q,\\\
\partial_{s}{\eta}-\Delta\eta+{\eta}-\tilde{f}\,{\eta}\,1_{\Omega_{c}}-\kappa\,\nabla\cdot(\tilde{u}\nabla{\lambda})&=&{\alpha}\,\gamma_{v}(\tilde{v}-v_{d})\quad\mbox{
in }Q,\\\ {\lambda}(0)=0,\ {\eta}(0)&=&0\quad\mbox{ in }\Omega,\\\
\dfrac{\partial{\lambda}}{\partial{\boldsymbol{n}}}=0,\
\dfrac{\partial{\eta}}{\partial{\boldsymbol{n}}}&=&0\quad\mbox{ on
}(0,T)\times\partial\Omega.\end{array}\right.$

Now, by taking $\lambda-\Delta\lambda\in L^{2}(Q)$ as test function in (5.28)1
and $\eta\in L^{2}(H^{1})$ as test function in (5.28)2, then the following
bound is obtained via the Gronwall Lemma

$\|(\lambda,\eta)\|_{X_{2}\times W_{2}}\leq
C(\|\tilde{u}\|_{W_{2}},\|\tilde{v}\|_{X_{2}},\|\tilde{f}\|_{L^{2+}(Q_{c})},\|u_{d}\|_{L^{2}(Q)},\|v_{d}\|_{L^{2}(Q)}).$

Therefore, by applying Leray-Schauder fixed-point theorem, the existence of a
solution of problem (5.23), $(\lambda,\eta)\in X_{2}\times W_{2}$, is
obtained. The uniqueness of solution is directly deduced from the linearity of
problem (5.23). ∎

In the following result, more regularity and uniqueness of the Lagrange
multiplier $(\lambda,\eta)$ than provided by Theorem 5.10 will be obtained via
uniqueness of the problem (5.23).

######  Theorem 5.14.

Let $\tilde{s}=(\tilde{u},\tilde{v},\tilde{f})\in\mathcal{S}_{ad}$ be a local
optimal solution for the control problem (1.2). Then the Lagrange multiplier,
provided by Theorem 5.10, is unique and satisfies $(\lambda,\eta)\in
X_{2}\times W_{2}$.

###### Proof.

Let $(\lambda,\eta)\in L^{2}(H^{1})\times L^{2-}(Q)$ be a Lagrange multiplier
given in Theorem 5.10, which is a very weak solution of problem (5.23). In
particular, $(\lambda,\eta)$ satisfies (5.20)-(5.21). On the other hand, from
Theorem 5.13, system (5.23) has a unique solution
$(\overline{\lambda},\overline{\eta})\in X_{2}\times W_{2}$. Then, it suffices
to identify $(\lambda,\eta)$ with $(\overline{\lambda},\overline{\eta})$.

With this objective, for any $(U,V)\in\widehat{W}_{2}\times\widehat{X}_{2+}$,
we writte (5.23) for $(\overline{\lambda},\overline{\eta})$ (instead of
$(\lambda,\eta)$), testing the first equation by $U$, and the second one by
$V$, and integrating by parts in $\Omega$, it is obtained

(5.29)
$\int_{0}^{T}\langle\partial_{t}U,\overline{\lambda}\rangle+\int_{0}^{T}\int_{\Omega}(\nabla
U-\kappa\,U\nabla\tilde{v})\cdot\nabla\overline{\lambda}+(-rU+2\mu\tilde{u}U)\overline{\lambda}-U\overline{\eta}=\gamma_{u}\int_{0}^{T}\int_{\Omega}(\tilde{u}-u_{d})U,$
(5.30) $\int_{0}^{T}\int_{\Omega}\bigg{(}\partial_{t}V-\Delta
V+V-\tilde{f}V1_{\Omega_{c}}\bigg{)}\overline{\eta}+\kappa\,\tilde{u}\nabla
V\cdot\nabla\overline{\lambda}=\gamma_{v}\int_{0}^{T}\int_{\Omega}(\tilde{v}-v_{d})V.$

