an angiogenesis model with nonlinear chemotactic response and flux at the tumor boundary

Nonlinear Analysis 72 (2010) 330–347

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

An angiogenesis model with nonlinear chemotactic response and flux atthe tumor boundaryManuel Delgado ∗, Inmaculada Gayte, Cristian Morales-Rodrigo, Antonio SuárezDpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain

a r t i c l e i n f o

Article history:Received 16 December 2008Accepted 15 June 2009

Keywords:AngiogenesisChemotaxisSemigroup theoryBifurcation methodsCoexistence states

a b s t r a c t

In this paper we consider a parabolic problem as well as its stationary counterpart ofa model arising in angiogenesis. The problem includes a chemotaxis type term and anonlinear boundary condition at the tumor boundary.We show that the parabolic problemadmits a unique positive global in time solution. Moreover, by bifurcation methods, weshow the existence of coexistence states and also we study the local stability of the semi-trivial states.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper we analyze a systemmodeling a crucial step in the tumor growth process: the angiogenesis. We suggest tothe interested reader the paper [1] to knowmultiple aspects of angiogenesis. We only focus our attention on the behavior oftwo populations involved in such process: the endothelial cells (ECs), denoted by u, which move and reproduce to generatea new vascular net attracted by a chemical substance generated by the tumor (TAF), denoted by v. They interact in a regionΩ ⊂ RN , N ≥ 1, that is assumed to be bounded and connected and with a regular boundary ∂Ω . Specifically, we considerthe case

∂Ω = Γ1 ∪ Γ2,

with Γ1 ∩ Γ2 = ∅, being Γi closed and open in the relative topology of ∂Ω . We assume that Γ2 is the tumor boundaryand Γ1 is the blood vessel boundary, see Fig. 1 where we have represented a particular situation, in this case the tumor issurrounded by the vessel.We assume Neumann homogeneous boundary conditions in both variables at Γ1, and also for the variable u at Γ2.

However, and as one of the principal novelties of this model, we consider that the tumor generates a quantity of TAFdepending nonlinearly on the preexisting one. Specifically, we assume that at Γ2

∂v

∂n= µ

v

1+ v

being µ a real number, although in the real application µ will be a positive constant. In such a case, µ represents the rateof TAF production. Here n = (n1, . . . , nN) stands for the normal outward vector to ∂Ω . We are assuming that the tumorgenerates TAF with a production term of a Michaelis–Menten type, so we suppose a saturation effect on the boundary, incontrast with the model in [2] where this term is linear.

∗ Corresponding author.E-mail addresses:[email protected] (M. Delgado), [email protected] (I. Gayte), [email protected] (C. Morales-Rodrigo), [email protected] (A. Suárez).

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M. Delgado et al. / Nonlinear Analysis 72 (2010) 330–347 331

Ω Γ2

Γ1

Fig. 1. A particular example of domainΩ .

Hence, we study the following parabolic problem and its stationary counterpart

ut −1u = −div(V (u)∇v)+ λu− u2 inΩ × (0, T ),vt −1v = −v − cuv inΩ × (0, T ),∂u∂n=∂v

∂n= 0 on Γ1 × (0, T ),

∂u∂n= 0,

∂v

∂n= µ

v

1+ von Γ2 × (0, T ),

u(x, 0) = u0(x), v(x, 0) = v0(x) inΩ ,

(1)

where 0 < T ≤ +∞, λ,µ ∈ R, c > 0 and

V ∈ C1(R), V > 0 in (0,∞)with V (0) = 0; (2)

and u0 and v0 are non-negative and non-trivial given functions.Let us explain the model. We are assuming that u is affected by two kind of movements: an undirect that is given by the

usual Laplace operator and a direct one induced by the gradient of v. Here, V models the chemotactic response of the ECs tothe chemoattractant TAF, and in this case this response depends on the density of u on a nonlinear way. This is a density-dependent sensitivity model in the terminology of [3]. Also, we suppose that ECs grow following a logistic law. On the otherhand, we assume that the TAF has a typically linear degradation,−v, it is also affected by a competition term with u,−cuvi.e. the TAF is consumed by the ECs and this consumption does not have a direct effect but through the chemotaxis term. Ina certain sense, the tumor attracts the ECs, in fact, they move towards Γ2 where the highest gradient of TAF is produced.Similar models to (1) with Neumann or non-flux boundary conditions have been studied extensively in the last years,

