WELL-POSEDNESS OF A NONLINEAR MODEL OF PROLIFERATING CELL POPULATIONS WITH INHERITED CYCLE LENGTH∗

Abdul-Majeed AL-IZERIKhalid LATRACHUniversit´e Blaise Pascal(Clermont II)Laboratoire de Math´ematiques,CNRS UMR 6620 Campus des C´ezeaux-B.P.80026,63171 Aubi`ere Cedex France E-mail:AbdulMajeed.Alizeri@math.univ-bpclermont.fr;Khalid.Latrach@math.univ-bpclermont.fr

WELL-POSEDNESS OF A NONLINEAR MODEL OF PROLIFERATING CELL POPULATIONS WITH INHERITED CYCLE LENGTH∗

Abdul-Majeed AL-IZERIKhalid LATRACH†Universit´e Blaise Pascal(Clermont II)Laboratoire de Math´ematiques,CNRS UMR 6620 Campus des C´ezeaux-B.P.80026,63171 Aubi`ere Cedex France E-mail:Abdul_Majeed.Al_izeri@math.univ-bpclermont.fr;Khalid.Latrach@math.univ-bpclermont.fr

This paper deals with a nonlinear initial boundary values problem derived from a modified version of the so called Lebowitz and Rubinow’s model[16]discussed in[8,9] modeling a proliferating age structured cell population with inherited properties.We give existence and uniqueness results on appropriate weighted Lp-spaces with 1≤p<∞in the case where the rate of cells mortality σ and the transition rate k are depending on the total density of population.General local and nonlocal reproduction rules are considered.

evolution equation;local and nonlocal boundary conditions;quasi-accretive operators;mild solutions,strong solutions;local and global solutions 2010 MR Subject Classification47H06;34A12;35F20

1　Introduction

In the works[8,9],the author discussed the well-posedness and various mathematical aspects of solution to the Cauchy problem

t>o,o

where K is a bounded linear operator on suitable function spaces.

As it was observed by Rotenberg[19],it seems that the linear model is not adequate.Indeed, the cells under consideration are in contact with a nutrient environment which is not part of the mathematical formulation.Fluctuations in nutrient concentration and other density-dependent effects such as contact inhibition of growth make the transition rates functions of the population density,thus creating a nonlinear problem.On the other hand,the biological boundaries at l1and l2are fixed and tightly coupled through out mitosis.The conditions present at the boundaries are left throughout the system and cannot be remote.This phenomena suggests that at mitosis the daughter cells and parent cell are related by a nonlinear reproduction rule. At mitosis,the daughter and mother cells are related by a nonlinear reproduction rule which describes the boundary conditions.

We point out that the well-posedness of nonlinear initial boundary value problems derived from Rotenberg model were already discussed in[12,2o]and[1].However,it seems that the wellposedness of nonlinear time dependent versions derived from(1.1)has not yet been investigated. The main goal of this work is to present and to discuss two nonlinear versions of the model (1.1)in a weighted Lp-spaces,1≤p<∞,on the set Ω:={(a,l);o

The structure of this work is as follows.In the next section we introduce the functional setting of the problem and the main assumptions.In Section 3 we study the initial boundary value problem(3.1)in the case where σ(·,·,·)and k(·,·,·,·,·)are nonlinear functions of the density of population f,and K is a nonlinear operator on suitable trace spaces modeling the biological rule.Following the same strategy as in the works[12,2o]for Lebowitz-Rubinow’s model(also see[4]and[13])we discuss existence and uniqueness of solutions to problem(3.1). The main result of this section is Theorem 3.5 which asserts that,under reasonable assumptions, problem(3.1)has a unique mild solution on appropriate weight Lpspaces where 1≤p<∞. If p>1,then this solution is also a weak solution of the problem.Further,if the initial data belongs to the domain of(see Section 2 for the definition ofthen we obtain a strong solution.Further,we derive sufficient conditions guaranteeing that problem(3.1)possesses a unique global strong solution.

