ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

NULL CONTROLLABILITY OF A MODEL IN POPULATION DYNAMICS

YOUNES ECHARROUDI, LAHCEN MANIAR

Abstract. In this article, we study the null controllability of a linear model with degenerate diffusion in population dynamics. We develop first a Carleman type inequality for the adjoint system of an intermediate model, and then an observability inequality. By a fixed point technique, we establish the existence of a control acting on a subset of the space domain that leads the population of a certain age to extinction in a finite time.

1. Introduction We consider the linear population dynamics model

∂y

∂t +∂y

∂a−(k(x)yx)x+µ(t, a, x)y=ϑχω inQ, y(t, a,1) =y(t, a,0) = 0 on (0, T)×(0, A),

y(0, a, x) =y_{0}(a, x) inQ_{A},
y(t,0, x) =

Z A

0

β(t, a, x)y(t, a, x)da in Q_{T},

(1.1)

where Q= (0, T)×(0, A)×(0,1), QA = (0, A)×(0,1), QT = (0, T)×(0,1) and
we will denote q = (0, T)×(0, A)×ω. The system (1.1) models the dispersion
of a gene in a given population. In this case, x represents the gene type and
y(t, a, x) is the distribution of individuals of age a at time t and of gene type x
of the population. The parametersβ(t, a, x),µ(t, a, x) are respectively the natural
fertility and mortality rates of individuals of ageaat timetand of gene typex,Ais
the maximal age of life of population, andkis the gene dispersion coefficient. The
subsetω= (x_{1}, x_{2})b(0,1) is the region where a control ϑis acting. This control
corresponds to an external supply or to removal of individuals on the subdomain
ω. Finally,RA

0 β(t, a, x)y(t, a, x)dais the distribution of the newborns of population that are of gene typexat timet. The variablexcan also represent a space variable, as in some diffusion population models studied in the literature.

The question of null controllability is widely investigated in many papers, among them we find [1, 2, 3, 4, 17] and the references therein. In [3, 4], the authors

2000Mathematics Subject Classification. 35K65, 92D25, 93B05, 93B07.

Key words and phrases. Population dynamics; Carleman estimate; observability inequality;

null controllability.

c

2014 Texas State University - San Marcos.

Submitted May 7, 2014. Published November 17, 2014.

1

proved the existence of a control that leads the population to its steady state ys. This is equivalent to show the null controllability for the system satisfied by y−ys. To reach this goal, the authors took the adjoint system as a collection of parabolic equations along characteristic lines, and used Carleman and observability inequalities for the heat equation proved in [12]. In [1, 2, 17], following the same strategy of [12], the authors showed a direct Carleman estimate for the backward adjoint system of the population model (1.1) and deduced its null controllability by showing adequate observability inequalities. Note that in [17], Traore considered a nonlinear distribution of newborns under the form F(RA

0 β(t, a, x)y(t, a, x)da). In this contribution and contrary to the previous works, we consider that the dispersion coefficient k in our problem depend on x and degenerate at the left boundary;

i.e., k(0) = 0, e.g. k(x) = x^{α}. In this case, we say that the model (1.1) is a
degenerate population dynamics system. Genetically speaking, this assumption is
naturel because it means that if each population is not of a gene type, then this
gene can not be transmitted to its offspring.

In this context of degeneracy, we will study the null controllability of the de-
generate model (1.1) at each fixed timeT >0. More exactly, we show that for all
y_{0} ∈ L^{2}(Q_{A}) and any δ ∈ (0, A), there exists a control ϑ ∈ L^{2}(q) such that the
associated solution of (1.1) verifies

y(T, a, x) = 0, a.e. in (δ, A)×(0,1). (1.2) Such a control does not depend only on the initial distributiony0, but also on the parameterδ. As in [2] and [17], we prove this result by developing a new Carleman estimate. This will be obtained by following the method of the work done in [6] for degenerate heat equation.

The remainder of this article is organized as follows: in Section 3, we give the functional framework in which system (1.1) is wellposed and provide the proof of the Carleman inequality for an intermediate trivial adjoint system. With the help of this inequality, we establish the observability inequality and show the null controllability of the intermediate system. Using a generalization of the Leray- Schauder fixed point theorem, we will deduce in Section 4 the main result of null controllability of (1.1). The last section is an appendix which is devoted to the proof of a Caccioppoli’s inequality which plays a crucial rule in the proof of the Carleman estimate.

2. Well-posedness result

In this article, we assume that the dispersion coefficientksatisfies the hypotheses
k∈C([0,1])∩C^{1}((0,1]), k >0 in (0,1] andk(0) = 0,

∃γ∈[0,1) :xk^{0}(x)≤γk(x), x∈[0,1]. (2.1)
The above hypothesis on k means in the case of k(x) = x^{α} that 0 ≤ α < 1.

Similarly, all results of this chapter can be obtained also in the case of 1≤α <2 taking, instead of Dirichlet condition, the Newmann condition (k(x)ux)(0) = 0 on x= 0.

On the other hand, we assume that the ratesµandβ satisfy
µ∈L^{∞}(Q), µ≥0 a.e. inQ,

β∈C^{2}([0, T]×[0, A]×[0,1]), β ≥0 a.e. inQ. (2.2)

To prove the well-posedness of (1.1), we introduce the following weighted Sobolev spaces

H_{k}^{1}(0,1) :=

u∈L^{2}(0,1) :uis abs. cont. in [0,1],

√

kux∈L^{2}(0,1), u(1) =u(0) = 0 ,
H_{k}^{2}(0,1) :=

u∈H_{k}^{1}(0,1) :k(x)ux∈H^{1}(0,1) ,

(2.3) endowed respectively with the norms

kuk^{2}_{H}1

k(0,1):=kuk^{2}_{L}2(0,1)+k√

ku_{x}k^{2}_{L}2(0,1), u∈H_{k}^{1}(0,1),
kuk^{2}_{H}2

k

:=kuk^{2}_{H}1

k(0,1)+k(k(x)u_{x})_{x}k^{2}_{L}2(0,1), u∈H_{k}^{2}(0,1). (2.4)
We recall from [10, 11] that the operatorCu:= (k(x)u_{x})_{x},u∈D(C) =H_{k}^{2}(0,1),
is closed self-adjoint and negative with dense domain inL^{2}(0,1).