we refer to the recent review paper [3].Model (1) has three main difficulties, due basically to the nonlinearities: the reaction terms, the chemotactic response

and the boundary condition. The logistic term has been already used tomodel the cell growth and death. Also, the nonlinearchemotactic sensitivity has been considered in different papers, see for instance [4–9] and references therein.Wewould liketo mention that in [4] the function V is bounded and negative for large values of u, which provides bounds of the solutionand so prevents overcrowding. In all the previous papers a logistic growth is not considered for u. For this kind of growthwith V (u) = uwe refer to the papers [10–12] for parabolic problems and [13,14] for the stationary ones.However, the nonlinear term at the boundary of the tumor has not been used extensively in the literature, in fact we are

just aware about the paper [15] that combines a nonlinear boundary condition with a chemotaxis type term. The presenceof these terms implies a more involved and realistic model (see [16] pg. 30–31 and [17] Chapter 4 for an explanation andexamples of models with nonlinear boundary conditions). We think that the understanding of the behavior of the solutionscan help to decide if a system like (1) can be included in a suitable model for a complex biological phenomenon.We summarize our main results. With respect to the parabolic problem, we construct local solutions with the use of [18]

and global solutions via uniform bounds in time in the L∞(Ω) norm. More precisely, we show:

• There exists a unique local in time positive solution of (1).• If V is bounded, there exists a unique global in time positive solution of (1).

With respect to the stationary problem associated to (1), it is clear that there exist three kinds of solutions: the trivialone, the semi-trivial solutions (u, 0) and (0, v) and the solutionswith both components positive, the coexistence states (u, v).Basically, the trivial solution always exists, and:

332 M. Delgado et al. / Nonlinear Analysis 72 (2010) 330–347

• The semi-trivial solution (u, 0) exists if and only if λ > 0. In fact, in this case the semi-trivial solution is (λ, 0).• There exists a value µ1 > 0 such that the semi-trivial solution (0, v) exists if and only if µ > µ1.

With respect to the existence of coexistence states, there exist two functions F : (0,+∞) 7→ R, λ 7→ F(λ) andΛ : (µ1,+∞) 7→ R, µ 7→ Λ(µ) such that:

• If λ ≤ 0 or µ ≤ µ1 there does not exist any coexistence state.• Assuming that V ′(0) > 0, there exists at least a coexistence state if

(µ− F(λ))(λ−Λ(µ)) > 0. (3)

• Assuming that V ′(0) = 0, there exists at least a coexistence state of if λ > 0 and

µ− F(λ) > 0. (4)

With respect to the local stability of the semi-trivial solutions, we show that

• the trivial solution is locally stable if λ < 0 and µ < µ1 and unstable if λ > 0 or µ > µ1.• (u, 0) is locally stable if µ < F(λ), and unstable if µ > F(λ).• (0, v) is locally stable if λ < Λ(µ) (resp. λ < 0 if V ′(0) = 0), and unstable if λ > Λ(µ) (resp. λ > 0 if V ′(0) = 0).

So, when both semi-trivial solutions are stable or unstable, there exists at least one coexistence state. Hence, the curvesµ = F(λ) and λ = Λ(µ) appearing in (3), (4) are crucial in the study of existence of positive solutions and we will studythem in detail in this paper.In order to prove these results we mainly use bifurcation and sub and supersolution methods.The structure of this paper is as follows. In Section 2, we introduce some results of functional analysis. In Section 3,

we prove the existence of local solution in time and then the global existence in time. The next section is devoted to thestationary problem. Finally, in the last section, we briefly discuss some biological implications of our results.

2. Some results of functional analysis

We recall some facts about Sobolev spaces, the interpolation theory and the functional settingwhich is used in this paperfor reader’s convenience. This frame is a particular case of [18].(1) For Sobolev spaces with non-integer index, the so-called the Slobodeckii spaces, it holds

Theorem 2.1. Let Ω ⊂ RN be a bounded C2-domain.