Making the difference between (5.20) for $(\lambda,\eta)$ and (5.29) for
$(\overline{\lambda},\overline{\eta})$, and between (5.21) and (5.30), and
then adding the respective equations, since the right-hand side terms vanish,
it can be deduced

$\displaystyle\int_{0}^{T}\langle\partial_{t}U,\lambda-\overline{\lambda}\rangle_{(H^{1})^{\prime}}+\int_{0}^{T}\int_{\Omega}(\nabla
U-\kappa\,U\nabla\tilde{v}-\kappa\,\tilde{u}\nabla{V})\cdot\nabla(\lambda-\overline{\lambda})$
$\displaystyle+\int_{0}^{T}\int_{\Omega}(-rU+2\mu\tilde{u}U,\lambda-\overline{\lambda})$
$\displaystyle+\int_{0}^{T}\int_{\Omega}\bigg{(}\partial_{t}V-\Delta
V+V-U-\tilde{f}V1_{\Omega_{c}}\bigg{)}(\eta-\overline{\eta})=0.$

Then, if $(U,V)\in\widehat{W}_{2}\times\widehat{X}_{2+}$ is the unique
solution of linear system (5.9) associated to any $(g_{u},g_{v})\in
L^{2}((H^{1})^{\prime})\times L^{2+}(Q)$ (given by Lemma 5.9), we arrive at

(5.31) $\int_{0}^{T}\langle
g_{u},\lambda-\overline{\lambda}\rangle_{(H^{1})^{\prime}}+\int_{0}^{T}\int_{\Omega}g_{v}(\eta-\overline{\eta})=0.$

By density arguments, it is easy to deduce that $\lambda-\overline{\lambda}=0$
and $\eta-\overline{\eta}=0$, which implies that
$(\lambda,\eta)=(\overline{\lambda},\overline{\eta})$. As a consequence of the
regularity of $(\overline{\lambda},\overline{\eta})$, it holds that
$(\lambda,\eta)\in X_{2}\times W_{2}$. ∎

All previous arguments of Section 5 prove Theorem 1.4.

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# Terms of Lucas sequences having a large smooth divisor

Nikhil Balaji  Department of Computer Science and Engineering
Indian Institute of Technology Delhi
New Delhi 110016
India  and  Florian Luca  School of Maths Wits University
1 Jan Smuts, Braamfontein
Johannesburg 2000
South Africa  Research Group in Algebric Structures and Applications
King Abdulaziz University
Abdulah Sulayman
Jeddah 22254
Saudi Arabia  Centro de Ciencias Matemáticas UNAM
Morelia
Mexico
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We show that the $Kn$–smooth part of $a^{n}-1$ for an integer $a>1$ is
$a^{o(n)}$ for most positive integers $n$.

F. L. worked on this paper while visiting the Max Planck Institute for
Software Systems in Saarbrücken, Germany in Fall of 2020. This author thanks
the Institute for hospitality and support

##  1\. Introduction

It is known that if for every $n$, the sequence $\binom{2n}{n}$ can be
computed in $O(\log^{k}{n})$ arithmetic operations for a fixed constant $k$,
then integers can be factored efficiently [4, 6]. We ask if there exist
linearly recurrent sequences which contain many small factors like
$\binom{2n}{n}$. If such sequences exist, they can be used instead of
$\binom{2n}{n}$ to factor integers. This is because the $n$-th term of any
linearly recurrent sequence can be computed in $O(\log{n})$ arithmetic
operations using repeated squaring of the companion matrix [1]. We first set
up some notation to formally state our question.

Let $P(n)$ be the largest prime factor of $n$ and $s_{y}(n)$ be the largest
divisor $d$ of $n$ with $P(d)\leq y$. Thus, $s_{y}(n)$ is the $y$-smooth part
of $n$. Given a sequence ${\bf u}=(u_{n})_{n\geq 0}$ of positive integers we
ask whether we can find $c>1$ and $K$ such that

${\mathcal{A}}_{K,c,{\bf u}}=\\{n:s_{Kn}(u_{n})>c^{n}\\}$

contains many elements. For example, if $u_{n}=\binom{2n}{n}$ is the sequence
of middle binomial coefficients, then ${\mathcal{A}}_{2,2,{\bf u}}$ contains
all the positive integers. The main question we tackle in this paper can be
formally stated as follows.