Section 4 focuses on problem(4.1)where σ1(·,·,·)is a nonlinear function of〈f〉(t)withdadl(the usual L1norm of L1(Ω))and k1(·,·,·,·,·)is a nonlinear function of the density of population f.Here the boundary conditions are modeled by

where σ2(·,·,·)and k2(·,·,·,·)are nonlinear functions of both f(t,·,·)(l)and〈f〉(t).After giving some preparatory results,we state the main result of this section(Theorem 4.6).We establish that under appropriate conditions,the existence of a local weak solution on weighted Lp-spaces with 1

For the sake of completeness,in Section 5,we recall and gather some facts from functional analysis required throughout of the paper.

2　Notations and Preliminaries

The goal of this section is to first fix the notations and to introduce the functional setting of the problem.To do so,let 1≤p<∞and set

where Ω:={(a,l);oo be an arbitrary real number and define the weighted spaceby

where the weight function hω(·,·)is given by

is a Banach space.

Let us now recall the following lemma established in[1].

According to Lemme 2.1 in[1],(resp.So,we can define the space

Before going further we shall prove the following lemma which is required below.

Lemma 2.2Let 1≤p<+∞.If f∈Xp,thenand conversely.In particular we have

where q denotes the conjugate exponent of p.

ProofLet f∈Xp.Since l≥a,we have henceConversely,letOne can write

This proves the first assertion and (i).

applying H¨older’s inequality,we get

This ends the proof.

For simplicity,we shall identifywith the space

Accordingly,the boundary operator K may be viewed as a map from Ypinto itself.

We close this section by recalling that the function sgno(·)is defined by

and the definition of the symbol[·,·]sis given in Section 5(see also[3,pp.1o2-1o3]).

3　Local Boundary Conditions

The goal of this section is to discuss existence and uniqueness results for the following initial boundary value problem

where σ(·,·,·)and k(·,·,·,·,·)are nonlinear functions of the density of population f,fostands for the initial data and K denote a nonlinear operator from Ypinto itself modeling the transition biological rule.

We now introduce the following hypotheses required below.

·(A1)There exists κ>o such that,for all f1,f2∈Yp,we have

·(A2)The functions σ(·,·,·)and k(·,·,·,·,·)are measurables and there exist ζ∈L∞(Ω) and ρ∈L∞(Ω×Ω)such that

Further,σ(·,·,·)is assumed to be a Carath´eodory function.

In the remainder of this section,the real ω will satisfy the condition

where κ is the constant appearing in hypothesis(A1).

The function σ(·,·,·)is assumed to be a Carath´eodory function(see Section 5),so its generates a Nemytskii operator Nσgiven by

Let B be the operator defined by

Now problem(3.1)may be written abstractly as

For the sake of completeness,we shall recall the following result established in[1].

Lemma 3.1If condition(A1)is satisfied,thenis a ω-m-accretive operator on

Remark 3.2Note that the result of Lemma 3.1 holds also true for p=1(see Theorem 2 in[12]).

Lemma 3.3Let p∈[1,+∞)and assume that condition(A1)is satisfied.Then

ProofIt is enough to prove thatThe proof is similar to that of Theorem 3 in[12]and then it is omitted.

Lemma 3.4Let p∈[1,+∞).If hypothesis(A2)holds true and Nσmapsinto itself, then there exists a constant C>o such that

Using(A2)together with H¨older’s inequality we get

Next,simple calculations using the estimate

Theorem 3.5Let ω be an arbitrary real satisfying ω>maxand let 1≤ p<∞.Assume that conditions(A1)and(A2)hold true and Nσmapsinto itself.Then problem(3.2)has a unique mild solution onFurther,if p∈(1,+∞),then this mild solution is in fact a weak solution and it is a strong solution whenever the initial data fobelongs to

ProofAccording to Lemma 3.1,Remark 3.2 and Lemma 3.3,is quasi-m-accretivewith dense domain.It follows from Lemma 3.4 thatis a Lipschitzian mapping onis also a quasi-m-accretive operator onThe first part of the theorem follows from Corollary 4.1 in[3](here the initial data fois taken in Xpbecauseand the fact that any function inbelongs to Xp(cf.Lemma 2.2)).We know that,for 1

of Theorem 5.1 shows that this weak solution is,in fact,a strong solution on

Remark 3.6(a)Recall that it is proved in[1,Lemma 1]that,if o≤κ<1,then TKis a m-accretive on Xp.Accordingly,if(A2)holds true and Nσmaps Xpinto itself,then the conculusion of Theorem 3.5 remains valid even if l1=o(cf.[1,Theorem 4.1]).Evidently,in this case we have ω=o.Note also that if κ>1,then the condition l1>o is necessary.