Using properties of the operatorC, one can show as in [13, 14, 19] the existence
of a unique solution of the model (1.1) and that this solution is generated by aC_{0}-
semigroup on the spaceL^{2}((0, A)×(0,1)). Moreover, this solution has additional
time, age and gene regularity. More precisely, the following well-posedness result
holds.

Theorem 2.1. Under the assumptions (2.1)and (2.2)and for allϑ ∈L^{2}(Q)and
y0 ∈L^{2}(QA), the system (1.1) admits a unique solution y. This solution belongs
to E := C([0, T], L^{2}((0, A)×(0,1)))∩C([0, A], L^{2}((0, T)×(0,1)))∩L^{2}((0, T)×
(0, A), H_{k}^{1}(0,1)). Moreover, the solution of (1.1)satisfies the inequality

sup

t∈[0,T]

ky(t)k^{2}_{L}2(QA)+ sup

a∈[0,A]

ky(a)k^{2}_{L}2(QT)+
Z 1

0

Z A

0

Z T

0

(p

k(x)yx)^{2}dt da dx

≤CZ

q

ϑ^{2}+
Z

Q_{A}

y_{0}^{2}da dx
.

The properties of operator C allow us also to define the root of the operator
B =−C denoted by B^{1/2}. On the other hand, by the definitions (2.3) and (2.4)
and following the same arguments used in the proofs of [18, Propositions 3.5.1,
3.6.1] one can show that D(B^{1/2}) = H_{k}^{1}(0,1). Moreover, the following result is
needed in the sequel. For the proof, see [18, Corollary 3.4.6].

Proposition 2.2. The operatorB defined above has a unique extension

B∈ L(H_{k}^{1}(0,1), H_{k}^{−1}(0,1)), (2.5)
where H_{k}^{−1}(0,1) denotes the dual space of H_{k}^{1}(0,1) with respect to the pivot space
L^{2}(0,1).

3. Null controllability of an intermediate system In this section, we investigate the null controllability of the system

∂y

∂t +∂y

∂a−(k(x)y_{x})_{x}+µ(t, a, x)y=ϑχ_{ω} inQ,
y(t, a,1) =y(t, a,0) = 0 in (0, T)×(0, A),

y(0, a, x) =y0(a, x) inQA,
y(t,0, x) =b(t, x) inQ_{T},

(3.1)

withb∈L^{2}(Q_{T}). To reach this target, we show first a Carleman estimate for the
adjoint system of (3.1).

3.1. Carleman inequalities results. Consider the adjoint system of (3.1),

∂w

∂t +∂w

∂a + (k(x)wx)x−µ(t, a, x)w= 0, w(t, a,1) =w(t, a,0) = 0,

w(T, a, x) =wT(a, x), w(t, A, x) = 0.

(3.2)

We assume thatµsatisfies (2.2),wT ∈L^{2}(QA) and that the coefficient of diffu-
sionksatisfies (2.1). Let us introduce the weight functions

Θ(t, a) := 1

(t(T−t))^{4}a^{4}, ψ(x) :=c1(
Z x

0

r

k(r)dr−c2), ϕ(t, a, x) := Θ(t, a)ψ(x).

(3.3)
For the moment, we suppose thatc2> _{k(1)(2−γ)}^{1} andc1>0. One can observe that
ψ(x)<0,x∈(0,1), or Θ(a, t)→+∞as t→0^{+}, T^{−} anda→0^{+}. The first result
of this paragraph is the following proposition.

Proposition 3.1. Consider the two following systems withh∈L^{2}(Q),

∂w

∂t +∂w

∂a + (k(x)w_{x})_{x}=h,
w(a, t,1) =w(a, t,0) = 0,

w(a, T, x) =wT(a, x), w(A, t, x) = 0,

(3.4)

∂w

∂t +∂w

∂a + (k(x)wx)x−µ(t, a, x)w=h, w(t, a,1) =w(t, a,0) = 0,

w(T, a, x) =wT(a, x), w(t, A, x) = 0.

(3.5)

Then, there exist C > 0 and s0 >0, such that every solutions of (3.4) or (3.5) satisfy, fors≥s0, the inequality

s^{3}
Z

Q

Θ^{3} x^{2}

k(x)w^{2}e^{2sϕ}dt da dx+s
Z

Q

Θk(x)w^{2}_{x}e^{2sϕ}dt da dx

≤CZ

Q

|h|^{2}e^{2sϕ}dt da dx+sk(1)
Z A

0

Z T

0

Θw^{2}_{x}(a, t,1)e^{2sϕ(a,t,1)}dt da
.

(3.6)

Proof. We establish the inequality (3.6) for every solution of system (3.4), and then
deduce the result for the model (3.5). Letwbe the solution of (3.4). The function
ν(t, a, x) :=e^{sϕ(t,a,x)}w(t, a, x) satisfies the system

L^{+}_{s}ν+L^{−}_{s}ν =e^{sϕ}h,

ν(t, a,1) =ν(t, a,0) =ν(T, a, x) =ν(0, a, x) =ν(t, A, x) =ν(t,0, x) = 0, (3.7) where

L^{+}_{s}ν := (k(x)νx)x−s(ϕa+ϕt)ν+s^{2}ϕ^{2}_{x}k(x)ν,
L^{−}_{s}ν:=ν_{t}+ν_{a}−2sk(x)ϕ_{x}ν_{x}−s(k(x)ϕ_{x})_{x}ν.

Passing to the norm in (3.7), one has

kL^{+}_{s}νk^{2}_{L}2(Q)+kL^{−}_{s}νk^{2}_{L}2(Q)+ 2hL^{+}_{s}ν, L^{−}_{s}νi_{L}2(Q)=ke^{sϕ(a,t,x)}hk^{2}_{L}2(Q).
Then, the proof of step one is based on the calculus of the inner producthL^{+}_{s}ν, L^{−}_{s}νi
whose a first expression is given in the following lemma.