######  Question 1.1.

Does there exist a linearly recurrent sequence u such that
${\mathcal{A}}_{K,c,{\bf u}}$ is infinite?

Here we address the problem in the simplest case namely $u_{n}=a^{n}-1$ for
some positive integer $a$. Our results are easily extendable to all Lucas
sequences, in particular, the sequence of Fibonacci numbers.

To start we recall the famous $ABC$-conjecture. Put

${\text{\rm rad}}(n)=\prod_{p\mid n}p$

for the algebraic radical of $n$.

######  Conjecture 1.1.

For all $\varepsilon>0$ there exists a constant $K_{\varepsilon}$ such that
whenever $A,B,C$ are coprime nonzero integers with $A+B=C$, then

$\max\\{|A|,|B|,|C|\\}\leq K_{\varepsilon}{\text{\rm
rad}}(ABC)^{1+\varepsilon}.$

Throughout this paper $a>1$ is an integer and $u_{n}=a^{n}-1$.

We have the following result.

######  Theorem 1.1.

Assume the $ABC$ conjecture. Then for any $K>0$, $c>1$, the set
${\mathcal{A}}_{K,c,{\bf u}}$ is finite.

One can ask what can one prove unconditionally. Maybe we cannot prove that
${\mathcal{A}}_{K,c,{\bf u}}$ is finite but maybe we can prove that it is
thin, that is that it does not contain too many integers. This is the content
of the next theorem.

######  Theorem 1.2.

We have

(1) $\\#({\mathcal{A}}_{K,c,{\bf u}}\cap[1,N])\ll N\exp\left(-\frac{\log
N}{156\log\log N}\right).$

In particular, if one wants to find for all large $N$ an interval starting at
$N$ of length $k$, that is $[N+1,\ldots,N+k]$ which has non-empty intersection
with ${\mathcal{A}}_{K,c,{\bf u}}$ then infinitely often one should take
$k>\exp(\log N/(157\log\log N))$. But if the $ABC$ conjecture is true, one
will no longer find elements of ${\mathcal{A}}_{K,c,{\bf u}}$ in the above
interval for large $N$ no matter how large $k$ is.

##  2\. Proofs

###  2.1. The proof of Theorem 1.1

We apply the $ABC$ conjecture to the equation

$a^{n}-1=st,\quad s:=s_{Kn}(u_{n}),\quad t=(a^{n}-1)/s$

for $n\in{\mathcal{A}}_{K,c,{\bf u}}$ with the obvious choices. Note that

${\text{\rm rad}}(s)=\prod_{\begin{subarray}{c}p\leq Kn\\\ p\mid
a^{n}-1\end{subarray}}p\quad{\text{\rm and}}\quad t<(a/c)^{n}.$

We then have

$a^{n}\ll_{\varepsilon}(a\cdot{\text{\rm
rad}}(s)t))^{1+\varepsilon}\ll\left(\prod_{\begin{subarray}{c}p\leq Kn\\\
p\mid a^{n}-1\end{subarray}}p\right)^{1+\varepsilon}(a/c)^{n(1+\varepsilon)}.$

We may of course assume that $1<c<a$. Then

$\sum_{\begin{subarray}{c}p\leq Kn\\\ p\mid a^{n}-1\end{subarray}}\log
p\geq\frac{n}{1+\varepsilon}(\log
a-(1+\varepsilon)\log(a/c))+O_{\varepsilon}(1).$

We choose $\varepsilon>0$ small enough so that $\log
a-(1+\varepsilon)\log(a/c)>0$. Then, we get

(2) $S_{a,K}(n):=\sum_{\begin{subarray}{c}p\leq Kn\\\ p\mid
a^{n}-1\end{subarray}}\log p\gg_{\varepsilon}n.$

The next lemma shows that the left–hand side above is $\leq n^{2/3+o(1)}$ as
$n\to\infty$. This is unconditional and finishes the proof of Theorem 1.1.

######  Lemma 2.1.

We have

$S_{K,a}(n)\leq K^{1/2}n^{1/2+o(1)}$

as $n\to\infty$.