(b)If P=o,i.e.,k(·,·,·,·,·)=o,then Theorem 3.5 is nothing else but Theorem 3 in[12] for p=1 and Theorem 4.2 in[1]for p∈(1,+∞).

We have also the following proposition.

Proposition 3.7Let p∈[1,+∞)and let ω be as in Theorem 3.5.Let f1,f2∈be two mild solutions to Problem(3.2)where T>o.Given∊>o,there exists β>o such that if‖f1(o)−f2(o)‖p,ω≤β,then

ProofIt is similar to the proof of Proposition 3.3 in[13],so it is omitted.□

4　Nonlocal Boundary Conditions

In this section we will discuss existence and uniqueness of solutions to the following initial boundary value problem

The function σ1(·,·,·)is a nonlinear function of〈f〉while σ2(·,·,·)and k2(·,·,·,·,·)are nonlinear functions of both the density of population f and〈f〉.Here fodenotes the initial data.

Before going further,we introduce the following hypotheses required in the sequel.

·(A3)The function σ1(·,·,·)is measurable and for any r>o,there exists Λr>o such that

for all(a,l)∈Ω,z1,z2∈[−r,r].

·(A4)There exist a function σ1:Ω×ℝ−→ℝ and two constantssuch that

·(A5)The function k1(·,·,·,·,·)is measurable and satisfies

·(A6)The functions σ2(·,·,·)and k2(·,·,·,·)are measurable and there exist two constants C1>o and C2>o such that

for all(l,l′)

Hence,problem(4.1)may be written abstractly as

where K denotes the following nonlocal boundary operator

In the remainder of this section,the spacewill be equipped with the normgiven by

Lemma 4.1Let r>o and p∈[1,+∞).If conditions(A3)–(A6)hold true.Then there exist two constants Cr>o and ϑ>o such that

Using(A5)together with H¨older’s inequality we get

where Cr:=2

(ii)Now,we shall show the second estimate.For,we can write

Using assumption(A6)together with Lemma 2.2(ii),we get

Applying H¨older’s inequality,we get

A simple calculations using the estimate

On the other hand,using(A6)and the estimate

This yields the inequality

where ϑ2=2C1maxPutting ϑ=max(ϑ1,ϑ2),we get

This proves(ii).

Remark 4.2In the following lemma,we will show thatis quasi-accretive onwhenever ω>maxwhere ϑ is the constant appearing in Lemma 4.1(ii).

Lemma 4.3Let p∈[1,+∞)and let ω be an arbitrary real number satisfying ω>If hypothesis(A6)holds true,then,there exist a constant δ:=δ(p)>o such that

(i)if p=1,then,for all g1,g2∈,we have

(ii)if p∈(1,+∞),then,for all φ1,φ2∈we have for some ψδ∈Jδ(φ1−φ2)and λ∈

Proof(i)Define the real δ(1)by δ(1)=ω+1 and let g1,g2be such that

Since ϑe−ωl1≤1,we get

(ii)Now,we consider the case p∈(1,+∞)with p>1.Let φ1,φ2be such that φ1/=φ2and consider the real δ(p)Putting φ=φ1−φ2,we can write

Since φ1,φ2applying Lemma 4.1(ii),we get

Since ϑpe−ωl1≤1,we get

For any λ>o,the solution of problem(4.3)is given

In order to write abstractly equations(4.4)and(4.5),we define the following linear operators For u∈Yp,we have

which implies

Similarly,for u∈Yp,we can write

Next,using H¨older’s inequality and the fact that 1≤eωl2e−ω(l−s),for g∈we have

In the same way we can write

Now using these operators and the fact that f must satisfy the boundary conditions,equations(4.4)and(4.5)may be written in the form

Accordingly,(f|Γ2,f)is a solution to the fixed point problem Fλ(v,w)=(v,w)on the product space

Lemma 4.4If assumption(A6)holds true.There exists a constant λp>o,such that

for any λ∈(o,λp)there exists a functionsuch that

ProofAs we have seen above,in order to solve problem(4.11),it suffices to prove that the fixed point problem Fλ(u,f)=(u,f)has a unique solution in where the operatorFλis defined by(4.1o).To do so,let λpbe the real defined by

where ϑ is the constant given in Lemma 4.1(ii).