1. ([18], pg. 25) If s0, s1 ∈ R+ \ N, s1 ≤ s0, then

W s0,p(Ω) → W s1,p(Ω) ∀p ∈ [1,∞].

2. ([19]. Th. 7.58) Let s0 > 0, 1 < p < q <∞ and s1 = s0 − Np +

Nq . If s1 ≥ 0, then

W s0,p(Ω) → W s1,q(Ω).

(2) With respect to the interpolation theory, we recall that if E0 and E1 are two normed spaces continuously embedded intoa topological space E , we can define the real interpolation for 0 < θ < 1 and 1 ≤ p ≤ ∞, which we denote (E0, E1)θ,p(see [20], Def. 22.1). It is true that

1. (E0, E1)θ,p ⊂ E0 + E1.2. E0 ∩ E1 → (E0, E1)θ,p.3. ([20] Lemma 22.2) If 0 < θ < 1 and 1 ≤ p ≤ q ≤ ∞, then

(E0, E1)θ,p → (E0, E1)θ,q.

Finally, due to (E0, E1)θ,1 ⊂ (E0, E1)θ,p ⊂ E0 + E1, it follows from [20], Lemma 25.2 (ii) that if (E0, E1)θ,p is a Banach space,then

∃C > 0, such , that ‖a‖(E0,E1)θ,p ≤ C‖a‖1−θE0‖a‖θE1 ∀a ∈ E0 ∩ E1. (5)

(3) Consider now a parabolic problem in the formzt +Az = F(z) inΩ × (0, T ),Bz = G(z) in ∂Ω × (0, T ),z(x, 0) = z0(x) inΩ,

(6)

where

Az := −N∑j,k=1

∂j(ajk∂kz + ajz)+ bj∂jz + a0z


is a general second order lineal operator in divergence form with L(Rr)-valued coefficients which we suppose to becontinuous on Ω , and

Bz :=N∑j,k=1

njγ (ajk∂kz + ajz)+ cγ (z)

is a boundary operator where γ denotes the trace operator and c is continuous on ∂Ω . HereΩ ⊂ RN is a bounded regulardomain and r is the number of equations of (6).Weknow that (A,B) is normally elliptic (see the definition of this concept in [18], pg. 18) since there exists a ∈ C(Ω,L(Rr))such that

ajk = a δjk (the Kronecker symbol), 1 ≤ j, k ≤ N,

the spectrum of a verifies

Re[σ(a(z))] > 0 ∀z ∈ Ω,

and the boundary condition is a Neumann boundary condition of each component of ∂Ω (see [18] pg. 21).The interpolation theory will be used in this paper for r = 1, that is, only for one equation. We write the results in this

case. If we denote

W s,γB :=

z ∈ W s,γ (Ω) : Bz = 0 if 1+

1γ< s ≤ 2

W s,γ (Ω) if − 1+ 1/γ < s < 1+ 1/γ(W−s,γ

′

(Ω))′ if − 2+ 1/γ < s ≤ −1+ 1/γ ,

(7)

then it holds.

Theorem 2.2 ([18], Th. 5.2; Th. 7.2). Suppose that (A,B) is a normally elliptic Neumann problem on Ω with C1-coefficients,1 < p <∞ and 0 < θ < 1. Then

1.

(Lp,W 2,pB )θ,p = W2θ,pB , 2θ ∈ (0, 2) \ 1, 1+ 1/p.

2. If −2+ 1/p < s0 < s1 < 1+ 1/p, and sθ := (1− θ)s0 + θs1 6∈ N, then,

(W s0,pB ,W s1,pB )θ,p = Wsθ ,pB .

When (A,B) is a normally elliptic problem on Ω with C1-coefficients, it is possible to construct an interpolation-extrapolation scale of spaces; we put E0 = Lp, E1 = W

2,pB , and E−1 a completion of E0 (see [18], pg. 29), we define

Eθ = (E0, E1)θ,p = W 2θp,B for 2θ ∈ (0, 2) \ 1, 1+ 1/p, (8)

andwe can extend the definition inductively for Ek+θ . Then, there exists a family of operators, Aθ ∈ L(E1+θ , Eθ ), being Aθ thenegative infinitesimal generator of an analytic semigroup on Eθ . The semigroup e−tAθ is defined e−tAθ : Eθ → Eθ (see [18],pg. 28-30).