###### Proof.

Let $\ell_{p}$ be the order of $a$ modulo $p$; that is the smallest positive
integer $k$ such that $a^{k}\equiv 1\pmod{p}$. Since primes $p$ participating
in $S_{K,a}(n)$ have $p\mid a^{n}-1$, it follows that $\ell_{p}\mid n$. Since
also such primes are $O(n)$, it follows that

$S_{K,a}\ll\\#P_{K,n}\log n,$

where $P_{K,a}(n):=\\{p\leq Kn:\ell_{p}\mid n\\}$. To estimate $P_{K,a}(n)$ we
fix a divisor $d$ of $n$ and look at primes $p\leq Kn$ such that $\ell_{p}=d$.
Such primes $p$ have the property that $p\equiv 1\pmod{d}$ by Fermat’s Little
Theorem. In particular, the number of such (without using results on primes in
progressions) is at most

$\left\lfloor\frac{Kn}{d}\right\rfloor\leq\frac{Kn}{d}.$

However, since these primes divide $a^{d}-1$, the number of them is $O(d)$.
Thus, for a fixed $d$ the number of such primes is

$\ll\min\left\\{\frac{Kn}{d},d\right\\}\ll(Kn)^{1/2}.$

Summing this up over all divisors $d$ of $n$ we get that

$\\#P_{K,a}(n)\ll d(n)(Kn)^{1/2}\leq K^{1/2}n^{1/2+o(1)}$

as $n\to\infty$, where we used $d(n)$ for the number of divisors of $n$ and
the well-known estimate $d(n)=n^{o(1)}$ as $n\to\infty$ (see Theorem 315 in
[3])). Hence,

$S_{K,a}(n)\ll\\#P_{K,a}(n)\log n\leq K^{1/2}n^{1/2+o(1)}$

as $n\to\infty$, which is what we wanted. ∎

######  Remark.

The current Lemma 2.1 was supplied by the referee. Our initial statement was
weaker. The combination between Lemma 2.1 and estimate (2) shows that we can
even take $K$ growing with $n$ such as $K=n^{1-\varepsilon}$ in the hypothesis
of Theorem 1.1 and retain its conclusion. This has been also noticed in [5].

###  2.2. The proof of Theorem 1.2

It is enough to prove an upper bound comparable to the upper bound from the
right–hand side of (1) for $\\#({\mathcal{A}}_{K,c,{\bf u}}\cap(N/2,N])$ as
then we can replace $N$ by $N/2$, then $N/4$, etc. and sum up the resulting
inequalities. So, assume that $n\in(N/2,N]$. We estimate

$Q_{N}:=\prod_{n\in(N/2,N]}s_{KN}(u_{n}).$

On the one hand, since $s_{KN}(u_{n})\geq s_{Kn}(u_{n})\geq c^{n}\geq c^{N/2}$
for all $n\in{\mathcal{A}}_{K,c,{\bf u}}$, we get that

$\log Q_{N}\gg N(\\#{\mathcal{A}}_{K,c,{\bf u}}\cap(N/2,N]).$

Next, writing $\nu_{p}(m)$ for the exponent of $p$ in the factorisation of
$m$, we have

(3) $\log Q_{N}=\sum_{n\in(N/2,N]}\sum_{p\leq KN}\nu_{p}(u_{n})\log
p\leq\sum_{p\leq KN}\log p\sum_{n\in(N/2,N]}\nu_{p}(u_{n}).$

Let $o_{p}:=\nu_{p}(u_{\ell_{p}})$. It is well-known that if $p$ is odd then

$\nu_{p}(u_{n})=\left\\{\begin{matrix}o_{p}+\nu_{p}(n)&{\text{\rm
if}}&\ell_{p}\mid n;\\\ 0&{\text{\rm otherwise}}&\leavevmode\nobreak\ \\\
\end{matrix}\right.$