Lemma 3.2. The identityhL^{+}_{s}ν, L^{−}_{s}νi=S1+S2 holds with
S_{1}=s

Z

Q

(k(x)ν_{x})^{2}ϕ_{xx}dt da dx−s^{3}
Z

Q

(k(x)ϕ_{x})_{x}k(x)ϕ^{2}_{x}ν^{2}dt da dx
+s^{2}

Z

Q

(ϕ_{a}+ϕ_{t})(k(x)ϕ_{x})_{x}ν^{2}dt da dx
+s

Z

Q

k(x)ν_{x}((k(x)ϕ_{x})_{xx}ν+ (k(x)ϕ_{x})_{x}ν_{x})dt da dx
+s^{3}

Z

Q

(k^{2}ϕ^{3}_{x})_{x}ν^{2}dt da dx−s^{2}
Z

Q

(k(x)(ϕ_{a}+ϕ_{t})ϕ_{x})_{x}ν^{2}dt da dx
+s

2 Z

Q

(ϕ_{at}+ϕ_{tt})ν^{2}dt da dx−s^{2}
2

Z

Q

(ϕ^{2}_{x})_{t}k(x)ν^{2}dt da dx
+s

2 Z

Q

(ϕ_{at}+ϕ_{aa})ν^{2}dt da dx−s^{2}
2

Z

Q

(ϕ^{2}_{x})_{a}k(x)ν^{2}dt da dx,
and

S2= Z A

0

Z T

0

[k(x)νxνa]^{1}_{0}dt da+
Z A

0

Z T

0

[k(x)νxνt]^{1}_{0}dt da
+s^{2}

Z A

0

Z T

0

[k(x)ϕx(ϕa+ϕt)ν^{2}]^{1}_{0}dt da−s^{3}
Z A

0

Z T

0

[k^{2}(x)ϕ^{3}_{x}ν^{2}]^{1}_{0}dt da

−s Z A

0

Z T

0

[k(x)ννx(k(x)ϕx)x]^{1}_{0}dt da−s
Z A

0

Z T

0

[(k(x)νx)^{2}ϕx]^{1}_{0}dt da.

Proof. We have I11=

Z

Q

(k(x)νx)xνtdt da dx= Z A

0

Z T

0

[k(x)νxνt]^{1}_{0}dt da−
Z 1

0

Z A

0

[k(x)

2 ν_{x}^{2}]^{T}_{0} da dx.

By the definition ofν, one has I11=

Z A

0

Z T

0

[k(x)νxνt]^{1}_{0}dt da,
I12=

Z

Q

(k(x)νx)xνadt da dx

= Z A

0

Z T

0

[k(x)ν_{x}ν_{a}]^{1}_{0}dt da−
Z

Q

k(x)ν_{x}ν_{xa}dt da dx

= Z A

0

Z T

0

[k(x)νxνa]^{1}_{0}dt da,

I13= Z

Q

−2sk(x)ϕxνx(k(x)νx)xdt da dx

= Z

Q

−sϕx((k(x)νx)^{2})xdt da dx

=−s Z A

0

Z T

0

[(k(x)ν_{x})^{2}ϕ_{x}]^{1}_{0}dt da+s
Z

Q

(k(x)ν_{x})^{2}ϕ_{xx}dt da dx.

I_{14}=
Z

Q

(−s(k(x)ϕx)_{x}ν)(k(x)ν_{x})_{x}dt da dx

=−s Z A

0

Z T

0

[k(x)νxν(k(x)ϕx)x]^{1}_{0}dt da
+s

Z

Q

k(x)νx(ν(k(x)ϕx)xx+νx(k(x)ϕx)x)dt da dx.

I21=−s Z

Q

(ϕa+ϕt)ννtdt da dx=−s 2

Z

Q

(ϕa+ϕt)(ν^{2})tdt da dx

=s 2

Z

Q

(ϕta+ϕtt)ν^{2}dt da dx,
I22=−s

Z

Q

(ϕa+ϕt)ννadt da dx=−s 2

Z

Q

(ϕa+ϕt)(ν^{2})adt da dx

= s 2 Z

Q

(ϕaa+ϕta)ν^{2}dt da dx,
I23=

Z

Q

(2sk(x)ϕxνx)(s(ϕa+ϕt)ν)dt da dx

=− Z

Q

s^{2}ν^{2}(k(x)(ϕa+ϕt)ϕx)xdt da dx
+s^{2}

Z A

0

Z T

0

[k(x)(ϕa+ϕt)ϕxν^{2}]^{1}_{0}dt da.

I24= Z

Q

(s(ϕa+ϕt)ν)(s(k(x)ϕx)xν)dt da dx

= Z

Q

s^{2}(ϕa+ϕt)(k(x)ϕx)xν^{2}dt da dx.

I31= Z

Q

s^{2}ϕ^{2}_{x}k(x)ννtdt da dx

= Z 1

0

Z A

0

[s^{2}

2ϕ^{2}_{x}k(x)ν^{2}]^{T}_{0} da dx−s^{2}
2

Z

Q

(ϕ^{2}_{x}k(x))_{t}ν^{2}dt da dx

=−s^{2}
2

Z

Q

(ϕ^{2}_{x}k(x))_{t}ν^{2}dt da dx,
I32=s^{2}

Z

Q

ϕ^{2}_{x}k(x)ννadt da dx=−s^{2}
2

Z

Q

(ϕ^{2}_{x}k(x))aν^{2}dt da dx.

I33= Z

Q

(−2sk(x)ϕxνx)(s^{2}ϕ^{2}_{x}k(x)ν)dt da dx

= Z

Q

−s^{3}k^{2}(x)ϕ^{3}_{x}(ν^{2})xdt da dx

=−s^{3}
Z A

0

Z T

0

[k^{2}(x)ϕ^{3}_{x}ν^{2}]^{1}_{0}dt da+s^{3}
Z

Q

(k^{2}(x)ϕ^{3}_{x})_{x}ν^{2}dt da dx.

I_{34}=
Z

Q

−(s(k(x)ϕx)_{x}ν)(s^{2}ϕ^{2}_{x}k(x)ν)dt da dx

=−s^{3}
Z

Q

(k(x)ϕ_{x})_{x}k(x)ϕ^{2}_{x}ν^{2}dt da dx.

By adding all these identities, the result follows.

Back to the proof of Proposition 3.1. Now, using the definitions ofϕandψgiven
in (3.3), the Dirichlet boundary conditions satisfied byν and the assumptionk(0) =
0, the expressions of S_{1} and S_{2} stated in the previous lemma can be simplified
follows,

S1= s 2 Z

Q

(Θaa+ Θtt)ψν^{2}dx dt da+s
Z

Q

Θtaψν^{2}dt da dx
+sc1

Z

Q

Θ(2k(x)−xk^{0}(x))ν_{x}^{2}dt da dx−2s^{2}
Z

Q

Θc^{2}_{1} x^{2}

k(x)(Θa+ Θt)ν^{2}dt da dx
+s^{3}

Z

Q

Θ^{3}c^{3}_{1}( x

k(x))^{2}(2k(x)−xk^{0}(x))ν^{2}dt da dx,
and

S2=−sc1k(1) Z A

0

Z T

0

Θν_{x}^{2}(a, t,1)dt da+ 2s^{3}
Z A

0

Z T

0

Θ^{3}c^{3}_{1}[ x^{3}

k(x)ν^{2}]x=0dt da.