3. The parabolic problem

3.1. Local existence

We are interested in the positive solutions of the system of PDEs (1), where c is a positive constant and λ,µ ∈ R.First, we will get the existence and uniqueness of a local positive solution. This will be proved by using the abstract theoryof quasilinear parabolic problems as done in [18]. After that, in the next section, we will study the global existence.

Theorem 3.1. Let p > N and suppose (2), that the initial data (u0, v0) ∈ (W 1,p(Ω))2 and u0 ≥ 0, v0 ≥ 0 a.e. in Ω . Then,problem (1) has a unique non-negative local in time classical solution

(u, v) ∈ (C([0, Tmax);W 1,p(Ω)) ∩ C2,1(Ω × (0, Tmax)))2,

where Tmax denotes the maximal existence time. Moreover, if there exists a function w : (0,+∞) → (0,+∞) such that, foreach T > 0,

‖(u(t), v(t))‖∞ ≤ w(T ), 0 < t < minT , Tmax, (9)

then Tmax = +∞.


Proof. We will prove that problem (1) is included in the frame of [18]. Let δ0 > 0 be and denote D0 := (−δ0,+∞) ×(−δ0,+∞) which is an open set containing the range of the solutions u and v. For r = 2 (number of equations) we definer2 functions ajk ∈ C2(D0;L(R2)) for 1 ≤ j, k ≤ r in the following way. For each (η1, η2) ∈ D0,

ajk

(η1η2

)=

(0 00 0

)if j 6= k; ajk

(η1η2

)=

(1 0

−V (η2) 1

)if j = k;

then, we put

A

(vu

)(vu

):= −

N∑j,k=1

∂j

(ajk

(vu

)(∂kv∂ku

))= −

(div (∇v)

div (∇u− V (u)∇v)

).

For the boundary conditions, we denote (cij)1 ≤ i, j ≤ 2 the following function matrix:

c11

(η1η2

)=

0 on Γ1

−µ

1+ η1on Γ2 c12

(η1η2

)=

0 on Γ10 on Γ2

c21

(η1η2

)=

0 on Γ1µV (η2)1+ η1

on Γ2c22

(η1η2

)=

0 on Γ10 on Γ2.

Then, we write

B

(vu

)(vu

):=

[N∑j=1

njγ(ajk

(vu

)(∂kv∂ku

))+ γ

((c11 c12c21 c22

)(vu

))]=

∂v

∂n−

µv

1+ v−V (u)

∂v

∂n+∂u∂n+µV (u)v1+ v

.This couple (A,B) is a linear boundary value problem normally elliptic.For the reaction term, we define the function f ∈ C2(D0;R2) by

f(η1η2

)=

(−η1 − cη1η2λη2 − η

22

).

Then, (1) can be written as the following quasilinear parabolic boundary value problem

∂t

(vu

)+A

(vu

)(vu

)= f

(vu

)inΩ × (0,∞),

B

(vu

)(vu

)=

(00

)on ∂Ω × (0,∞),(

v(x, 0)u(x, 0)

)=

(v0(x)u0(x)

)onΩ.

(10)

Then, Theorems 14.4 and 14.6 of [18] are applicable. The first one says that there exists a unique maximal weakW 1,p-solution which is a W 1,q-solution for each q ∈ (1,∞) ([18], Coroll. 14.5). The second one asserts that the solution is aclassical solution and the equation is verified point-wise because the boundary operator is equal to 0.The non-negativity of the solution follows from [18], Theorem 15.1. In fact, the hypothesis 15.3 is verified for r = 2

because V (0) = 0 and, so, the non-negativity of u holds. But, if u ≥ 0, then the maximum principle applied to the problem

vt −1v = −v − cuv inΩ × (0, T ),∂v

∂n= 0 on Γ1 × (0, T ),

∂v

∂n= µ

v

1+ von Γ2 × (0, T ),

v(x, 0) = v0(x) inΩ ,

(11)

implies that v ≥ 0.To reach the result about the global solution, we can invoke [18], Theorem 15.5 which is applicable because (A,B) is a

lower triangular system. For these systems and if ajk is a diagonal matrix, f is independent of the gradient and if there existsa functionw : R+ → R+ such that

‖(u(t), v(t))‖∞ ≤ w(T ), 0 ≤ t ≤ T <∞, t < Tmax,

then Tmax = ∞, suppose that the solution is bounded away from ∂Ω for each T > 0.


