Null controllability for a degenerate population model in divergence form via Carleman estimates

Genni Fragnelli

doi:10.1515/anona-2020-0034

Article Open Access

Null controllability for a degenerate population model in divergence form via Carleman estimates

Genni Fragnelli

Published/Copyright: October 15, 2019

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From the journal Advances in Nonlinear Analysis Volume 9 Issue 1

Abstract

In this paper we consider a degenerate population equation in divergence form depending on time, on age and on space and we prove a related null controllability result via Carleman estimates.

Keywords: population equation; degenerate equation; Carleman estimates; observability inequalities

MSC 2010: 35K65; 92D25; 93B05; 93B07

1 Introduction

We consider the following population model in divergence form describing the dynamics of a single species:

∂y∂t+∂y∂a−(k(x)yx)x+μ(t,a,x)y=f(t,a,x)χωin Q,y(t,a,1)=y(t,a,0)=0on QT,A,y(0,a,x)=y0(a,x)in QA,1,y(t,0,x)=∫0Aβ(a,x)y(t,a,x)dain QT,1. (1.1)

Here Q := (0, T) × (0, A) × (0, 1), Q_T,A := (0, T) × (0, A), Q_A,1: = (0, A) × (0, 1) and Q_T,1 := (0, T) × (0, 1). Moreover, y(t, a, x) is the distribution of certain individuals at location x ∈ (0, 1), at time t ∈ (0, T), where T is fixed, and of age a ∈ (0, A). A is the maximal age of life, while β and μ are the natural fertility and the natural death rates, respectively. Thus, the formula ∫0A βy da denotes the distribution of newborn individuals at time t and location x. In the model χ_ω is the characteristic function of the control region ω ⊂ (0, 1); the function k is the dispersion coefficient and we assume that it depends on the space variable x and degenerates at the boundary of the state space. We say that the function k is

Definition 1.1

Weakly degenerate (WD) if k ∈ W^1,1([0, 1]),

k>0 in (0,1) and k(0)=k(1)=0,

and there exist M₁, M₂ ∈ (0, 1) such that xk′(x) ≤ M₁k(x) and (x – 1)k′(x) ≤ M₂k(x) for a.e. x ∈ [0, 1].

Definition 1.2

Strongly degenerate (SD) if k ∈ W^1,∞([0, 1]),

k>0 in (0,1) and k(0)=k(1)=0,

and there exist M₁, M₂ ∈ [1, 2) such that xk′(x) ≤ M₁k(x) and (x – 1)k′(x) ≤ M₂k(x) for a.e. x ∈ [0, 1].

For example, as k one can consider k(x) = x^α(1 – x)^β, α, β > 0. Clearly, we say that k is weakly or strongly degenerate only at 0 if Definition 1.1 is satisfied only at 0, i.e. k ∈ W^1,1([0, 1]), k > 0 in(0, 1], k(0) = 0 and, there exists M₁ ∈ (0, 1) or M₁ ∈ [1, 2) such that xk′(x) ≤ M₁k(x) for a.e. x ∈ [0, 1]. Analogously at 1.

In the last centuries, population models have been widely investigated by many authors from many points of view (see, for example, [5], [9], [14], [20]). From the general theory for the Lotka-McKendrick system, it is known that the asymptotic behavior of the solution depends on the so called net reproduction rate R₀: if R₀ > 1, the solution is exponentially growing; if R₀ < 1, the solution is exponentially decaying; if R₀ = 1, the solution tends to the steady state solution. Clearly, if R₀ > 1 and the system represents the distribution of a damaging insect population or of a pest population, it is very worrying. For example, in 2017 B. Zhong, C. Lv, W. Qin show that the net reproduction rate for the Tirathaba rufivena (which causes a lot of damages for the crop, for example, of fruits and flowers) depends on the temperature: it is 10.40 if the temperature is 28^∘ C and it is 4.13 is the temperature is 20^∘ C (see, for example, [25]); in 2011 S. S. Win, R. Muhamad, Z. A. M. Ahmad, N. A. Adam show that the net reproduction rate for the Nilaparvata lugens (which caused a lot of damages for the rice crop throughout South and South-East Asia since the early 1970’s) was about 10 (see, for example, [24]). For this reason, recently great attention is given to null controllability. For example in [21], where (1.1) models an insect growth, the control corresponds to a removal of individuals by using pesticides.

There are a lot of papers that deal with null controllability for (1.1) when the dispersion coefficient k is a constant or a strictly positive function (see, for example, [3]). If y is independent of a and k degenerates at the boundary or at an interior point of the domain we refer, for example, to [2], [15] and to [17], [18], [19] if μ is singular at the same point of k. To our best knowledge, [1] is the first paper where y depends on t, a and x and the dispersion coefficient k can degenerate. In particular, the authors assume that k degenerates at the boundary (for example k(x) = x^α, being x ∈ (0, 1) and α > 0). Using Carleman estimates for the adjoint problem, the authors prove null controllability for (1.1) under the condition T ≥ A. However, this assumption is not realistic when A is too large. To overcome this problem in [10], the authors used Carleman estimates and a fixed point method via the Leray - Schauder Theorem. However, in [10] the authors consider a dispersion coefficient that can degenerate only at a point of the boundary and they use the fixed point technique in which the birth rate β must be in C²(Q) specially in the proof of [10, Proposition 4.2]. In the recent paper [13], we studied null controllability for (1.1) in non divergence form and with a diffusion coefficient degenerating at a one point of the boundary domain or in an interior point. Observe that, in the case of a boundary degeneracy, we cannot derive the null controllability for (1.1) by the one of the problem in non divergence form or vice versa, see [7]. For this reason here we study the null controllability for (1.1) assuming that k degenerates at the boundary of the domain and T < A completing [1]. We underline that here, contrary to [10] and [13], we assume also that k can degenerate at both points of the boundary domain (see Theorem 4.8) and β is only a continuous function. On the other hand, while in [10] the authors used Carleman estimates, a generalization of the Leray - Schauder fixed point Theorem and the multi-valued theory, here we use only Carleman estimates, some results of [13] and a technique based on cut off functions, making the proof slimmer and easier to read. Moreover, the technique that we use to prove Theorem 4.8 can be applied also to the problem in non divergence form considered in [13], generalizing [13, Theorem 4.8]. Finally, in the proof of the last theorem we make precise a calculation of [13, Theorem 4.8] which was not accurate. Observe that in this paper, as in [13], we do not consider the positivity of the solution, even if it is clearly interesting. This problem is related to the minimum time issue, i.e. given T cannot be arbitrarily small, but this study is still a work in progress, see [23] for related results in non degenerate cases.

A final comment on the notation: by c or C we shall denote universal strictly positive constants, which are allowed to vary from line to line.

2 Well posedness results

On the rates μ and β we assume:

Hypothesis 2.1

The functions μ and β are such that

∙β∈C(Q¯A,1) and β≥0 in QA,1,∙μ∈C(Q¯) and μ≥0 in Q. (2.1)

To prove well posedness of (1.1), we introduce, as in [2], the following Hilbert spaces

Hk1:={u∈L2(0,1)∣u absolutely continuous in [0,1], kux∈L2(0,1) and u(1)=u(0)=0}

and

Hk2:={u∈Hk1(0,1)|kux∈H1(0,1)}.

We have, as in [2] or [16], that the operator

A0u:=(kux)x,D(A0):=Hk2(0,1)

is self–adjoint, nonpositive and generates a strongly continuous semigroup on the space L²(0, 1).

Now, setting Aau:=∂u∂a, we have that

Au:=Aau−A0u,

for

u∈D(A)=u∈L2(0,A;D(A0)):∂u∂a∈L2(0,A;Hk1(0,1)),u(0,x)=∫0Aβ(a,x)u(a,x)da,

generates a strongly continuous semigroup on L²(Q_A,1) := L²(0, A; L²(0, 1)) (see also [4]). Moreover, the operator B(t) defined as

B(t)u:=μ(t,a,x)u,

for u ∈ D(𝓐), can be seen as a bounded perturbation of 𝓐 (see, for example, [2]); thus also (𝓐 + B(t), D(𝓐)) generates a strongly continuous semigroup.

Setting L²(Q) := L²(0, T; L²(Q_A,1)), the following well posedness result holds:

Theorem 2.1

Assume that k is weakly or strongly degenerate at 0 and/or at 1. For all f ∈ L²(Q) and y₀ ∈ L²(Q_A,1), the system (1.1) admits a unique solution

y∈U:=C([0,T];L2(QA,1)))∩L2(0,T;H1(0,A;Hk1(0,1)))

and

supt∈[0,T]∥y(t)∥L2(QA,1)2+∫0T∫0A∥kyx∥L2(0,1)2dadt≤C∥y0∥L2(QA,1)2+C∥f∥L2(Q)2, (2.2)

where C is a positive constant independent of k, y₀ and f.

In addition, if f ≡ 0, then y ∈ C¹([0, T]; L²(Q_A,1)).

For the existence of the solution and the regularity of it we refer, for example, to [11] and [22]. On the other hand, we postpone the proof of (2.2) to the Appendix.