Let(u1,f1),(u2,f2) Using Lemma 4.1(ii)together with estimates(4.6)–(4.9),

Taking into acount of the definition of the real λp,it is clear that,for each λ∈(o,λp),the

operator Fλis then a strict contraction mapping.Applying the fixed point theorem of Banach

we infer that,for such λ,problem(4.11)has a unique solution in

Remark 4.5It follows from Lemmas 4.3 and 4.4 thatis a quasi-m-accretive operator.Note also that,in this case,is a dense subset of(see,for example[12,13]).

The main result of this section is the following.

(i)If p∈(1,+∞),then,for each fo∈Xp,problem(4.2)has a local weak solution.If the initial data fobelongs tothen problem(4.2)has a local strong solution.Assume, further,that(A7)is satisfied,then problem(4.2)has a unique global strong solution for eachand it is a unique global weak solution if fo/∈

(ii)If p=1,then,for each fo∈X1,problem(4.2)has a local mild solution.

Proof(i)According to Remark 4.5,is quasi-m-accretive onIt follows from

Next,let fo/∈there exists a sequence(fn)n∈ℕcontained insuch that fn→foas n→∞and put M:=sup{‖fn‖p,ω:n=o,1,···}.

A similar proof to that of Theorem 5.2 allows us deduce the existence of real TM>o such thatfor each n∈ℕ,the problem

possesses a unique strong solution,say gn,on the interval(o,TM).To prove the existence of a local weak solution,we have only to check that the sequence(gn)n∈ℕis a Cauchy sequence in the Banach spaceThis follows immediately from(5.3).

To discuss the existence of a unique global strong solution to problem(4.2),we assume that condition(A7)is satisfied.If g∈J(f),then one can write

We know from Lemmas 4.3 and 4.4(see also Remark 4.5)thatis ω-m-accretive.The use Lemma 4.1(i)together with[11]shows that the function-Lipschitz onwhere Cris the Lipschitz constant(cf.Lemma 4.1(i)).Consequently,the operatoris quasi-accretiveApplying Corollary 4.1 in[3]we infer that the problem

has a unique mild solution f on

The use of(5.2)yields

for o≤t≤Trwhere Tr>o is suitably chosen.This shows that u(·)is a mild solution to problem(4.2)in the interval[o,Tr]which completes the proof.

Remark 4.7It should be noticed that in the case where the perturbation is trivial(P= o),problem(4.1)coincide with problem(DE)–(DC)in[2o].So,Theorem 4.6 extends Theorems 5.5 and 6.5 in[2o]to a more general framework.

5　Annex

Let(X,‖·‖)be a real Banach space.An operator A:D(A)⊆X→2Xis said to be accretive if the inequality‖x−y+λ(u−v)‖≥‖x−y‖holds for all λ≥o,x,y∈D(A)and u∈Ax,v∈Ay.If,in addition,R(I+λA)(i.e.,the range of the operator I+λA),is for one,hence for all,λ>o,precisely X,then A is called m-accretive.Accretive operators were introduced by Browder[1o]and Kato[14]independently.

Finally,A is said to be quasi-accretive(quasi-m-accretive),if there exists w∈ℝ such that A+wI is accretive(respectively m-accretive),in this case we say also that A is w-accretive (w-m-accretive respectively).Notice that A is accretive if and only if A is quasi-accretive with w=o.

The operators which are quasi-m-accretive play an important role in the study of nonlinear partial differential equations.

Consider the Cauchy problem

where A is quasi-m-accretive on X and f∈L1(o,T,X).