From the third condition in assumptions (2.1), we deduce that the functionx7→_{k(x)}^{x}^{3}
is nondecreasing in (0,1], and then, 0< _{k(x)}^{x}^{3} ≤_{k(1)}^{1} , 0≤_{k(x)}^{x}^{3} ν^{2}≤ _{k(1)}^{1} ν^{2},x∈(0,1].

Hence, lim_{x→0}+ x^{3}

k(x)ν^{2}= 0. Accordingly,
hL^{+}_{s}ν, L^{−}_{s}νi

= s 2 Z

Q

(Θaa+ Θtt)ψν^{2}dt da dx+s
Z

Q

Θtaψν^{2}dt da dx
+sc_{1}

Z

Q

Θ(2k(x)−xk^{0}(x))ν_{x}^{2}dt da dx−2s^{2}
Z

Q

Θc^{2}_{1} x^{2}

k(x)(Θ_{a}+ Θ_{t})ν^{2}dt da dx
+s^{3}

Z

Q

Θ^{3}c^{3}_{1}( x

k(x))^{2}(2k(x)−xk^{0}(x))ν^{2}dt da dx

−sc1k(1) Z A

0

Z T

0

Θν_{x}^{2}(a, t,1)dt da.

Thanks to the third assumption in (2.1), we have S1≥ s

2 Z

Q

(Θaa+ Θtt)ψν^{2}dt da dx+s
Z

Q

Θtaψν^{2}dt da dx
+sc_{1}

Z

Q

Θk(x)ν_{x}^{2}dt da dx−2s^{2}
Z

Q

Θc^{2}_{1} x^{2}

k(x)(Θ_{a}+ Θ_{t})ν^{2}dt da dx
+s^{3}

Z

Q

Θ^{3}c^{3}_{1} x^{2}

k(x)ν^{2}dt da dx.

(3.8)

Now, using the|Θ(Θa+ Θt)| ≤cΘ^{3},we infer for squite large that

| −2s^{2}
Z

Q

Θc^{2}_{1} x^{2}

k(x)(Θa+ Θt)ν^{2}dt da dx|

≤2s^{2}c^{2}_{1}c
Z

Q

x^{2}

k(x)Θ^{3}ν^{2}dt da dx≤ c^{3}_{1}
4s^{3}

Z

Q

x^{2}

k(x)Θ^{3}ν^{2}dt da dx.

(3.9)

On the other hand, we have

|ψ(x)|=|c1l(x)−c1c2| ≤c1| Z x

0

r

k(r)dr|+c1c2≤ c_{1}

(2−γ)k(1) +c1c2, (3.10) and this yields

|s 2

Z

Q

(Θaa+ Θtt)ψν^{2}dt da dx+s
Z

Q

Θtaψν^{2}dt da dx|

≤s c1

(2−γ)k(1)+c1c2

Z

Q

Θaa+ Θtt

2 +|Θta|

ν^{2}dt da dx

≤M s c1

(2−γ)k(1)+c1c2

Z

Q

Θ^{3/2}ν^{2}dt da dx.

(3.11)

By H¨older, Young and Hardy-Poincar´e inequalities (see [6]) and the fact that

∃M1>0 such that Θ^{2}≤M1Θ^{3}, (3.12)
we conclude that

Z 1

0

Θ^{3/2}ν^{2}dx=
Z 1

0

(Θ^{1/2}ν

√k x )(Θν x

√ k)dx

≤CZ 1 0

Θk(x)ν_{x}^{2}dx^{1/2}Z 1
0

Θ^{2}ν^{2} x^{2}

k(x)dx^{1/2}

≤C Z 1

0

Θk(x)ν_{x}^{2}dx+C1

4 Z 1

0

Θ^{3} x^{2}
k(x)ν^{2}dx.

By this and (3.11), we infer that

s 2

Z

Q

(Θaa+ Θtt)ψν^{2}dt da dx+s
Z

Q

Θtaψν^{2}dt da dx

≤sc1C Z

Q

Θk(x)ν_{x}^{2}dt da dx+sc1

4 C2

Z

Q

Θ^{3} x^{2}

k(x)ν^{2}dt da dx.

(3.13)

Takingsmall enough andsquite large, we conclude that

s 2 Z

Q

(Θ_{aa}+ Θ_{tt})ψν^{2}dx dt da+s
Z

Q

Θ_{ta}ψν^{2}dt da dx

≤sc1

4 Z

Q

Θk(x)ν_{x}^{2}dt da dx+c^{3}_{1}s^{3}
4

Z

Q

Θ^{3} x^{2}

k(x)ν^{2}dt da dx.

(3.14)

This involves, combining with the inequalities (3.8) and (3.9) that
S_{1}≥K_{1}s^{3}

Z

Q

Θ^{3} x^{2}

k(x)ν^{2}dt da dx+K_{2}s
Z

Q

Θk(x)ν_{x}^{2}dt da dx.

Therefore,

2hL^{+}_{s}ν, L^{−}_{s}νi ≥m
s^{3}

Z

Q

Θ^{3} x^{2}

k(x)ν^{2}dt da dx+s
Z

Q

Θk(x)ν_{x}^{2}dt da dx

−2sc1k(1) Z A

0

Z T

0

Θν_{x}^{2}(a, t,1)dt da.

Hence, we obtain the following Carleman estimate for (3.7)
s^{3}

Z

Q

Θ^{3} x^{2}

k(x)ν^{2}dt da dx+s
Z

Q

Θk(x)ν^{2}_{x}dt da dx

≤CZ

Q

h^{2}e^{2sϕ}dt da dx+sk(1)
Z A

0

Z T

0

Θν^{2}_{x}(a, t,1)dt da
.

To return to system (3.4), we use the function changeν(t, a, x) :=e^{sϕ(t,a,x)}w(t, a, x).

This implies that

ν_{x}=sϕ_{x}e^{sϕ}w+e^{sϕ}w_{x}, e^{2sϕ}w^{2}_{x}≤2(ν_{x}^{2}+s^{2}ϕ^{2}_{x}ν^{2}).