3 Carleman estimates

From the general theory, it is known that null controllability for a linear parabolic system is, roughly speaking, equivalent to the observability for the associated homogeneous adjoint problem (see, for example, [12]). Thus, the key point is to prove such an inequality. A usual strategy in showing the observability inequality is to prove that certain global Carleman estimates hold true for the adjoint operator. Hence, this section is devoted to obtain global Carleman estimates for the operator which is the adjoint of the given one in both the weakly and the strongly degenerate cases. In particular, we consider the following adjoint system associated to (1.1):

∂z∂t+∂z∂a+(k(x)zx)x−μ(t,a,x)z=f,(t,a,x)∈Q,z(t,a,0)=z(t,a,1)=0,(t,a)∈QT,A,z(t,A,x)=0,(t,x)∈QT,1. (3.1)

3.0.0.1 Carleman inequalities when the degeneracy is at 0

In this subsection we will consider the case when k(0) = 0 and we assume that μ satisfies (2.1). On the other hand, on k we make additional assumptions:

Hypothesis 3.1

The function k ∈ C⁰[0, 1] ⋂ C¹(0, 1] is such that k(0) = 0, k > 0 on (0, 1] and there exists M₁ ∈ (0, 2) such that xk′(x) ≤ M₁k(x) for all x ∈ [0, 1]. Moreover, if M₁ ≥ 1 one has to require that there exists θ ∈ (0, M₁], such that the function x↦k(x)xθ is nondecreasing near 0.

We remark that the assumption xa′ ≤ M₁a, with M₁ < 2, is essential in all our results and it is the same made, for example, in [2]. It implies that 1a ∈ L¹(0, 1) and, in particular, if K < 1, then 1a ∈ L¹(0, 1). Thus, the case M₁ ≥ 2 is excluded. Summing up, we will confine our analysis to the case of 1a ∈ L¹(0, 1). This is, however, the interesting case from the viewpoint of null controllability. In fact, if 1a ∉ L¹(0, 1) and y is independent of a, then (1.1) fails to be null controllable on the whole interval [0, 1], and regional null controllability is the only property that can be expected, see [8].

Moreover, the assumption “∃θ such that the function x→k(x)xθ is nondecreasing near 0” is just technical and it is equivalent to the following one: “∃θ such that θa ≤ xa′ near 0”. Clearly, the prototype is k(x) = x^α, α ∈ (0, 2).

Now, let us introduce the weight function

φ(t,a,x):=Θ(t,a)(p(x)−2∥p∥L∞(0,1)), (3.2)

where Θ is as in (4.7) and p(x):=∫0xyk(y)dy. Observe that φ(t, a, x) < 0 for all (t, x) ∈ Q and φ(t, a, x) → – ∞ as t → 0⁺, T^– or a → 0⁺. The following estimate holds:

Theorem 3.1

Assume that Hypothesis 3.1 is satisfied. Then, there exist two strictly positive constants C and s₀ such that every solution v of (3.1) in

V:=L2(QT,A;Hk2(0,1))∩H1(0,T;H1(0,A;Hk1(0,1)))

satisfies, for all s ≥ s₀,

∫QsΘkvx2+s3Θ3x2kv2e2sφdxdadt≤C∫Qf2e2sφdxdadt+sC∫0T∫0AΘ(t,a)[kvx2e2sφ](t,a,1)dadt.

Clearly the previous Carleman estimate holds for every function v that satisfies (3.1) in (0, T) × (0, A) × (B, C) as long as (0, 1) is substituted by (B, C) and k satisfies Hypothesis 3.1 in (B, C).

3.0.0.2 Proof of Theorem 3.1

As a first step assume that μ ≡ 0.

In order to prove Theorem 3.1, we define, for s > 0, the function

w(t,a,x):=esφ(t,a,x)v(t,a,x)

where v is the solution of (3.1) in 𝓥; observe that, since v ∈ 𝓥, w ∈ 𝓥. Clearly, one has that w satisfies

(e−sφw)t+(e−sφw)a+(k(e−sφw)x)x=f(t,a,x),(t,x)∈Q,w(0,a,x)=w(T,a,x)=0,(a,x)∈QA,1,w(t,A,x)=w(t,0,x)=0,(t,x)∈QT,1,w(t,a,0)=w(t,a,1)=0,(t,a)∈QT,A. (3.3)

Defining Lw := w_t + w_a + (kw_x)_x and L_sw := e^sφL(e^–sφw), the equation of (3.3) can be recast as follows

Lsw=Ls+w+Ls−w=esφf,

where

Ls+w:=(kwx)x−s(φt+φa)w+s2kφx2w,Ls−w:=wt+wa−2skφxwx−s(kφx)xw.

As usual, we compute the inner product <Ls+w,Ls−w>L2(Q) whose first expression is given in the following lemma

Lemma 3.1

Assume Hypothesis 3.1. The following identity holds

<Ls+w,Ls−w>L2(Q)=s2∫Q(φtt+φaa)w2dxdadt+s∫Qk(x)(k(x)φx)xxwwxdxdadt−2s2∫Qkφxφtxw2dxdadt−2s2∫Qkφxφxaw2dxdadt+s∫Q(2k2φxx+kk′φx)wx2dxdadt+s3∫Q(2kφxx+k′φx)kφx2w2dxdadt+s∫Qφatw2dxdadt.{D.T.} (3.4)

{B.T.}∫QT,A[kwxwt]01dadt+∫QT,A[kwxwa]01dadt−s2∫QA,1φaw20Tdxda.+∫QT,A[−sφx(k(x)wx)2+s2k(x)φtφxw2−s3k2φx3w2]01dadt+∫QT,A[−sk(x)(k(x)φx)xwwx]01dadt+s2∫QT,A[kφxφaw2]01dadt−12∫QT,1[kwx2]0Adxdt+12∫QT,1[(s2kφx2−s(φt+φa))w2]0Adxdt.

Proof

It results, integrating by parts,

<Ls+w,Ls−w>L2(Q)=I1+I2+I3+I4,

where

I1=∫Q(kwx)x(wt−2skφxwx−s(kφx)xw)dxdadt,I2=∫Q(−sφtw+s2kφx2w)(wt−2skφxwx−s(kφx)xw)dxdadt,I3=∫Q((kwx)x−s(φt+φa)w+s2kφx2w)wadxdadt

and I4=−s∫Qφaw(wt−2skφxwx−s(kφx)xw)dxdadt.

By [2, Lemma 3.1], we get

I1+I2:=s2∫Qφttw2dxdadt+s∫Qk(x)(k(x)φx)xxwwxdxdadt−2s2∫Qk(x)φxφtxw2dxdadt+s∫Q(2k2φxx+k(x)k′φx)wx2dxdadt+s3∫Q(2k(x)φxx+k′φx)k(x)φx2w2dxdadt+∫QT,A[k(x)wxwt]x=0x=1dadt+∫QT,A[−sφx(k(x)wx)2+s2k(x)φtφxw2−s3k2φx3w2]x=0x=1dadt+∫QT,A[−sk(x)(k(x)φx)xwwx]x=0x=1dadt. (3.5)

Next, we compute I₃ and I₄. Integrating by parts, we have

I3=−12∫0T∫01[kwx2]0Adxdt+∫0T∫0A[kwxwa]01dadt+12∫0T∫01[(s2kφx2−s(φt+φa))w2]0Adxdt+s2∫Qφaaw2dxdadt+s2∫Qφtaw2dxdadt−s2∫Qkφxφxaw2dxdadt. (3.6)

On the other hand

I4=s2∫Qφatw2dxdadt−s2∫Qkφxφaxw2dxdadt−s2∫0A∫01φaw20Tdxda+s2∫0T∫0A[kφxφaw2]01dadt. (3.7)

Adding (3.5) - (3.7), (3.4) follows immediately.□

As a consequence of the definition of φ, one has the next estimate:

Lemma 3.2

Assume Hypothesis 3.1. There exist two strictly positive constants C and s₀ such that, for all s ≥ s₀, all solutions w of (3.3) satisfy the following estimate

sC∫QΘkwx2dxdadt+s3C∫QΘ3x2kw2dxdadt≤{D.T.}.

Proof

The distributed terms of <Ls+w,Ls−w>L2(Q) take the form

{D.T.}=s2∫Q(Θtt+Θaa)(p−2∥p∥L∞(0,1))w2dxdadt−2s2∫QΘΘtx2kw2dxdadt−2s2∫QΘΘax2kw2dxdadt+s∫QΘ(2k−k′x)wx2dxdadt+s3∫QΘ3(2k−k′x)x2k2w2dxdadt+s∫QΘta(p−2∥p∥L∞(0,1))w2dxdadt. (3.8)

Now, observe that there exists c > 0 such that

Θμ≤cΘν if 0<μ<ν|ΘΘt|≤cΘ3,|ΘΘa|≤cΘ3,|Θaa|≤cΘ32,|Θtt|≤cΘ32 and |Θta|≤cΘ32. (3.9)

Hence, proceeding as in the proof of [2, Lemma 3.5], one can deduce

(3.8)≥s2∫Q(Θtt+Θaa)(p−2∥p∥L∞(0,1))w2dxdadt−s3C4∫QΘ3x2kw2dxdadt−s3C4∫QΘ3x2kw2dxdadt+Cs∫QΘkwx2dxdadt+s3C∫QΘ3x2kw2dxdadt+s∫QΘta(p−2∥p∥L∞(0,1))w2dxdadt. (3.10)

Now, it results

s∫QΘtt(p−2∥p∥L∞(0,1))≤C2s∫QΘ3/2b(x)w2dxdadt+sC2∫QΘ3/2w2dxdadt,

where b(x)=∫0xyk(y)dy. As in [2, Lemma 3.5], one can estimate the last two terms in the following way

s2∫QΘ3/2b(x)w2dxdadt≤C16s3∫QΘ3x2k(x)w2dxdadt,

for s large enough and

s2∫QΘ3/2w2dxdadt≤C4s∫QΘk(x)wx2dxdadt+C16s3∫QΘ3x2k(x)w2dxdadt.

Hence

s∫QΘttψ(x)w2dxdadt≤C4s∫01Θk(x)wx2dxdadt+C8s3∫QΘ3x2k(x)w2dxdadt.