Given∊>o.An∊-discretization on[o,T]of the equation u′(t)+A(u(t))∋f(t)consists of a partition o=to≤t1≤t2≤···≤tNof the interval[o,tN]and a finite sequencesolution to(5.1)is a piecewise constant function z:[o,tN]→X whose values zion(ti−1,ti]satisfy the finite difference equation

It is well known(see[3,Corollary 4.1])that(5.1)has a unique mild solution in the sense that there exists a unique continuous function u:such that u(o)=xo,and moreover,for each∊>o there is an∊-approximate solution z of u′+A(u)∋f on[o,T]such that‖u(t)−z(t)‖≤∊for all t∈[o,T]with u(o)=xo.

If u is the mild solution of problem(5.1),then for each(x,y)∈A and o≤s≤t≤T,we have

here the function[·,·]s:X×X→ℝ is defined by[y,x]s=sup{x∗(y):x∗∈J1(x)},where J1:X→2X∗is the duality mapping on X,i.e.,J1(x)={x∗∈X∗:x∗(x)=‖x‖,‖x∗‖=1}. This means that u is an integral solution to equation(5.1)in the sense of B´enilan[5]and moreover both concepts of solution coincide under our context.

If u,v are integral solutions of u′(t)+A(u(t))∋f(t)and v′(t)+A(v(t))∋g(t),respectively, with f,g∈L1(o,T,X),then

A strong solution of problem(5.1)is a function u∈W1,∞(o,T;X),i.e.,u is locally absolutely continuous and differentiable almost everywhere,and u′(t)+A(u(t))∋f(t)for almost all t∈[o,T].

Concerning the existence of strong solutions,the following theorem is known(see[3,Theorem 4.5]or[6,p.1o8]).

Theorem 5.1If X is a Banach space with the Radon-Nikodym property,A:D(A)⊆X→2Xis a quasi-m-accretive operator,and f∈BV(o,T;X),i.e.,f is a function of bounded variation on[o,T],then problem(5.1)has a unique strong solution whenever xo∈D(A).

Our results rely on the following theorem.

Theorem 5.2[3,p.15o]Let X be a reflexive Banach space and let A be a quasi-maccretive operator in X.Let F:X→X be locally Lipschitz.Then,for each yo∈D(A),there is a local strong solution to the problem

Then,the solution is global.

On the other hand,we say that u∈C(o,T;X)is a weak solution of problem(5.1)if there are sequences(un)⊆W1,∞(o,T;X)and(fn)⊆L1(o,T;X)satisfying the following four conditions:

With respect to the existence of weak solutions,the following result,which is an easy consequence of Theorem 5.1,is important.

Theorem 5.3Let X be a Banach space with the Radon-Nikodym property.Then problem(5.1)admits a unique weak solution which is the unique integral solution of this problem.

We now recall some important facts regarding accretive operators which will be used in our paper(see,for example[3]).

Proposition 5.4Let A:D(A)→2Xbe an operator on X.The following conditions are equivalent:

·A is an ω-accretive operator,

·the inequality[u−v,x−y]s≥−ω‖x−y‖,holds for every x,y∈D(A)and u∈Ax,v∈Ay,

·For each o<λ<1ωthe resolvent Jλ:=(I+λA)−1:R(I+λA)→D(A)is a single-valuedipschitzian mapping.

Let Σ be a subset of ℝN.A function g:Σ×ℂ−→ℂ is said to satisfy the Carath´eodory conditions on Σ×ℂ if

(1)the function t−→g(t,u)is measurable on Σ for all u∈ℂ,

(2)the function u−→g(t,u)is continuous on u for almost all t∈Σ.

If g satisfies the Carath´eodory conditions,then we can define an operator Ngon the set of functions ψ:Σ−→ℂ by

for every z∈Σ.The operator Ngis called the Nemytskii operator generated by g.In Lp-spaces the Nemytskii operator has been extensively investigated(see[2]and the references therein). However,we recall the following result due to Krasnoselskii which states a basic fact for the theory of these operators on Lp-spaces.

Proposition 5.5Assume that g is a Carath´eodory function.If the operator Ngacts from Lp1into Lp2,then Ngis continuous and takes bounded sets into bounded sets.

For the proof of this proposition we refer,for example,to[2].

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∗May 13,2015;revised September 2,2015.†

Khalid LATRACH.