Then, inequality (3.6) follows immediately for every solution of system (3.4). To
show this inequality for the solutions of (3.5), we apply the last inequality for the
functionh=h+µw. Hence, there are two positive constantsC ands0 such that,
for alls≥s_{0}, the following inequality holds

s^{3}
Z

Q

Θ^{3} x^{2}

k(x)w^{2}e^{2sϕ}dt da dx+s
Z

Q

Θk(x)w^{2}_{x}e^{2sϕ}dt da dx

≤CZ

Q

|h|^{2}e^{2sϕ}dt da dx+sk(1)
Z A

0

Z T

0

Θw_{x}^{2}(t, a,1)e^{2sϕ(t,a,1)}dt da
.

(3.15)

On the other hand, we have Z

Q

|h|^{2}e^{2sϕ}dt da dx≤2Z

Q

|h|^{2}e^{2sϕ}dt da dx+kµk^{2}_{∞}
Z

Q

|w|^{2}e^{2sϕ}dt da dx
.
Now, applying Hardy-Poincar´e inequality to the functionν :=e^{sϕ}w, we obtain

Z

Q

|w|^{2}e^{2sϕ}dt da dx≤ 1
k(1)

Z

Q

k(x)

x^{2} |w|^{2}e^{2sϕ}dt da dx

≤ C k(1)

Z

Q

k(x)ν_{x}^{2}dt da dx

≤ C k(1)

Z

Q

s^{2}c^{2}_{1}Θ^{2} x^{2}
k(x)ν^{2}+

Z

Q

k(x)e^{2sϕ}w_{x}^{2}dt da dx
.

Thus, Z

Q

|h|^{2}e^{2sϕ}dt da dx≤2hZ

Q

|h|^{2}e^{2sϕ}dt da dx+kµk^{2}_{∞} C
k(1)

Z

Q

s^{2}c^{2}_{1}Θ^{2} x^{2}
k(x)ν^{2}
+

Z

Q

k(x)e^{2sϕ}w^{2}_{x}dt da dxi
.

This implies, using again (3.12) and takingsquite large, that
s^{3}

Z

Q

Θ^{3} x^{2}

k(x)w^{2}e^{2sϕ}dt da dx+s
Z

Q

Θk(x)w^{2}_{x}e^{2sϕ}dt da dx

≤D(

Z

Q

|h|^{2}e^{2sϕ}dt da dx+sk(1)
Z A

0

Z T

0

Θw^{2}_{x}(t, a,1)e^{2sϕ(t,a,1)}dt da
+

Z

Q

s^{2}c^{2}_{1}Θ^{2} x^{2}

k(x)ν^{2}dt da dx+
Z

Q

k(x)e^{2sϕ}w^{2}_{x}dt da dx)

≤CZ

Q

|h|^{2}e^{2sϕ}dt da dx+sk(1)
Z A

0

Z T

0

Θw_{x}^{2}(t, a,1)e^{2sϕ(t,a,1)}dt da
.

This completes the proof.

Now, we can provide the main result of this section, namely an ω-Carleman estimate of the model (3.2).

Theorem 3.3. Assume that k satisfies hypotheses (2.1)and letA >0 andT >0 be given. Then there exist two positive constantsCands0, such that every solution wof (3.2)satisfies, for alls≥s0, the inequality

Z

Q

(sΘkw^{2}_{x}+s^{3}Θ^{3}x^{2}

k w^{2})e^{2sϕ}dt da dx≤C
Z

ω

Z A

0

Z T

0

w^{2}dt da dx. (3.16)
Proof. Let us introduce the smooth cut-off functionξ:R→Rdefined as follows

0≤ξ(x)≤1, ∀x∈R, ξ(x) = 0, x∈[x1+ 2x2

3 ,1], ξ(x) = 1, x∈[0,2x1+x2

3 ].

(3.17)

We define the functionv:=ξw, wherewis the solution of the system (3.2). Using the Carleman estimate obtained for the model (3.5) and Caccioppoli’s inequality stated in Lemma 5.1, one can prove the existence of C > 0 such the following estimate holds

Z 1

0

Z A

0

Z T

0

(sΘkv_{x}^{2}+s^{3}Θ^{3}x^{2}

kv^{2})e^{2sϕ}dt da dx≤C
Z

ω

Z A

0

Z T

0

w^{2}dt da dx. (3.18)
In (x1,1), let us consider the functionz:=ηwwithη= 1−ξ. Sincezis supported
by [0, T]×[0, A]×[x1,1] and in this interval the equation (3.5) is uniformly parabolic,
then we can replace the functionkby a positive function belonging toC^{1}([0,1]) and
which coincides with k on (x1,1) denoted also byk and this implies that (3.5) is
nondegenerate. Moreover, we can prove in a similar manner as in [2] the following
result.

Proposition 3.4. Let z be the solution of

∂z

∂t +∂z

∂a+ (k(x)zx)x−c(t, a, x)z=h inQb, z(t, a,1) =z(t, a,0) = 0 on (0, T)×(0, A),

(3.19)
with h ∈ L^{2}(Q) and k ∈ C^{1}([0,1]) is a positive function. Then, there exist two
positive constants cands_{0}, such that for anys≥s_{0},z satisfies the estimate

Z

Q

(s^{3}φ^{3}z^{2}+sφz_{x}^{2})e^{2sΦ}dt da dx

≤cZ

Q

h^{2}e^{2sΦ}dt da dx+
Z

ω

Z A

0

Z T

0

s^{3}φ^{3}z^{2}e^{2sΦ}dt da dx
,

(3.20)

whereQ:= (0, T)×(0, A)×(0,1), the functions φandΦare defined as follows
φ(t, a, x) = Θ(t, a)e^{κσ(x)}, Θ(t, a) = 1

t^{4}(T−t)^{4}a^{4},
Φ(a, t, x) = Θ(t, a)Ψ(x), Ψ(x) =e^{κσ(x)}−e^{2κkσk}^{∞},

(3.21) (t, a, x)∈Q,κ >0,σis a function satisfying

σ∈C^{2}([0,1]), σ(x)>0 in (0,1), σ(0) =σ(1) = 0,

σx(x)6= 0 in[0,1]\ω0, (3.22)
whereω_{0}bω is an open subset.