The same estimate holds also for

∫QΘaa(p−2∥p∥L∞(0,1))dxdadt and ∫QΘta(p−2∥p∥L∞(0,1))dxdadt.

Using the above estimates in (3.10) the thesis follows immediately for s₀ large enough.□

The next lemma holds.

Lemma 3.3

Assume Hypothesis 3.1. The boundary terms in (3.4) become

{B.T.}=−∫QT,A[sΘxkwx2]01dadt. (3.11)

Proof

Using the definition of φ, [2, Lemma 3.6], the boundary conditions of w and proceeding as in [13, Lemma 3.2], the boundary terms of <Ls+w,Ls−w>L2(Q) become

{B.T.}=∫QT,A[kwxwa]01dadt−s2∫QA,1φaw20Tdxda−∫QT,A[sΘxkwx2]01dadt+s2∫QT,A[kφxφaw2]01dadt−12∫QT,1[kwx2]0Adxdt+12∫QT,1[(s2kφx2−s(φt+φa))w2]0Adxdt=−∫QT,A[sΘxkwx2]01dadt.

□

As a consequence of Lemmas 3.3 and 3.2, we have

Proposition 3.1

Assume Hypothesis 3.1. There exist two strictly positive constants C and s₀ such that, for all s ≥ s₀, all solutions w of (3.3) in 𝓥 satisfy

sC∫QΘkwx2dxdadt+s3C∫QΘ3x2kw2dxdadt≤C∫Qf2e2sφ dxdadt+∫QT,A[sΘkwx2](t,a,1)dadt.

Recalling the definition of w, we have v = e^–sφw and v_x = (w_x – sφ_xw)e^–sφ. Thus, Theorem 3.1 follows immediately by Proposition 3.1 when μ ≡ 0.

Now, we assume that μ ≢ 0.

To complete the proof of Theorem 3.1 we consider the function f = f + μv. Hence, there are two strictly positive constants C and s₀ such that, for all s ≥ s₀, the following inequality holds

∫QsΘkvx2+s3Θ3x2kv2e2sφdxdadt≤C∫Qf¯2e2sφdxdadt+sC∫0T∫0AΘ(t,a)[kvx2e2sφ](t,a,1)dadt. (3.12)

On the other hand, we have

∫Q∣f¯∣2e2sφdxdadt≤2(∫Q|f|2e2sφdxdadt+∫Q|μ|2|v|2e2sφdxdadt). (3.13)

Now, if M₁ < 1, applying the Hardy-Poincaré proved in [2, Proposition 2.1] to the function ν := e^sφv, we obtain

∫Q|μ|2|v|2e2sφdxdadt≤∥μ∥∞2∫Qν2dxdadt≤C∫Qν2k(x)x2dxdadt≤C∫Qk(x)νx2dxdadt≤C∫Qk(x)e2sφvx2dxdadt+Cs2∫QΘ2e2sφx2kv2dxdadt. (3.14)

If M₁ ≥ 1, using the Young’s inequality to the function ν := e^sφv, we have

∫Q|μ|2|v|2e2sφdxdadt≤∥μ∥∞2∫Qν2dxdadt≤C∫Qk1/3x2/3ν23/4x2kν21/4dxdadt≤C∫Qk1/3x2/3ν2dxdadt+C∫Qx2kν2dxdadt. (3.15)

Now, consider the function y(x) = (k(x)x⁴)^1/3. Clearly, y(x)=k(x)x2k(x)2/3≤Ck(x) and k1/3x2/3=y(x)x2. Moreover, using Hypothesis 3.1, one has that the function y(x)xq=k(x)xθ13, where q:=4+ϑ3 ∈ (1, 2), is nondecreasing near 0. The Hardy-Poincaré inequality (see [2, Proposition 2.1.]) implies

∫Qk1/3x2/3ν2dxdadt=∫Qyx2ν2dxdadt≤C∫Qy(νx)2dxdadt≤C∫Qk(x)νx2dxdadt≤C∫Qk(x)e2sφvx2dxdadt+Cs2∫QΘ2e2sφx2kv2dxdadt. (3.16)

In any case, by (3.14), (3.15) and (3.16),

∫Q|μ|2|v|2e2sφdxdadt≤C∫Qk(x)e2sφvx2dxdadt+Cs2∫QΘ2e2sφx2kv2dxdadt.

Using this last inequality in (3.13), it follows

∫Q|f¯|2~e2sφ dxdadt≤2∫Q|f|2~e2sφ dxdadt+C∫Qk(x)e2sφvx2dxdadt+Cs2∫QΘ2e2sφx2kv2dxdadt≤C∫Q|f|2~e2sφ dxdadt+C∫QΘk(x)e2sφvx2dxdadt+Cs2∫QΘ3e2sφx2kv2dxdadt. (3.17)

Substituting in (3.12), one can conclude

∫QsΘkvx2+s3Θ3x2kv2e2sφdxdadt≤C(∫Q|f|2 e2sφk dxdadt+s∫0T∫0AΘ(t,a)[kvx2e2sφ](t,a,1)dadt),

for all s large enough.

3.0.0.3 Carleman inequalities when the degeneracy is at 1

In this subsection we will consider the case when k(1) = 0. Again μ satisfies (2.1) and on k we make the following assumption:

Hypothesis 3.2

The function k ∈ C⁰[0, 1] ⋂ C¹[0, 1) is such that k(1) = 0, k > 0 on [0, 1) and there exists M₂ ∈ (0, 2) such that (x – 1) k′(x) ≤ M₂k(x) for all x ∈ [0, 1]. Moreover, if M₂ ≤ 1 one has to require that there exists θ ∈ (0, M₂], such that the function x↦k(x)|1−x|θ is nonincreasing near 1.

For Hypothesis 3.2 we can make the same considerations made for Hypothesis 3.1.

As in the previous subsection, let us introduce the weight function

φ¯(t,a,x):=Θ(t,a)(p¯(x)−2∥p¯∥L∞(0,1)), (3.18)

where Θ is as in (4.7) and p¯(x):=∫0xy−1k(y)dy. As before, φ̄(t, a, x) < 0 for all (t, x) ∈ Q and φ̄(t, a, x) → –∞ as t → 0⁺, T^– or a → 0⁺. The following estimate holds:

Theorem 3.2

Assume that Hypothesis 3.2 is satisfied. Then, there exist two strictly positive constants C and s₀ such that every solution v of (3.1) in 𝓥 satisfies, for all s ≥ s₀,

∫QskΘvx2+s3Θ3(x−1)2kv2e2sφ¯dxdadt≤C∫Qf2e2sφ¯dxdadt+sC∫0T∫0AΘ(t,a)[(1−x)vx2e2sφ¯](t,a,0)dadt.

The previous Carleman estimate holds for every function v that satisfies (3.1) in (0, T) × (0, A) × (B, C) as long as (0, 1) is substituted by (B, C) and k satisfies Hypothesis 3.2 in (B, C).

The proof of Theorem 3.2 is analogous to one of Theorem 3.1 so we omit it. However, we underline that in the proof of Theorem 3.1 we use [2, Lemma 3.1] which is proved only if k degenerates at 0; actually we observe that the proof of [2, Lemma 3.1] does not depend on the degeneracy point; hence, it holds also if k(1) = 0. Instead, Lemma 3.3, if k(1) = 0, becomes

−∫QT,A[sΘ(x−1)kwx2]01dadt=−∫QT,A[sΘkwx2](t,a,0)dadt;

thus, Proposition 3.1 can be rewritten in the following way

sC∫QΘkwx2dxdadt+s3C∫QΘ3(x−1)2kw2dxdadt≤C∫Qf2e2sφ¯ dxdadt+∫QT,A[sΘkwx2](t,a,0)dadt.

If μ ≢ 0, we can proceed as in the proof of Theorem 3.1. However, while in that case we use the Hardy-Poincaré inequality proved in [2, Proposition 2.1] which holds only if k(0) = 0, in this case we have to use the following inequality whose proof we postpone to the Appendix.

Proposition 3.2

(Hardy-Poincaré inequalities). Assume that k : [0, 1] ⟶ ℝ₊ is in 𝓒([0, 1]), k(1) = 0, k > 0 on [0, 1)

Case (i):

Hypothesis (HP1): Assume that k is such that there exists θ ∈ (0, 1) such that the function

x⟶k(x)(1−x)θ is nondecreasing in neighbourhood of x=1.

Then, there is a constant C > 0 such that for any function w, locally absolutely continuous on [0, 1), continuous at 1 and satisfying

w(1)=0,and∫01k(x)|w′(x)|2dx<+∞,

the following inequality holds

∫01k(x)(1−x)2w2(x)dx≤C∫01k(x)|w′(x)|2dx. (3.19)

If hypothesis (HP1) is replaced by

Hypothesis (HP1)’: Assume that k is such that there exists θ ∈ (0, 1) such that the function

x⟶k(x)(1−x)θ is nondecreasing in [0,1),

Then, for any function w, locally absolutely continuous on [0, 1), continuous at 1 and satisfying

w(1)=0,and∫01k(x)|w′(x)|2dx<+∞,

the inequality (3.19) holds with the explicit constant C=4(1−θ)2.

Case (ii):

Hypothesis (HP2): Assume that k is such that there exists θ ∈ (1, 2) such that the function

x⟶k(x)(1−x)θ is nonincreasing in a neighbourhood of x=1,

Then, there is a constant C > 0 such that for any function w, locally absolutely continuous on [0, 1) satisfying

w(0)=0,and∫01k(x)|w′(x)|2dx<+∞,

the inequality (3.19) holds.

If hypothesis (HP2) is replaced by

Hypothesis (HP2)’: Assume that k is such that there exists θ ∈ (1, 2) such that the function

x⟶k(x)(1−x)θ is nonincreasing in [0,1).