(see, for example, (66) in [7]). In particular, if $p\mid u_{n}$, then
$p^{o_{p}}\mid u_{n}$. Furthermore, for each $k\geq 0$, the exact power of $p$
in $u_{n}$ is $o_{p}+k$ if and only if $\ell_{p}p^{k}$ divides $n$ and
$\ell_{p}p^{k+1}$ does not divide $n$. When $p=2$, we may assume that $a$ is
odd (otherwise $\nu_{2}(u_{n})=0$ for all $n\geq 1$), and the right–hand side
of the above forrnula needs to be ammended to

$\nu_{2}(u_{n})=\left\\{\begin{matrix}o_{2}&{\text{\rm if}}&2\nmid n;\\\
o_{p}+\nu_{2}(a+1)+\nu_{2}(n/2)&{\text{\rm if}}&2\mid n.\\\
\end{matrix}\right.$

Thus, for odd $p$,

(4) $\sum_{n\in(N/2,N]}\nu_{p}(u_{n})=o(p)\\#\\{N/2<n\leq N:\ell_{p}\mid
n\\}+\sum_{k\geq 1}\\#\\{N/2<n\leq N:\ell_{p}p^{k}\mid n\\}.$

A similar formula holds for $p=2$. In particular, for $p=2$, we have

$\sum_{n\in(N/2,N]}\nu_{2}(u_{n})=O(N).$

Thus, the prime $p=2$ contributes a summand of size $O(N)$ to the right–hand
side of (3). From now on, we assume that $p$ is odd. The first cardinality in
the right–hand side of formula (4) above is

$\\#\\{N/2<n\leq N:\ell_{p}\mid
n\\}\leq\left\lfloor\frac{N}{2\ell_{p}}\right\rfloor+1\ll\frac{N}{\ell_{p}}.$

The remaining cardinalities on the right–above can be bounded as

$\\#\\{N/2<n\leq N:\ell_{p}p^{k}\mid
n\\}\leq\left\lfloor\frac{N}{2\ell_{p}p^{k}}\right\rfloor+1\ll\frac{N}{\ell_{p}p^{k}}.$

Thus,

$\sum_{n\in(N/2,N]}\nu_{p}(u_{n})\ll\frac{No_{p}}{\ell_{p}}+\sum_{k\geq
1}\frac{N}{\ell_{p}p^{k}}\ll\frac{No_{p}}{\ell_{p}}+\frac{N}{\ell_{p}p}.$

We thus get

$\log Q_{N}\ll N\sum_{p\leq Kn}\frac{o_{p}\log p}{\ell_{p}}+N\sum_{p\leq
Kn}\frac{\log p}{\ell_{p}p}\ll N\sum_{p\leq Kn}\frac{o_{p}\log
p}{\ell_{p}}:=S.$

It remains to bound $S$. Since $p^{o_{p}}\mid a^{\ell_{p}}-1$, we get that
$p^{o_{p}}<a^{\ell_{p}}$ so $o_{p}\log p\ll\ell_{p}$. Hence,

$S=N\sum_{p\leq KN}\frac{o_{p}\log p}{\ell_{p}}\ll
N\pi(KN)\ll_{K}\frac{N^{2}}{\log N}.$

We get the first non–trivial upper bound on $\\#({\mathcal{A}}_{K,c,{\bf
u}}\cap(N/2,N]$, namely

$N\\#({\mathcal{A}}_{K,c,{\bf u}}\cap(N/2,N])\ll\log Q_{N}\ll
S\ll\frac{N^{2}}{\log N}+N\log\log N\ll_{K}\frac{N^{2}}{\log N},$

so

$\\#({\mathcal{A}}_{K,c,{\bf u}}\cap(N/2,N])\ll_{K}\frac{N}{\log N}.$

To do better, we need to look more closely at $o_{p}\log p/\ell_{p}$ for
primes $p\leq KN$. We split the sum $S$ over primes $p\leq KN$ in two subsums.
The first is over the primes in the set $Q_{1}$ consisting of $p$ such that
$o_{p}\log p/\ell_{p}<1/y_{N}$, where $y_{N}$ is some function of $N$ which we
will determine later. We let $Q_{2}$ be the complement of $Q_{1}$ in the set
of primes $p\leq Kn$. The sum over primes $p\in Q_{1}$ is