The existence of the function σ is proved in [12]. Hence, applying Proposition
3.4 to the functionzandh= (kη_{x}w)_{x}+kη_{x}w_{x}, using the definitions ofη,σ,φand
Φ and thanks again to the Caccioppoli’s inequality we obtain the estimate

Z

Q

(s^{3}φ^{3}z^{2}+sφz_{x}^{2})e^{2sΦ}dt da dx≤C
Z

ω

Z A

0

Z T

0

w^{2}dt da dx. (3.23)
Taking into account thatw=v+zand using the inequality (3.18), we obtain

Z

Q

(sΘkw_{x}^{2}+s^{3}Θ^{3}x^{2}

k w^{2})e^{2sϕ}dt da dx

≤2 Z

Q

(s^{3}Θ^{3} x^{2}

k(x)z^{2}+sΘk(x)z^{2}_{x})e^{2sϕ}dt da dx
+ 2

Z

Q

(s^{3}Θ^{3} x^{2}

k(x)v^{2}+sΘk(x)v_{x}^{2})e^{2sϕ}dt da dx

≤C Z

ω

Z A

0

Z T

0

w^{2}dt da dx+ 2
Z

Q

(s^{3}Θ^{3} x^{2}

k(x)z^{2}+sΘk(x)z_{x}^{2})e^{2sϕ}dt da dx.

(3.24)

On the other hand, by the definition ofϕ, taking
c1≥ k(1)(2−γ)(e^{2κkσk}^{∞}−1)

c2k(1)(2−γ)−1 ,

one can prove the existence ofς >0, such that, for all (t, a, x)∈[0, T]×[0, A]×[x1,1], we have

Θk(x)e^{2sϕ}≤ςφe^{2sΦ},Θ^{3} x^{2}

k(x)e^{2sϕ}≤ςφ^{3}e^{2sΦ}.

Using this and the relation (3.23) it follows that Z

Q

(s^{3}Θ^{3} x^{2}

k(x)z^{2}+sΘk(x)z^{2}_{x})e^{2sϕ}dt da dx

= Z 1

x_{1}

Z A

0

Z T

0

(s^{3}Θ^{3} x^{2}

k(x)z^{2}+sΘk(x)z_{x}^{2})e^{2sϕ}dt da dx

≤ς Z 1

x_{1}

Z A

0

Z T

0

(s^{3}φ^{3}z^{2}+sφz^{2}_{x})e^{2sΦ}dt da dx

≤ςC Z

ω

Z A

0

Z T

0

w^{2}dt da dx.

Finally, using the last inequality and (3.24) we obtain the Carleman estimate (3.16).

3.2. An observability inequality result. This paragraph is devoted to the ob- servability inequality of the system (3.2). This inequality is obtained by using our Carleman estimate (3.16) and Hardy-Poincar´e inequality, see [6].

Proposition 3.5. Assume that ksatisfies the hypotheses (2.1). LetA >0,T >0
and0< δ≤min(T, A). Take w_{T} such that

w_{T}(a, x) = 0 a.e. in(0, δ)×(0,1). (3.25)
Then, there isCδ>0 such that every solutionwof (3.2)satisfies the observability
inequality

Z 1

0

Z T

0

w^{2}(t,0, x)dt dx+
Z 1

0

Z A

0

w^{2}(0, a, x)da dx

≤Cδ

Z

ω

Z A

0

Z T

0

w^{2}(t, a, x)dt da dx.

(3.26)

For the proof, we need to show a crucial technical result. For this, consider the following wholes, see [17],

N1={(t, a)∈(0, T)×(0, A);t≥a+T−δ},
N2={(t, a)∈(0, T)×(0, A);t≤a+δ−A},
D_{1}={(t, a)∈(0, T)×(0, A);t≤ −T−^{δ}_{2}

A−^{δ}_{2}a+T −δ
2},
D2={(t, a)∈(0, T)×(0, A);a≥ −A−^{δ}_{2}

T−^{δ}_{2}t+A−δ(δ−2A)
2(2T−δ)},
D3= (0, T)×(0, A)−(D1∪D2), D4={(t, a)∈D3; (a, t)∈/N1∪N2}.

(3.27)

See Figure 1.

Lemma 3.6. Suppose that (3.25) holds. Then all solutions of (3.2)satisfy w(t, a, x) = 0, a.e. in(N1∪N2)×(0,1).

Proof. Let (t_{0}, a_{0})∈N_{1}. Then, we havet_{0}=a_{0}+T−δ+dwith 0≤d < δ. Therefore
a_{0}< δ−d. LetS_{d}={(t0+r, a_{0}+r), r∈(0, δ−d−a_{0})}be a characteristic line in

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

↑

→ t

T

T−δ

δ N1

D2

D_{4}

D_{1}

N2

A−δ A a

Figure 1. Decomposition of the region (0, T)×(0, A)

N_{1}. Settingw(r, x) =w(t_{0}+r, a_{0}+r, x) andeµ(r, x) =µ(t_{0}+r, a_{0}+r, x), wherew
is the solution of (3.2). Then,wsolves

∂w

∂r + (k(x)w_{x})_{x}−eµ(r, x)w= 0, in (0, δ−d−a_{0})×(0,1),
w(r,1) =w(r,0) = 0, on (0, δ−d−a0),

w(δ−d−a_{0}, x) =w(T, δ−d, x) =w_{T}(δ−d, x), in (0,1).

(3.28)

Hence,wis given by

w(r,·) =L(δ−d−a0−r)w(δ−d−a0,·), (3.29)
where (L(l))_{l≥0} is the semigroup generated by the operator Cw = (kw_{x})_{x}−µw.e
Therefore, (3.25) and (3.29) lead tow= 0. Thus, for a. e. d∈(0, δ),w= 0 onS_{d}.
Subsequently,w= 0 inN_{1}×(0,1). Arguing in the same way forN_{2} and the fact
thatw(t, A, x) = 0 in (0, T)×(0,1), we can show thatw= 0 inN_{2}×(0,1) and this

achieves the proof.

Proof of Proposition 3.5. Consider a smooth cut-off function ρ_{1} ∈ C_{0}^{∞}(R^{2},[0,1])
stated as follows ρ1(t, a) = 1,(t, a) ∈D1, ρ1(t, a) = 0,(t, a) ∈D2, ρ1 > 0,(t, a) ∈
D3. The functionwe=ρ1wsatisfies the system

∂we

∂t +∂we

∂a + (k(x)wex)x−µ(t, a, x)we= (∂ρ1

∂t +∂ρ1

∂a)w in Q, w(t, a,e 1) =w(t, a,e 0) = 0 on (0, T)×(0, A),

w(T, a, x) = 0e inQA, w(t, A, x) = 0e in QT.