Then, for any function w, locally absolutely continuous on [0, 1) satisfying

w(0)=0,and∫01k(x)|w′(x)|2dx<+∞,

the inequality (3.19) holds with the explicit constant C=4(1−θ)2.

4 Observability and controllability

In this section we will prove, as a consequence of the Carleman estimates established in Section 3, observability inequalities for the associated adjoint problem of (1.1). From now on, we assume that the control set ω is such that

ω=(α,ρ)⊂⊂(0,1). (4.1)

Moreover, on k and β we assume the following assumptions:

Hypothesis 4.1

The function k ∈ C⁰[0, 1]⋂ C¹(0, 1) is such that k(0) = 0 = k(1), k > 0 on (0, 1) and there exist M₁, M₂ ∈ (0, 2) such that xk′(x) ≤ M₁k(x) and (x – 1)k′(x) ≤ M₂k(x) for all x ∈ [0, 1]. Moreover, one has to require that:

if M₁ ≥ 1, there exists θ ∈ (0, M₁], such that the function x↦k(x)xθ is nondecreasing near 0;
if M₂ ≤ 1, there exists y ∈ (0, M₂], such that the function x↦k(x)|1−x|y is nonincreasing near 1.

Hypothesis 4.2

Assume T < A and suppose that there exists ā ≤ T such that

β(a,x)=0 for all (a,x)∈[0,a¯]×[0,1]. (4.2)

Observe that Hypothesis 4.2 is the biological meaningful one. Indeed, ā is the minimal age in which the female of the population become fertile, thus it is natural that before ā there are no newborns. For other comments on Hypothesis 4.2 we refer to [12].

Under the previous hypotheses, the following observability inequality holds:

Proposition 4.1

Suppose that Hypotheses 3.1 or 3.2 or 4.1 and 4.2 hold. Then, for every δ ∈ (T, A), there exists a strictly positive constant C = C(δ) such that every solution v ∈ 𝓤 of

∂v∂t+∂v∂a+(k(x)vx)x−μ(t,a,x)v+β(a,x)v(t,0,x)=0,(t,x,a)∈Q,v(t,a,0)=v(t,a,1)=0,(t,a)∈QT,A,v(T,a,x)=vT(a,x)∈L2(QA,1),(a,x)∈QA,1v(t,A,x)=0,(t,x)∈QT,1, (4.3)

satisfies

∫0A∫01v2(T−a¯,a,x)dxda≤C∫0δ∫01vT2(a,x)dxda+∫0T∫0A∫ωv2dxdadt. (4.4)

Here v_T(a, x) is such that v_T(A, x) = 0 in (0, 1).

Remark 1

If T = ā, the observability inequality given in the previous proposition is the corresponding of [1, Proposition 3.1], where the authors proved it under different assumptions and with T ≥ A.
Moreover, as in [13], observe that in (4.4) the presence of the integral ∫0δ∫01vT2(a,x)dxda is related to the presence of the term β(a, x) v(t, 0, x) in the equation of (4.3). In fact, estimating such a term using the method of characteristic lines, we obtain the previous integral. Obviously, if v_T(a, x) = 0 a.e. in (0, δ) × (0, 1), we obtain the classical observability inequality.

Before proving Proposition 4.1 we will give some results that will be very helpful. As a first step we introduce the following class of functions

W:={vsolution of (4.3)|vT∈D(A2)},

where D(𝓐²) = {u ∈ D(𝓐) | 𝓐u ∈ D(𝓐)} is densely defined in D(𝓐) (see, for example, [6, Lemma 7.2]) and hence in L²(Q_A,1). Obviously,

W=C1([0,T];D(A))⊂V:=L2(QT,A;Hk2(0,1))∩H1(0,T;H1(0,A;Hk1(0,1)))⊂U.

Proposition 4.2

(Caccioppoli’s inequality). Let ω′ and ω two open subintervals of (0, 1) such that ω′ ⊂⊂ ω ⊂⊂ (0, 1). Let ψ(t, a, x) := Θ(t, a)Ψ(x), where Θ is defined in (4.7) and Ψ ∈ C¹(0, 1) is a strictly negative function. Then, there exist two strictly positive constants C and s₀ such that, for all s ≥ s₀,

∫0T∫0A∫ω′vx2e2sψdxdadt≤C∫0T∫0A∫ωv2dxdadt+∫Qf2e2sψdxdadt, (4.5)

for every solution v of (3.1).

The proof of the previous proposition is similar to the one given in [13], but we repeat it in the Appendix for the reader’s convenience.

Moreover, the following non degenerate inequality proved in [13] is crucial:

Theorem 4.1

[see [13, Theorem 3.2]] Let z ∈ 𝓩 be the solution of (3.1), where f ∈ L²(Q), k ∈ C¹([0, 1]) is a strictly positive function and

Z:=L2(QT,A;H2(0,1)∩H01(0,1))∩H1(0,T;H1(0,A;H01(0,1))).

Then, there exist two strictly positive constants C and s₀, such that, for any s ≥ s₀, z satisfies the estimate

∫Q(s3ϕ3z2+sϕzx2)e2sΦdxdadt≤C(∫Qf2e2sΦdxdadt−sκ∫0T∫0Ake2sΦϕ(zx)2x=0x=1dadt), (4.6)

where the functions ϕ and Φ are defined as follows

ϕ(t,a,x)=Θ(t,a)eκσ(x),Θ(t,a)=1t4(T−t)4a4,Φ(a,t,x)=Θ(t,a)Ψ(x),Ψ(x)=eκσ(x)−e2κ∥σ∥∞, (4.7)

(t, a, x) ∈ Q, κ > 0 and σ(x):=d∫x11k(t)dt, where 𝔡 = ∥k′∥_L^∞(0,1).

Remark 2

The previous Theorem still holds under the weaker assumption k ∈ W^1,∞(0, 1) without any additional assumption.

On the other hand, if we require k ∈ W^1,1(0, 1) then we have to add the following hypothesis: there exist two functions 𝔤 ∈ L¹(0, 1), 𝔥 ∈ W^1,∞(0, 1) and two strictly positive constants 𝔤₀, 𝔥₀ such that 𝔤(x) ≥ 𝔤₀ and

−k′(x)2k(x)∫x1g(t)dt+h0+k(x)g(x)=h(x)fora.e. x∈[0,1].

In this case, i.e. if k ∈ W^1,1(0, 1), the function Ψ in (4.7) becomes

Ψ(x):=−r∫0x1k(t)∫t1g(s)dsdt+∫0xh0k(t)dt−c, (4.8)

where r and 𝔠 are suitable strictly positive functions. For other comments on Theorem 4.1 we refer to [13].

With the aid of Theorems 3.1, 3.2, 4.1 and Proposition 4.2, we can now show ω-local Carleman estimates for (3.1).

Theorem 4.2

Assume Hypothesis 3.1. Then, there exist two strictly positive constants C and s₀ such that every solution v of (3.1) in 𝓥 satisfies, for all s ≥ s₀,

∫QsΘkvx2+s3Θ3x2kv2e2sφdxdadt≤C∫Qf2dxdadt+∫0T∫0A∫ωv2dxdadt.

Proof

Let us consider a smooth function ξ : [0, 1] → ℝ such that

0≤ξ(x)≤1, for all x∈[0,1],ξ(x)=1,x∈[0,(2α+ρ)/3],ξ(x)=0,x∈[(α+2ρ)/3,1].

We define w(t, a, x) := ξ(x)v(t, a, x) where v ∈ 𝓥 satisfies (3.1). Then w satisfies

wt+wa+(kwx)x−μw=ξf+(kξxv)x+ξxkvx=:h,(t,a,x)∈Q,w(t,a,0)=w(t,a,1)=0,(t,a)∈QT,A.

Thus, applying Theorem 3.1 and Proposition 4.2,

∫0T∫0A∫02α+ρ3sΘkvx2+s3Θ3x2kv2e2sφdxdadt=∫0T∫0A∫02α+ρ3sΘkwx2+s3Θ3x2kw2e2sφdxdadt≤∫QsΘkwx2+s3Θ3x2kw2e2sφdxdadt≤C∫Qh2e2sφ dxdadt≤C∫Qf2e2sφdxdadt+∫0T∫0A∫ω′v2dxdadt+∫0T∫0A∫ω′vx2e2sφdxdadt≤C∫Qf2e2sφdxdadt+∫0T∫0A∫ωv2dxdadt, (4.9)

where ω′:=2α+ρ3,α+2ρ3.

Now, consider z = ηv, where η = 1 – ξ and take ᾱ ∈ (0, α). Then z satisfies

zt+za+(kzx)x−μz=ηf+(kηxv)x+ηxkvx=:h,(t,a,x)∈QT,A×(α¯,1)=:Q¯,z(t,a,α¯)=z(t,a,1)=0,(t,a)∈QT,A. (4.10)

Clearly the equation satisfied by z is not degenerate, thus applying Theorem 4.1 and Proposition 4.2, one has

∫Q¯(s3ϕ3z2+sϕzx2)e2sΦdxdadt≤C∫Q¯h2e2sΦdxdadt≤C∫Q¯f2e2sΦdxdadt+∫0T∫0A∫ω′(v2+vx2)e2sΦdxdadt≤C∫Qf2e2sΦdxdadt+∫0T∫0A∫ωv2dxdadt.