$S_{1}=N\sum_{p\in Q_{1}}\frac{o_{p}\log
p}{\ell_{p}}\leq\frac{N}{y_{N}}\pi(KN)\ll_{K}\frac{N^{2}}{y_{N}\log N}.$

For $Q_{2}$, we use the trivial estimate

$S_{2}=N\sum_{p\in Q_{2}}\frac{o_{p}\log p}{\ell_{p}}\ll N\\#Q_{2},$

and it remains to estimate the cardinality of $Q_{2}$. Note that $Q_{2}$
consists of primes $p$ such that $o_{p}>\ell_{p}/(y_{N}\log
p)\gg\ell_{p}/(y_{N}\log N)$. We put $\ell_{p}$ in dyadic intervals. That is
$\ell_{p}\in(2^{i},2^{i+1}]$ for some $i\geq 0$. Then primes $p\leq KN$ in
$Q_{2}$ with such $\ell_{p}$ have the property that $o_{p}\gg 2^{i}/(y_{N}\log
N)$. Hence,

$\displaystyle\frac{2^{i}\\#(Q_{2}\cap(2^{i},2^{i+1}])}{y_{N}\log N}$
$\displaystyle\ll$ $\displaystyle\sum_{p\in
Q_{2}\cap(2^{i},2^{i+1}]}\nu_{p}(a^{\ell_{p}}-1)\log
p\leq\sum_{\ell\in(2^{i},2^{i+1}]}\log(a^{\ell}-1)$ $\displaystyle\ll$
$\displaystyle\sum_{\ell\in(2^{i},2^{i+1}]}\ell\ll 2^{2i},$

which gives

$\\#(Q_{2}\cap(2^{i},2^{i+1}])\ll 2^{i}y_{N}\log N.$

Summing up over all the $i$, we get

$\\#Q_{2}\leq 2^{I}y_{N}\log N,$

where $I$ is maximal such that $(2^{I},2^{I+1}]$ contains an element $p$ of
$Q_{2}$. By a result of Stewart (see Lemma 4.3 in [7]),

$\displaystyle 2^{I}$ $\displaystyle<$ $\displaystyle\ell_{p}<o_{p}y_{N}\log
N<p\exp\left(-\frac{\log p}{51.9\log\log p}\right)y_{N}\log N\log\ell_{p}$
$\displaystyle\ll$ $\displaystyle
KN\exp\left(-\frac{\log(Kn)}{51.9\log\log(KN)}\right)y_{N}\log(KN)^{2}$
$\displaystyle\ll_{K}$ $\displaystyle N\exp\left(-\frac{\log N}{51.95\log\log
N}\right)y_{N}(\log N)^{2}.$

Thus,

$\displaystyle\\#Q_{2}$ $\displaystyle\ll$ $\displaystyle 2^{I}y_{N}\log
N\ll_{K}N\exp\left(-\frac{\log N}{51.95\log\log N}\right)y_{N}^{2}(\log
N)^{3}$ $\displaystyle\ll$ $\displaystyle N\exp\left(-\frac{\log N}{52\log\log
N}\right)y_{N}^{2}.$

Choosing $y_{N}:={\displaystyle{\exp\left(c\frac{\log N}{\log\log N}\right)}}$
with a positive constant $c$ to be determined later, we get

$\displaystyle N\\#({\mathcal{A}}_{K,c,{\bf u}}\cap(N/2,N])$
$\displaystyle\ll$ $\displaystyle N\\#Q_{2}+\frac{N}{y_{N}\log N}$
$\displaystyle\ll_{K}$ $\displaystyle
N\left(\exp\left(\left(2c-\frac{1}{52}\right)\frac{\log N}{\log\log
N}\right)+\exp\left(-\frac{c\log N}{\log\log N}\right)\right).$

Choosing $c:=1/156$, we get

$\\#({\mathcal{A}}_{K,c,{\bf u}}\cap(N/2,N])\ll N\log N\exp\left(-\frac{\log
N}{156\log\log N}\right),$

which is what we wanted.

### Acknowledgements

We thank the referee for suggesting the current Lemma 2.1 with its proof and
for pointing out reference [5].

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