(3.30)

Multiplying (3.30) byw, integrating overe Q, using the definition ofρ1 and Lemma 3.6, we obtain

Z 1

0

Z A−δ

0

w^{2}(0, a, x)da dx+
Z 1

0

Z T−δ

0

w^{2}(t,0, x)dt dx

≤ −2 Z

Q

(∂ρ1

∂t +∂ρ1

∂a)ρ_{1}w^{2}dt da dx

≤M_{δ}
Z 1

0

Z

D_{4}

w^{2}dt da dx.

Thanks to Hardy-Poincar´e inequality we conclude that Z 1

0

Z A−δ

0

w^{2}(0, a, x)da dx+
Z 1

0

Z T−δ

0

w^{2}(t,0, x)dt dx

≤dδ

Z 1

0

Z

D4

k(x)w^{2}_{x}dt da dx,

(3.31)

withdδ =^{CM}_{k(1)}^{δ}. Observe that Θ is bounded inD4to infer that
Z 1

0

Z A

0

w^{2}(0, a, x)da dx+
Z 1

0

Z T

0

w^{2}(t,0, x)dt dx

≤Cδ

Z 1

0

Z

D4

Θk(x)w_{x}^{2}e^{2sϕ}dt da dx.

(3.32)

Taking slarge and thanks to the Carleman inequality stated in Theorem 3.3, we obtain the observability inequality of system (3.2).

3.3. Null controllability of the intermediate system. This paragraph is de- voted to study the null controllability of system (3.1). For this, let > 0 and consider the following cost function

J(ϑ) = 1 2

Z 1

0

Z A

δ

y^{2}(T, a, x)da dx+1
2

Z

q

ϑ^{2}(t, a, x)dt da dx.

We can prove that J is continuous, convex and coercive. Then, it admits at least one minimizerϑ and, arguing as in [5] or [7, Chapter 5], we have

ϑ=−w(t, a, x)χω(x) in Q, (3.33) withwis a solution of the following system

∂w

∂t +∂w

∂a + (k(x)(w)x)x−µ(t, a, x)w= 0 inQ,
w_{}(t, a,1) =w_{}(t, a,0) = 0 on (0, T)×(0, A),

w(T, a, x) = 1

y(T, a, x)χ_{(δ,A)}(a) inQA,
w(t, A, x) = 0 in QT,

(3.34)

andy is the solution of system (3.1) associated to the controlϑ.

Multiplying (3.34) byy_{}, integrating overQ, using (3.33) and Young inequality
we obtain that

1

Z 1

0

Z A

δ

y_{}^{2}(T, a, x)da dx+
Z

q

ϑ^{2}_{}(t, a, x)dx dt da

= Z

QT

b(t, x)w(t,0, x)dt dx+ Z

QA

y0(a, x)w(0, a, x)da dx

≤ 1 4Cδ

Z

Q_{T}

w^{2}_{}(t,0, x)dt dx+
Z

Q_{A}

w^{2}_{}(0, a, x)da dx

+Cδ

Z

QT

b^{2}(t, x)dt dx+
Z

QA

y_{0}^{2}(a, x)da dx
,

with C_{δ} is the constant given in Proposition 3.5. Hence, by the observability in-
equality (3.26), we conclude that

1

Z 1

0

Z A

δ

y^{2}_{}(T, a, x)da dx+
Z

q

ϑ^{2}_{}(t, a, x)dt da dx

≤1 4

Z

q

w^{2}_{}(t, a, x)dt da dx+Cδ

Z

Q_{T}

b^{2}(t, x)dt dx+
Z

Q_{A}

y^{2}_{0}(a, x)da dx
.
Hence, (3.33) yields

1

Z 1

0

Z A

δ

y_{}^{2}(T, a, x)da dx+3
4

Z

q

ϑ^{2}_{}(t, a, x)dt da dx

≤C_{δ}Z

Q_{T}

b^{2}(t, x)dt dx+
Z

Q_{A}

y^{2}_{0}(a, x)da dx
,

(3.35)

and this yields Z 1

0

Z A

δ

y_{}^{2}(T, a, x)dx da≤C_{δ}Z

Q_{T}

b^{2}(t, x)dx dt+
Z

Q_{A}

y^{2}_{0}(a, x)dx da
,
Z

q

ϑ^{2}_{}(t, a, x)dt da dx≤4C_{δ}
3

Z

Q_{T}

b^{2}(t, x)dt dx+
Z

Q_{A}

y_{0}^{2}(a, x)da dx
.

(3.36)

Then, we can extract two subsequences of y_{} and ϑ_{} denoted also by ϑ_{} and y_{}
that converge weakly towards ϑ and y in L^{2}(q) and L^{2}((0, T)×(0, A);H_{k}^{1}(0,1))
respectively. Furthermore,y is the unique solution of (3.1) that satisfies (1.2). In
summary, we showed the following proposition.

Proposition 3.7. For anyδ >0assumed to be small enough, for ally0∈L^{2}(QA),
there exists a control ϑ ∈ L^{2}(q) such that the associated solution of system (3.1)
verifies (1.2).

4. Main null controllability result

Now, after establishing the null controllability of system (3.1) we are ready to provide the one of the model (1.1). More precisely, we have the following theorem.

Theorem 4.1. For any δ >0 assumed to be small enough, for all y0 ∈L^{2}(QA),
there exists a control ϑ ∈ L^{2}(q) such that the associated solution of system (1.1)
verifies (1.2).

To prove this result, letλbe a positive constant. A more precise restriction will
be given later. Putye=e^{−λt}y. Thenyesolves

∂ye

∂t +∂ey

∂a−(k(x)yex)x+µ1(t, a, x)ey=ϑχe ω in Q, y(t, a,e 1) =ey(t, a,0) = 0 on (0, T)×(0, A),

ey(0, a, x) =y0(a, x) inQA, ey(t,0, x) =

Z A

0

β(t, a, x)ey(t, a, x)da in QT,

(4.1)

withϑe=e^{−λt}ϑandµ1=µ+λ. Now, consider the system

∂ye

∂t +∂ey

∂a−(k(x)ye_{x})_{x}+µ_{1}(t, a, x)ey=ϑχe _{ω} in Q,
y(t, a,e 1) =ey(t, a,0) = 0 on (0, T)×(0, A),

ey(0, a, x) =y0(a, x) inQA,
y(t,e 0, x) =b(t, x) inQ_{T},

(4.2)

with b ∈ L^{2}(Q_{T}). Thus, showing Theorem 4.1 is equivalent to show the null
controllability of system (4.1). For this, we consider the following multi-valued
mapping

Λδ:L^{2}(QT)→ P(L^{2}(QT))
defined, for every smallδ >0 andR∈L^{2}(QT), by

Λ_{δ}(R) =
Z A

0

βydae :yesatisfies (1.2) and (4.2) forb=R, andϑesatisfies (3.36) . To prove that model (4.1) is null controllable, it is sufficient to prove that the multivalued mapping admits a fixed point and this by using a generalization of the Leray-Schauder fixed point theorem stated in [8]. To use this generalization, we introduce the set

Nδ ={R∈L^{2}(QT) :∃ρ∈(0,1), R∈ρΛδ(R)}. (4.3)
The existence of a fixed point of the multi-valued mapping Λδ is an immediate
consequence of the following proposition.