Hence

∫0T∫0A∫α+2ρ31(s3ϕ3v2+sϕvx2)e2sΦdxdadt=∫0T∫0A∫α+2ρ31(s3ϕ3z2+sϕzx2)e2sΦdxdadt≤C∫Qf2e2sΦdxdadt+∫0T∫0A∫ωv2dxdadt,

for a strictly positive constant C. Proceeding, for example, as in [16] one can prove the existence of ς > 0, such that, for all (t, a, x) ∈ [0, T] × [0, A] × [ᾱ, 1], we have

e2sφ≤ςe2sΦ,x2k(x)e2sφ≤ςe2sΦ. (4.11)

Thus, for a strictly positive constant C,

∫0T∫0A∫α+2ρ31sΘkvx2+s3Θ3x2kv2e2sφdxdadt≤C∫0T∫0A∫α+2ρ31(s3ϕ3v2+sϕvx2)e2sΦdxdadt≤C∫Qf2e2sΦdxdadt+∫0T∫0A∫ωv2dxdadt. (4.12)

Now, consider α̃ ∈ (α, (2α + ρ)/3), ρ̃ ∈ ((α + 2ρ)/3, ρ) and a smooth function τ : [0, 1] → ℝ such that

0≤τ(x)≤1, for all x∈[0,1],τ(x)=1,x∈[(2α+ρ)/3,(α+2ρ)/3],τ(x)=0,x∈[0,α~]∪[ρ~,1],

and define ζ(t, a, x) := τ(x)v(t, a, x). Clearly, ζ satisfies (4.10) with h := τf + (kτ_xv)_x + τ_xkv_x. Observe that in this case τ_x, τ_xx ≢ 0 in ω¯:=α~,2α+ρ3∪α+2ρ3,ρ~. As before, by Theorem 4.1, Proposition 4.2 and (4.11), we have

∫0T∫0A∫2α+ρ3α+2ρ3sΘkvx2+s3Θ3x2kv2e2sφdxdadt≤C∫0T∫0A∫2α+ρ3α+2ρ3(s3ϕ3v2+sϕvx2)e2sΦdxdadt=C∫0T∫0A∫2α+ρ3α+2ρ3(s3ϕ3ζ2+sϕζx2)e2sΦdxdadt≤C∫Qf2e2sΦdxdadt+∫0T∫0A∫ωv2dxdadt. (4.13)

Adding (4.9), (4.12) and (4.13), the thesis follows.□

Proceeding as before one can prove

Theorem 4.3

Assume Hypothesis 3.2. Then, there exist two strictly positive constants C and s₀ such that every solution v of (3.1) in 𝓥 satisfies, for all s ≥ s₀,

∫QsΘkvx2+s3Θ3(1−x)2kv2e2sφdxdadt≤C∫Qf2dxdadt+∫0T∫0A∫ωv2dxdadt.

Remark 3

Observe that the results of Theorems 4.2 and 4.3 still hold true if we substitute the domain (0, T) × (0, A) with a general domain (T₁, T₂) × (y, A), provided that μ and β satisfy the required assumptions. In this case, in place of the function Θ defined in (4.7), we have to consider the weight function

Θ~(t,a):=1(t−T1)4(T2−t)4(a−y)4. (4.14)

Using the previous local Carleman estimates one can prove the next observability inequalities.

Theorem 4.4

Assume Hypotheses 3.1 or 3.2 and 4.2 with T > ā. Then, for every δ ∈ (0, A), there exists a strictly positive constant C = C(δ) such that every solution v of (4.3) in 𝓥 satisfies

∫0A∫01v2(T−a¯,a,x)dxda≤C∫0T∫0δ∫01v2(t,a,x)dxdadt+C∫0a¯∫01vT2(a,x)dxda+∫0T∫0A∫ωv2dxdadt.

Moreover, if v_T(a, x) = 0 for all (a, x) ∈ (0, ā) × (0, 1), one has

∫0A∫01v2(T−a¯,a,x)dxda≤C∫0T∫0δ∫01v2(t,a,x)dxdadt+∫0T∫0A∫ωv2dxdadt.

Proof

As in [13] and using the method of characteristic lines, one can prove the following implicit formula for v solution of (4.3):

S(T−t)vT(T+a−t,⋅), (4.15)

if t ≥ T̃ + a and

v(t,a,⋅)=S(T−t)vT(T+a−t,⋅)+∫aT+a−tS(s−a)β(s,⋅)v(s+t−a,0,⋅)ds,Γ=a¯∫aAS(s−a)β(s,⋅)v(s+t−a,0,⋅)ds,Γ=ΓA,T, (4.16)

otherwise. Here (S(t))_t≥0 is the semigroup generated by the operator 𝓐₀ – μId for all u ∈ D(𝓐₀) (Id is the identity operator), Γ_A,T := A – a + t – T̃ and

Γ:=min{a¯,ΓA,T}. (4.17)

In particular, it results

v(t,0,⋅):=S(T−t)vT(T−t,⋅), (4.18)

if t ≥ T – ā.

Proceeding as in [13, Theorem 4.4], with suitable changes, one has that there exists a positive constant C such that:

∫QA,1v2(T~,a,x)dxda≤C∫T−a¯2T−a¯4∫QA,1v2(t,a,x)dxdadt. (4.19)

However, here we make all the calculations in order to make precise some steps in [13, Theorem 4.4], where there is a misprint. Indeed in (4.19) the integration is made on T−a¯2,T−a¯4 in place of T4,3T4 as in [13]. The integration on T4,3T4 is correct if T = ā as in the next Corollary or in [13, Corollary 4.1].

Indeed, define, for ς > 0, the function w = e^ςtv, where v solves (4.3). Then w satisfies

∂w∂t+∂w∂a+(k(x)wx)x−(μ(t,a,x)+ς)w=−β(a,x)w(t,0,x),(t,x,a)∈Q~,w(t,a,0)=w(t,a,1)=0,(t,a)∈Q~T,A,w(T,a,x)=eςTvT(a,x),(a,x)∈QA,1,w(t,A,x)=0,(t,x)∈Q~T,1, (4.20)

where Q̃ := (T̃, T) × Q_A,1, Q̃_T,A := (T̃, T) × (0, A) and Q̃_T,1 := (T̃, T) × (0, 1). Multiplying the equation of (4.20) by –w and integrating by parts on Q_t := (T̃, t) × (0, A) × (0, 1), it results

−12∫QA,1w2(t,a,x)dxda+eςT~2∫QA,1v2(T~,a,x)dxda+12∫T~t∫01w2(τ,0,x)dxdτ+ς∫Qtw2(τ,a,x)dxdadτ≤∫Qtβw(τ,0,x)wdxdadτ≤∥β∥L∞(Q)1ϵ∫Qtw2dxdadτ+ϵA∥β∥L∞(Q)∫T~t∫01w2(τ,0,x)dxdτ, (4.21)

for ϵ > 0. Choosing ϵ=12∥β∥L∞(Q)A and ς=∥β∥L∞(Q)ϵ, we have

∫QA,1v2(T~,a,x)dxda≤C∫QA,1w2(t,a,x)dxda≤C∫QA,1v2(t,a,x)dxda.

Then, integrating over T−a¯2,T−a¯4, we have (4.19).

Now, take δ ∈ (0, A). By (4.19), we have

∫QA,1v2(T~,a,x)dxda≤C∫T−a¯2T−a¯4∫0δ+∫δA∫01v2(t,a,x)dxdadt. (4.22)

Consider the term ∫T−a¯2T−a¯4∫δA∫01v2(t,a,x)dxdadt. If Hypothesis 3.1 holds, proceeding as in [2, Lemma 3.2], one has

∫01v2dx≤C∫01kvx2dx+∫01x2kv2dx, (4.23)

for a strictly positive constant C. Hence,

∫T−a¯2T−a¯4∫δA∫01v2(t,a,x)dxdadt≤C∫T−a¯2T−a¯4∫δA∫01Θ~kvx2e2sφ~dxdadt+C∫T−a¯2T−a¯4∫δA∫01Θ~3x2kv2e2sφ~dxdadt,

where Θ̃ is defined in (4.14) with T₁ := T – ā, T₂ := T, y = 0 and φ̃ is the function associated to Θ̃ according to (3.2). Analogously, if Hypothesis 3.2 holds, then we obtain

∫T−a¯2T−a¯4∫δA∫01v2(t,a,x)dxdadt≤C∫T−a¯2T−a¯4∫δA∫01Θ~kvx2e2sφ¯~dxdadt+C∫T−a¯2T−a¯4∫δA∫01Θ~3(1−x)2kv2e2sφ¯~dxdadt,

where φ̄̃ is the function associated to Θ̃ according to (3.18). Thus, by Theorem 4.2 or 4.3 applied to Q̄ := (T – ā, T) × (0, A) × (0, 1) and Remark 3,

∫T−a¯2T−a¯4∫δA∫01v2(t,a,x)dxdadt≤C∫Q¯f2dxdadt+∫0T∫0A∫ωv2dxdadt,

where, in this case, f(t, a, x) := –β(a, x)v(t, 0, x). Thus

∫T−a¯2T−a¯4∫δA∫01v2(t,a,x)dxdadt≤C∥β∥L∞(Q)2∫Q¯v2(t,0,x)dxdadt+∫0T∫0A∫ωv2dxdadt, (4.24)

for a strictly positive constant C. By (4.18), (4.24) and proceeding as in [13], we have

∫T−a¯2T−a¯4∫δA∫01v2(t,a,x)dxdadt≤C∥β∥L∞(Q)2∫Qa¯,1vT2(a,x)dxda+∫0T∫0A∫ωv2dxdadt, (4.25)

for a strictly positive constant C. By (4.22) and (4.25), it results

∫QA,1v2(T~,a,x)dxda≤C∫0T∫0δ∫01v2(t,a,x)dxdadt+C∫Qa¯,1vT2(a,x)dxda+∫0T∫0A∫ωv2dxdadt. (2.26)

□

Corollary 4.1

Assume ā = T, Hypotheses 3.1 or 3.2 and 4.2. Then, for every δ ∈ (0, A), there exists a strictly positive constant C = C(δ) such that every solution v of (4.3) in 𝓥 satisfies

∫0A∫01v2(0,a,x)dxda≤C∫0T∫0δ∫01v2(t,a,x)dxdadt+C∫0a¯∫01vT2(a,x)dxda+∫0T∫0A∫ωv2dxdadt.

Moreover, if v_T(a, x) = 0 for all (a, x) ∈ (0, ā) × (0, 1), one has

∫0A∫01v2(0,a,x)dxda≤C∫0T∫0δ∫01v2(t,a,x)dxdadt+∫0T∫0A∫ωv2dxdadt.

Actually, proceeding as in [13] with suitable changes, we can improve the previous results in the following way:

Theorem 4.5

Assume Hypotheses 3.1 or 3.2 and 4.2. Then, for every δ ∈ (T, A), there exists a strictly positive constant C = C(δ) such that every solution v of (4.3) in 𝓥 satisfies

∫0A∫01v2(T−a¯,a,x)dxda≤C∫0δ∫01vT2(a,x)dxda+∫0T∫0A∫ωv2dxdadt.

By Theorem 4.5 and using a density argument, one can deduce Proposition 4.1. As a consequence one can prove, as in [13], the following null controllability results:

Theorem 4.6

Assume Hypotheses 3.1 or 3.2 or 4.1 and 4.2. Then, given T > 0 and y₀ ∈ L²(Q_A,1), for every δ ∈ (T, A), there exists a control f_δ ∈ L²(Q̃) such that the solution y_δ ∈ 𝓤 of

∂y∂t+∂y∂a−(k(x)yx)x+μ(t,a,x)y=fδ(t,x,a)χωin Q~,y(t,a,1)=y(t,a,0)=0on Q~T,A,y(T~,a,x)=y0(a,x)in QA,1,y(t,0,x)=∫0Aβ(a,x)y(t,a,x)dain Q~T,1, (4.27)

satisfies

yδ(T,a,x)=0a.e. (a,x)∈(δ,A)×(0,1).

Moreover, there exists C = C(δ) > 0 such that

∥fδ∥L2(Q~)≤C∥y0∥L2(QA,1). (4.28)

Here, we recall, Q̃ = (T̃, T) × (0, A) × (0, 1), Q̃_T,A = (T̃, T) × (0, A) and Q̃_T,1 = (T̃, T) × (0, 1).

Observe that if T = ā, Theorem 4.6 is exactly the null controllability result that we expect. Indeed, in this case (4.27) coincide with (1.1). On the other hand, if T > ā, the null controllability for (1.1) is given in the next theorem and it is based on the previous result:

Theorem 4.7

Assume Hypotheses 3.1 or 3.2 and 4.2. Then, given T ∈ (0, A) and y₀ ∈ L²(Q_A,1), for every δ ∈ (T, A), there exists a control f_δ ∈ L²(Q) such that the solution y_δ of (1.1) satisfies

yδ(T,a,x)=0a.e. (a,x)∈(δ,A)×(0,1).

Moreover, there exists C = C(δ) > 0 such that

∥fδ∥L2(Q)≤C∥y0∥L2(QA,1). (4.29)

Proof

The proof is similar to the one of [13, Theorem 4.8]. However, here we make all the calculations in order to make precise some steps in [13, Theorem 4.8], where there is a misprint and a term was missing.

As a first step, set T̃ := T – ā ∈ (0, T). By Theorem 2.1, there exists a unique solution u of

∂u∂t+∂u∂a−(k(x)ux)x+μ(t,a,x)u=0in (0,T~)×(0,A)×(0,1),u(t,a,1)=u(t,a,0)=0on (0,T~)×(0,A),u(0,a,x)=y0(a,x)in (0,A)×(0,1),u(t,0,x)=∫0Aβ(a,x)u(t,a,x)dain (0,T~)×(0,1). (4.30)

Set ỹ₀(a, x) := u(T̃, a, x); clearly ỹ₀ ∈ L²(Q_A,1). Now, consider

∂w∂t+∂w∂a−(k(x)wx)x+μ(t,a,x)w=h(t,x,a)χωin Q~,w(t,a,1)=w(t,a,0)=0on Q~T,A,w(T~,a,x)=y~0(a,x)in QA,1,w(t,0,x)=∫0Aβ(a,x)w(t,a,x)dain Q~T,1. (4.31)

Again, by Theorem 2.1, there exists a unique solution w_δ of (4.31) and, by the previous Theorem, there exists a control h_δ ∈ L²(Q̃) such that

wδ(T,a,x)=0 a.e. (a,x)∈(δ,A)×(0,1)

and

∥hδ∥L2(Q~)≤C∥y~0∥L2(QA,1),

for a positive constant C.

Now, define y_δ and f_δ by

yδ:=u,in[0,T~],wδ,in[T~,T]andfδ:=0,in[0,T~],hδ,in[T~,T].

Then y_δ satisfies (1.1) and f_δ ∈ L²(Q) is such that

yδ(T,a,x)=0 a.e. (a,x)∈(δ,A)×(0,1).

Indeed y_δ(T, a, x) = w_δ(T, a, x) = 0 a.e. (a, x) ∈ (δ, A) × (0, 1).

Now, we prove (4.29). As a first step, as in [13], we multiply the equation of (4.30) by u. Then, integrating over Q_A,1, we obtain:

12ddt∫0A∫01u2dxda+12∫01u2(t,A,x)dx+∫0A∫01kux2dxda+∫0A∫01μu2dxda=12∫01u2(t,0,x)dx.

Hence, using the fact that u(t, 0, x) = ∫0A β(a, x) u(t, a, x)da, we have

12ddt∫0A∫01u2dxda≤12∫01∫0Aβ(a,x)u(t,a,x)da2dx≤C2∫0A∫01u2dxda.

Setting F(t):=∥u(t)∥L2(QA,1)2 and multiplying the previous inequality by e^–Ct, it results

ddte−CtF(t)≤0.

Integrating over (0, t), for all t ∈ [0, T], we obtain

∫0A∫01u2(t,a,x)dxda≤eCT∫0A∫01u2(0,a,x)dxda=eCT∫0A∫01y02(a,x)dxda.

In particular,

∫0A∫01u2(T~,a,x)dxda≤eCT∫0A∫01y02(a,x)dxda.

Thus,

∥fδ∥L2(Q)2=∫T~T∫0A∫01hδ2dxdadt≤C∥y~0∥L2(QA,1)2=C∫0A∫01u2(T~,a,x)dxda≤C∫0A∫01y02(a,x)dxda, (4.32)

for a strictly positive constant C. Hence, (4.29) follows.□

As a consequence of the previous theorem, we obtain the null controllability property if the coefficient k degenerates at 0 and at 1 at the same time.

Theorem 4.8

Assume Hypotheses 4.1 and 4.2. Then, given T ∈ (0, A) and y₀ ∈ L²(Q_A,1), for every δ ∈ (T, A), there exists a control f_δ ∈ L²(Q) such that the solution y_δ of (1.1) satisfies

yδ(T,a,x)=0a.e. (a,x)∈(δ,A)×(0,1).

Moreover, there exists C = C(δ) > 0 such that

∥fδ∥L2(Q)≤C∥y0∥L2(QA,1). (4.33)

Proof

Fix y₀ ∈ L²(Q_A,1) and consider the two problems

(P1)∂y∂t+∂y∂a−(k(x)yx)x+μ(t,a,x)y=f(t,a,x)χωin (0,T)×(0,A)×(0,β¯),y(t,a,β¯)=y(t,a,0)=0on QT,A,y(0,a,x)=y0(a,x)in (0,A)×(0,β¯),y(t,0,x)=∫0Aβ(a,x)y(t,a,x)dain (0,T)×(0,β¯), (4.34)

and

(P2)∂y∂t+∂y∂a−(k(x)yx)x+μ(t,a,x)y=f(t,a,x)χωin (0,T)×(0,A)×(α¯,1),y(t,a,1)=y(t,a,α¯)=0on QT,A,y(0,a,x)=y0(a,x)in (0,A)×(α¯,1),y(t,0,x)=∫0Aβ(a,x)y(t,a,x)dain (0,T)×(α¯,1), (4.35)

where ᾱ ∈ (0, α) and β̄ ∈ (β, 1). Thus, by Theorem 4.7, there exist two controls h_1,δ and h_2,δ such that the solutions u_1,δ and u_2,δ of (P₁) and (P₂), associated to h_1,δ and h_2,δ, respectively, satisfy

u1,δ(T,a,x)=0a.e. (a,x)∈(δ,A)×(0,β¯),

and

u2,δ(T,a,x)=0a.e. (a,x)∈(δ,A)×(α¯,1).

Moreover, there exists C > 0 such that

∫0T∫0A∫0β¯h1,δ2dxdadt≤C∥y0∥L2(QA,1)2

and

∫0T∫0A∫α¯1h2,δ2dxdadt≤C∥y0∥L2(QA,1)2.