Proposition 4.2. (i) for allR∈L^{2}(QT),Λδ(R)is a closed and convex set.

(ii) Λ_{δ} is upper semi-continuous onL^{2}(Q_{T}).

((iii) Λ_{δ} :L^{2}(Q_{T})→P(L^{2}(Q_{T}))is a compact multivalued mapping.

(iv) N_{δ} is bounded inL^{2}(Q_{T}).

Proof. The proofs of (i) and (ii) are similar to the ones of (ii) and (iv) in [17],
with R (respectively R_{n}) instead ofe^{−λ}^{0}^{t}F(e^{λ}^{0}^{t}R) (respectively e^{−λ}^{0}^{t}F(e^{λ}^{0}^{t}R_{n}))
and the convergence space of the subsequence ofyen isL^{2}((0, A)×(0, T), H_{k}^{1}(0,1))
instead of the spaceL^{2}((0, A)×(0, T), H_{0}^{1}(0,1)).

Now, we address the proof of (iii). Let R ∈ L^{2}(QT) such that kRk_{L}2(QT) ≤
K, K >0. We have to prove that any sequence of elements of Λδ(R) admits a
convergent subsequence. Let (ρn)n ⊆Λδ(R). From the definition of Λδ, for all n
there exists (ϑen,eyn)∈L^{2}(q)×L^{2}(Q) such thatρn =RA

0 βyenda, ϑen verifies (3.36) andyen, the associated solution of (4.2) verifies (1.2). Then, by (3.36) we have

Z

q

ϑe^{2}_{n}(t, a, x)dt da dx≤4C_{δ}
3

Z

Q_{T}

R^{2}(t, x)dt dx+
Z

Q_{A}

y_{0}^{2}(a, x)da dx

≤4C_{δ}
3

K^{2}+
Z

Q_{A}

y_{0}^{2}(a, x)da dx
.

(4.4)

Hence, ϑe_{n} is bounded in L^{2}(q). Thus, there exists a subsequence of ϑe_{n} denoted
byϑe_{n}_{k} that converges weakly towardsϑeinL^{2}(q). On the other hand, multiplying

(4.2) byyen, integrating overQ, using Young inequality, we infer that Z

Q

k(x)(ye_{n})^{2}_{x}dt da dx+λ
Z

Qye_{n}^{2}dt da dx

≤ 1 2λ

Z

q

ϑe^{2}_{n}(t, a, x)dt da dx+λ
2
Z

Qey^{2}_{n}dt da dx
+1

2 Z

Q_{A}

y_{0}^{2}(a, x)da dx+
Z

Q_{T}

R^{2}(t, x)da dx
.

(4.5)

This implies Z

Q

k(x)(yen)^{2}_{x}dt da dx+λ
2
Z

Qey^{2}dt da dx

≤ 1 2λ

Z

q

ϑe^{2}_{n}(t, a, x)dt da dx+1
2

Z

QA

y_{0}^{2}(a, x)da dx+
Z

QT

R^{2}(t, x)da dx
.

(4.6)

Takingλ≥2 and using (4.4), (4.6) becomes Z

Q

k(x)(ye_{n})^{2}_{x}dt da dx+

Z

Qye^{2}dt da dx≤1
2+Cδ

3

K^{2}+
Z

QA

y^{2}_{0}(a, x)da dx
. (4.7)
Therefore,yen is bounded inL^{2}((0, T)×(0, A), H_{k}^{1}(0,1)). Hence, we can extract a
subsequence ofyen denoted byyen_{k}_{1} that converges weakly toward ey in L^{2}((0, T)×
(0, A), H_{k}^{1}(0,1)). Now, we considerρn_{k}_{1} =RA

0 βyen_{k}_{1}dathe subsequence ofρn asso-
ciated toye_{n}_{k}

1. Using (2.2), we conclude thatρ_{n}_{k}

1 satisfies the system

∂ρn_{k}_{1}

∂t −(k(x)(ρn_{k}_{1})x)x+
Z A

0

βµ1yen_{k}_{1}da=zn_{k}_{1} inQT,
ρn_{k}_{1}(t,1) =ρn_{k}_{1}(t,0) = 0 on (0, T),

ρn_{k}_{1}(0, x) =
Z A

0

β(0, a, x)y0(a, x)da in (0,1),

(4.8)

with,

z_{n}_{k}

1 = Z A

0

βϑe_{n}_{k}

1χ_{ω}da+
Z A

0

(β_{t}+β_{a})ey_{n}_{k}

1da

−Z A 0

k(x)βx(yen_{k}_{1})xda+
Z A

0

(k(x)βxyen_{k}_{1})xda
.

Taking into account the assumptions onk, using Hardy-Poincar´e and Minkowski’s
inequalities and exploiting the inequalities (4.4) and (4.7) forϑen_{k}_{1} andeyn_{k}_{1} respec-
tively, we deduce that

kznk1k^{2}_{L}2(QT)≤D_{δ}(K^{2}+
Z

Q_{A}

y^{2}_{0}(a, x)da dx). (4.9)
Now, multiplying the first equation of system (4.8) by ρn_{k}_{1}, integrating over QT

and using Young inequality, we obtain Z

QT

k(x)(ρn_{k}_{1})^{2}_{x}dt dx+λ
2
Z

QT

ρ^{2}_{n}_{k}

1dt dx

≤ 1 2λ

Z

Q_{T}

z^{2}_{n}_{k}

1dt dx+Cβ

2 Z

Q_{A}

y_{0}^{2}(a, x)da dx.

(4.10)