Denote with u₁ and h₁ (respectively u₂ and h₂) the trivial extensions of u_1,δ and h_1,δ (respectively u_2,δ and h_2,δ) to [β̄, 1] (respectively [0, ᾱ]), so that all functions are defined in the interval [0, 1]. Clearly, they depends always on δ and

∥hi∥L2(Q)≤C∥y0∥L2(QA,1),i=1,2. (4.36)

Now, let u₃ be the solution of

∂y∂t+∂y∂a−(k(x)yx)x+μ(t,a,x)y=0in (0,T)×(0,A)×(0,1),y(t,a,1)=y(t,a,0)=0on QT,A,y(0,a,x)=y0(a,x)in (0,A)×(0,1),y(t,0,x)=∫0Aβ(a,x)y(t,a,x)dain (0,T)×(0,1), (4.37)

and consider the three smooth cut off functions ξ, η, ϕ : [0, 1] → ℝ defined as

0≤ξ(x)≤1, for all x∈[0,1],ξ(x)=1,x∈[0,(2α+ρ)/3],ξ(x)=0,x∈[(α+2ρ)/3,1],0≤η(x)≤1, for all x∈[0,1],η(x)=0,x∈[0,(2α+ρ)/3],η(x)=1,x∈[(α+2ρ)/3,1]

and ϕ := 1 – ξ – η. Finally, take

y(t,a,x)=ξu1+ηu2+F(t)ϕu3,

where F(t):=T−tT.

It is easy to verify that y(t, a, 0) = y(t, a, 1) = 0, y(0, a, x) = y₀(a, x) (since F(0) = 1) and y(t, 0, x) = ∫0A β(a, x)y(t, a, x)da. Moreover,

y(T,a,x)=0a.e. (a,x)∈(δ,A)×(0,1)

and y satisfies the equation of (1.1) with

fδ=ξh1χω+ηh2χω−1Tϕu3−F(t)kϕ′u3,x−F(t)(kϕ′u3)x−kξ′u1,x−(kξ′u1)x−kη′u2,x−(kη′u2)x.

Obviously, the support of f_δ is contained in ω and, since k ∈ C¹(ω), the terms (kϕ′u₃)_x, (kξ′u₁)_x and (kη′u₂)_x are L²(0, 1) (recall that ϕ′(x) = ξ′(x) = η′(x) = 0 for all x ∈ (0, 1) ∖ ω); thus f_δ ∈ L²(Q) as required. As in [19], estimate (4.33) follows by the definition of f_δ, (4.36) and (2.2) for u_i, i = 1, 2, 3.□

Observe that the previous result can be proved also for the problem in non divergence form considered in [13].

A Appendix

A.1 Proof of (2.2)

Multiplying the equation of (1.1) by y and integrating over (0, A) × (0, 1), we obtain

12ddt∥y(t)∥L2(QA,1)2+12∫01y2(t,A,x)dx−12∫01y2(t,0,x)dx+∫0A∫01kyx2dxda=−∫0A∫01μy2dxda+∫0A∫ωfydxda.

Hence, using the initial condition y(t, 0, x) = ∫0A β(a, x) y(t, a, x)da, the assumptions on β and μ and the inequality of Jensen, one has

12ddt∥y(t)∥L2(QA,1)2+12∫01y2(t,A,x)dx+∫0A∫01kyx2dxda≤C2∫0A∫01y2(t,a,x)dxda+12∫0A∫01f2dxda, (A.1)

where C is a positive constant. Since ∫01y2(t,A,x)dx and ∫0A∫01kyx2dxda are positive, we deduce

ddt∥y(t)∥L2(QA,1)2≤C∥y(t)∥L2(QA,1)2+∥f(t)∥L2(QA,1)2.

Setting F(t):=∥y(t)∥L2(QA,1)2 and multiplying the previous inequality by e^–Ct, one has

ddt(e−CtF(t))≤e−Ct∥f(t)∥L2(QA,1)2. (A.2)

Integrating (A.2) over (0, t), for all t ∈ [0, T] it follows

e−CtF(t)≤F(0)+∫0te−Cτ∥f(τ)∥L2(QA,1)2dτ.

Hence, for all t ∈ [0, T],

F(t)≤eCTF(0)+∫0T∥f(τ)∥L2(QA,1)2dτ

and

supt∈[0,T]∥y(t)∥L2(QA,1)2≤C∥y0∥L2(QA,1)2+∥f∥L2(Q)2dτ. (A.3)

Therefore, by (A.1), it follows

12ddt∥y(t)∥L2(QA,1)2+∫0A∫01kyx2dxda≤C2∫0A∫01y2(t,a,x)dxda+12∫0A∫01f2dxda.

Integrating over (0, T), we have

12∥y(T)∥L2(QA,1)2+∫0T∫0A∫01kyx2dxdadt≤12∥y0∥L2(QA,1)2+C2∫0T∫0A∫01y2(t,a,x)dxdadt+12∫0T∫0A∫01f2dxdadt.

Hence, by (A.3),

∫0T∫0A∥kyx∥L2(0,1)2dadt≤∥y0∥L2(QA,1)2+C∫0T∥y(t)∥L2(QA,1)2dt+∥f∥L2(Q)2≤C∥y0∥L2(QA,1)2+∥f∥L2(Q)2dτ (A.4)

and (2.2) follows by (A.3) and (A.4).

A.2 Proof of Proposition 3.2

We consider case (i), Hypothesis (HP1). Fix β ∈ (θ, 1) arbitrarily for the moment. Since w(1) = 0, we have

∫01k(x)(1−x)2w2(x)dx=∫01k(x)(1−x)2(∫x1(1−y)β/2w′(y)(1−y)−β/2dy)2dx.

This implies

∫01k(x)(1−x)2w2(x)dx≤∫01k(x)(1−x)2(∫x1(1−y)β|w′(y)|2dy∫x1(1−y)−βdy)dx.

Hence, we have

∫01k(x)(1−x)2w2(x)dx≤11−β∫01k(x)(1−x)1+β(∫x1(1−y)β|w′(y)|2dy)dx.

By the Theorem of Fubini, it follows

∫01k(x)(1−x)2w2(x)dx≤11−β∫01(1−y)β|w′(y)|2(∫0yk(x)(1−x)1+βdx)dy. (A.5)

Now, divide the right hand side of (A.5) into three parts, i.e.

∫01(1−y)β|w′(y)|2(∫0yk(x)(1−x)1+βdx)dy=Lϵ+Mϵ+Nϵ,

where

Lϵ=∫01−ϵ(1−y)β|w′(y)|2(∫0yk(x)(1−x)1+βdx)dy,Mϵ=∫1−ϵ1(1−y)β|w′(y)|2(∫01−ϵk(x)(1−x)1+βdx)dy,

and

Nϵ=∫1−ϵ1(1−y)β|w′(y)|2(∫1−ϵyk(x)(1−x)1+βdx)dy.

Thanks to our hypothesis, there exists ϵ > 0 such that the function

x⟶k(x)(1−x)θ is nondecreasing on [1−ϵ,1);

thus, for N_ϵ, we have

Nϵ=∫1−ϵ1(1−y)β|w′(y)|2(∫1−ϵyk(x)(1−x)1+βdx)dy≤∫1−ϵ1(1−y)β−θ|w′(y)|2k(y)∫1−ϵy(1−x)θ−β−1dxdy≤1(β−θ)∫1−ϵ1k(y)|w′(y)|2dy. (A.6)

For M_ϵ, we have

Mϵ≤sup[0,1−ϵ]k∫1−ϵ1(1−y)βk(y)k(y)|w′(y)|2(∫01−ϵ(1−x)−(1+β)dx)dy≤ϵθ−ββk(1−ϵ)sup[0,1−ϵ]k∫1−ϵ1(1−y)β−θk(y)|w′(y)|2≤C∫1−ϵ1k(x)|w′(x)|2dx. (A.7)

Proceeding in a similar way, we obtain

Lϵ≤C∫01−ϵk(x)|w′(x)|2dx. (A.8)

Using (A.6), (A.7) and (A.8) in (A.5), we obtain

∫01k(x)(1−x)2w2(x)dx≤C∫01k(x)|w′(x)|2dx, (A.9)

where the constant C depends on a, ϵ, θ and β. If one assumes that Hypothesis (HP1)’ holds, that is

x⟶k(x)(1−x)θ is nondecreasing on [0,1),

then, one can take ϵ = 1 in the above computations, so that L_ϵ = M_ϵ = 0. Using then (A.6) in (A.5), with ϵ = 1, one obtains

∫01k(x)(1−x)2w2(x)dx≤1(1−β)(β−θ)∫01k(x)|w′(x)|2dx.

We then remark that this last estimate is optimal for β=θ+12, which gives the desired result.

We now consider case (ii), Hypothesis (HP2). Fix β ∈ (1, θ) arbitrarily for the moment. As before

∫01k(x)(1−x)2w2(x)dx=∫01k(x)(1−x)2(∫0x(1−y)β/2w′(y)(1−y)−β/2dy)2dx;

so that

∫01k(x)(1−x)2w2(x)dx≤∫01k(x)(1−x)2(∫0x(1−y)β|w′(y)|2dy∫0x(1−y)−βdy)dx.

It follows that

∫01k(x)(1−x)2w2(x)dx≤1β−1∫01k(x)(1−x)1+β(∫0x(1−y)β|w′(y)|2dy)dx.

Applying the Theorem of Fubini, we have

∫01k(x)(1−x)2w2(x)dx≤1β−1∫01(1−y)β|w′(y)|2(∫y1k(x)(1−x)1+βdx)dy. (A.10)

As before, we rewrite

∫01(1−y)β|w′(y)|2(∫y1k(x)(1−x)1+βdx)dy=Iϵ+Jϵ+Kϵ,

where

Iϵ=∫01−ϵ(1−y)β|w′(y)|2(∫y1−ϵk(x)(1−x)1+βdx)dy,Jϵ=∫01−ϵ(1−y)β|w′(y)|2(∫1−ϵ1k(x)(1−x)1+βdx)dy

and

Kϵ=∫1−ϵ1(1−y)β|w′(y)|2(∫y1k(x)(1−x)1+βdx)dy.

Thanks to our hypothesis, there exists ϵ > 0 such that the function

x⟶k(x)(1−x)θ is nonincreasing on [1−ϵ,1),

thus K_ϵ can be estimated in the following way:

Kϵ=∫1−ϵ1(1−y)β|w′(y)|2(∫y1k(x)(1−x)1+βdx)dy≤∫1−ϵ1(1−y)β|w′(y)|2k(y)(1−y)θ∫y1(1−x)θ−β−1dx≤1(θ−β)∫1−ϵ1k(x)|w′(x)|2dx. (A.11)

For J_ϵ, we can proceed in a similar way, obtaining

Jϵ=∫01−ϵ(1−y)β|w′(y)|2(∫1−ϵ1k(x)(1−x)1+βdx)dy≤ϵ−βk(1−ϵ)inf[0,1−ϵ]k∫01−ϵk(y)|w′(y)|2dy. (A.12)

For I_ϵ, we have

Iϵ=∫01−ϵ(1−y)β|w′(y)|2k(y)k(y)(∫y1−ϵk(x)(1−x)1+βdx)dy≤sup[0,1−ϵ]kβinf[0,1−ϵ]k∫01−ϵk(y)|w′(y)|2(1−y)βϵ−βdy≤C∫01−ϵk(x)|w′(x)|2dx. (A.13)

Using (A.11), (A.12) and (A.13) in (A.10), we obtain

∫01k(x)(1−x)2w2(x)dx≤C∫01k(x)|w′(x)|2dx, (A.14)

where the constant C depends on a, ϵ, θ and β. If one assumes that hypothesis (HP2)’ holds, that is

x⟶k(x)(1−x)θ is noninreasing on [0,1),

then, one can take ϵ = 1 in the above computations, so that J_ϵ = I_ϵ = 0. Using then (2.1) in (A.10), with ϵ = 1, one obtains

∫01k(x)(1−x)2w2(x)dx≤1(β−1)(θ−β)∫01k(x)|w′(x)|2dx.

We then remark that this last estimate is optimal for β=θ+12, which gives the desired result.

A.3 Proof of Proposition 4.2

Let us consider a smooth function ξ : [0, 1] → ℝ such that

0≤ξ(x)≤1, for all x∈[0,1],ξ(x)=1,x∈ω′,ξ(x)=0,x∈(0,1)∖ω.

Then, integrating by parts one has

0=∫0Tddt∫0A∫01(ξesψ)2v2dxdadt=∫Q2sψt(ξesψ)2v2+2(ξesψ)2v(−va−(kvx)x+μv+f)dxdadt=2s∫Qψt(ξesψ)2v2dxdadt+2s∫Qψa(ξesψ)2v2dxdadt+2∫Qξ2e2sψxkvvxdxdadt+2∫Q(ξ2e2sψk)vx2dxdadt+2∫Qξ2e2sψμv2dxdadt+2∫Qξ2e2sψfvdxdadt.

Hence, using Young’s inequality

2∫Qξ2e2sψkvx2dxdadt=−2s∫Qψtξesψ2v2dxdadt−2s∫Qψa(ξesψ)2v2dxdadt−2∫Qξ2e2sψxkvvxdxdadt−2∫Qξ2e2sψμv2dxdadt−2∫Qξ2e2sψfvdxdadt≤−2s∫Qψt(ξesψ)2v2dxdadt−2s∫Qψa(ξesψ)2v2dxdadt+∫Qk(ξ2e2sφ)xξesφv2dxdadt+∫Qξ2e2sφkvx2dxdadt.+(2∥μ∥L∞(Q)+1)∫Qξ2v2dxdadt+∫Qξ2e2sψf2dxdadt.

Thus,

infω′{k}∫0T∫0A∫ω′e2sψvx2dxdadt≤supω×(0,T){4kξesψx2−2s(ψt+ψa)(ξesψ)2}+2∥μ∥L∞(Q)+1∫0T∫0A∫ωv2dxdadt+∫Qf2e2sψdxdadt.

Acknowledgments

The author is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and she is supported by the FFABR “Fondo per il finanziamento delle attività base di ricerca” 2017, by the INdAM- GNAMPA Project 2019 ”Controllabilità di PDE in modelli fisici e in scienze della vita” and by PRIN 2017-2019 Qualitative and quantitative aspects of nonlinear PDEs.

References

[1] B. Ainseba, Y. Echarroudi, L. Maniar Null controllability of population dynamics with degenerate diffusion, Differential Integral Equations (2013), 1397–1410.10.57262/die/1378327432Search in Google Scholar

[2] F. Alabau-Boussouira, P. Cannarsa, G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ. 6 (2006), 161–204.10.1007/s00028-006-0222-6Search in Google Scholar

[3] S. Aniţa, Analysis and control of age-dependent population dynamics, Mathematical Modelling: Theory and Applications 11 (2000), Kluwer Academic Publishers, Dordrecht.10.1007/978-94-015-9436-3Search in Google Scholar

[4] V. Barbu, M. Iannelli, M. Martcheva, On the controllability of the Lotka-McKendrick model of population dynamics, J. Math. Anal. Appl. 253 (2001), 142—165.10.1006/jmaa.2000.7075Search in Google Scholar

[5] R. Borges, A. Calsina, S. Cuadrado, O. Diekmann, Delay equation formulation of a cyclin-structured cell population model, J. Evol. Equ. 14 (2014), 841–862.10.1007/s00028-014-0241-7Search in Google Scholar

[6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science+Business Media, LLC 2011.10.1007/978-0-387-70914-7Search in Google Scholar

[7] P. Cannarsa, G. Fragnelli, D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ. 8 (2008), 583–616.10.1007/s00028-008-0353-34Search in Google Scholar

[8] P. Cannarsa, P. Martinez, J. Vancostenoble, Persistent regional controllability for a class of degenerate parabolic equations, Commun. Pure Appl. Anal. 3 (2004), 607–635.10.3934/cpaa.2004.3.607Search in Google Scholar

[9] O. Diekmann, Ph. Getto, Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations, J. Differential Equations 215 (2005), 268–319.10.1016/j.jde.2004.10.025Search in Google Scholar

[10] Y. Echarroudi, L. Maniar, Null controllability of a model in population dynamics, Electron. J. Differential Equations 2014 (2014), 1–20.Search in Google Scholar

[11] K.J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, Berlin, Heidelberg, 1999.Search in Google Scholar

[12] E. Fernández-Cara, S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim. 45, 1395-1446.s10.1137/S0363012904439696Search in Google Scholar

[13] G. Fragnelli, Carleman estimates and null controllability for a degenerate population model, Journal de Mathématiques Pures et Appliqués 115 (2018), 74—126.10.1016/j.matpur.2018.01.003Search in Google Scholar

[14] G. Fragnelli, P. Martinez, J. Vancostenoble, Qualitative properties of a population dynamics system describing pregnancy, Math. Models Methods Appl. Sci. 15 (2005), 507–554.10.1142/S0218202505000455Search in Google Scholar

[15] G. Fragnelli, D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Advances in Nonlinear Analysis 2 (2013), 339–378.10.1515/anona-2013-0015Search in Google Scholar

[16] G. Fragnelli, D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, Mem. Amer. Math. Soc. 242 (2016), v+84 pp. Corrigendum, to appear.10.1090/memo/1146Search in Google Scholar

[17] G. Fragnelli, D. Mugnai, Carleman estimates for singular parabolic equations with interior degeneracy and non smooth coefficients, Adv. Nonlinear Anal. 6 (2017), 61–84.10.1515/anona-2015-0163Search in Google Scholar

[18] G. Fragnelli, D. Mugnai, Controllability of strongly degenerate parabolic problems with strongly singular potentials, Electron. J. Qual. Theory Differ. Equ. 50 (2018), 1–11.10.14232/ejqtde.2018.1.50Search in Google Scholar

[19] G. Fragnelli, D. Mugnai, Controllability of degenerate and singular parabolic problems: the double strong case with Neumann boundary conditions, Opuscula Math. 39 (2019), 207—225.10.7494/OpMath.2019.39.2.207Search in Google Scholar

[20] G. Fragnelli, L. Tonetto, A population equation with diffusion, J. Math. Anal. Appl. 289 (2004), 90–99.10.1016/j.jmaa.2003.08.047Search in Google Scholar

[21] Y. He, B. Ainseba, Exact null controllability of the Lobesia botrana model with diffusion, J. Math. Anal. Appl. 409 (2014), 530–543.10.1016/j.jmaa.2013.07.020Search in Google Scholar

[22] J.L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I. Translated from the French by P. Kenneth. Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg (1972).Search in Google Scholar

[23] D. Maity, M. Tucsnak and E. Zuazua, Controllability and positivity constraints in population dynamics with age structuring and diffusion, Journal de Mathématiques Pures et Appliqués. In press, 10.1016/j.matpur.2018.12.006.10.1016/j.matpur.2018.12.006Search in Google Scholar

[24] S. S. Win, R. Muhamad, Z. A. M. Ahmad, N. A. Adam, Population Fluctuations of Brown Plant Hopper Nilaparvata lugens Stal. and White Backed Plant Hopper Sogatella furcifera Horvath on Rica, Jounal of Entomology 8 (2011), 183–190.10.3923/je.2011.183.190Search in Google Scholar

[25] B. Zhong, C. Lv and W. Qin, Effect of Temperature on the Population Growth of Tirathaba rufivena (Lepidoptera: Pyralidae) on Areca catechu (Arecaceae), Florida Entomologist 100 (2017), 578–582.10.1653/024.100.0314Search in Google Scholar

Received: 2018-12-29

Accepted: 2019-05-16

Published Online: 2019-10-15

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/anona-2020-0034

Keywords for this article

population equation; degenerate equation; Carleman estimates; observability inequalities

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