3.2. Global existence

In this subsection we are going to prove that the local positive solution (u, v) of the system (1) exists for every t > 0,i.e., it is a global solution. For that, in the following lemmas our purpose is to prove the criterium (9). Sincewe are prolongingthe local solutions obtained in the previous section, during this section we assume that

(u0, v0) ∈ W 1,p(Ω),

for some p > N . In order to get uniform bounds in time, that will also exclude the possibility of a blow-up at infinity, weneed to assume

V ∈ L∞(R). (12)

Theproof of theuniformbounds in time is carried out by a recursive procedure (see for instance [7],where a similar argumentis used.)From now, C will denote a constant that may vary from line to line and does not depend on time.The following lemma states that v is uniformly bounded in time via the well-known method of sub and supersolutions

(see for example [17]).

Lemma 3.2. There exists a constant C > 0 such that the v-solutions of (1) satisfy

‖v(t)‖C(Ω) ≤ C, ∀t ∈ [0, Tmax).

Proof. Observe that the v-solutions of the problem (1) are subsolutions of

wt −1w = −w inΩ × (0, Tmax),∂w

∂n= 0 on Γ1 × (0, Tmax),

∂w

∂n= |µ| on Γ2 × (0, Tmax),

v(x, 0) = v0(x) inΩ.

(13)

Now, let ϕ be the solution of the stationary problem−1ϕ + ϕ = 0 inΩ∂ϕ

∂n= 0 on Γ1,

∂ϕ

∂n= |µ| on Γ2.

It is well known that ϕ > 0 inΩ . Taking K > 0 big enough, Kϕ is a supersolution of (13). Therefore, v ≤ w ≤ Kϕ and theconclusion easily follows.

Lemma 3.3. The solution of the u-equation of (1) satisfies

‖u(t)‖2 ≤ C, ∀t ∈ [0, Tmax),

where C a positive constant.

Proof. Multiplying the v-equation by v and integrating inΩ , we obtain

d2dt

∫Ω

v2 +

∫Ω

v2 +

∫Ω

|∇v|2 =

∫Γ2

µv2

1+ v−

∫Ω

cuv2.

Adding the term− Cε2∫Ωv2 on both sides of the equality, taking into account Lemma 3.2 andmultiplying the before equality

by e(2−Cε)t , we have

ddt

(e(2−Cε)t

∫Ω

v2)+ 2e(2−Cε)t

∫Ω

|∇v|2 ≤ 2|µ|C |Γ2|e(2−Cε)t ,

where |Γ2| denotes the N − 1 dimensional Lebesgue measure of Γ2. Therefore, integrating in (0, t) and multiplying bye−(2−Cε)t we get∫

Ω

v2(t)+ e−(2−Cε)t2∫ t

0

(e(2−Cε)s

∫Ω

|∇v|2)ds ≤

(2|µ|C |∂Ω|

∫ t

0e(2−Cε)sds

)e−(2−Cε)t +

∫Ω

v20e−(2−Cε)t . (14)

In particular, from (14) we obtain

e−(2−Cε)t∫ t

0

(e(2−Cε)s

∫Ω

|∇v|2)ds ≤ C(‖v0‖2). (15)


Next, we multiply the u-equation by u and we integrate in the space variable

ddt

∫Ω

u2 = −2∫Ω

|∇u|2 + 2∫Ω

V (u)∇u · ∇v − 2∫Γ2

V (u)µuv1+ v

+

∫Ω

2λu2 − 2∫Ω

u3.

Adding 2∫Ωu2 on both sides of the equality we have

ddt

∫Ω

u2 + 2∫Ω

u2 = −2∫Ω

|∇u|2 + 2∫Ω

V (u)∇u · ∇v − 2∫Γ2

V (u)µuv1+ v

+ 2∫Ω

((λ+ 1)u2 − u3).

Owing to |V (s)| ≤ C for almost every s ∈ R and the inequality (λ+ 1)s2 − s3 ≤ C = C(λ) for all s ≥ 0, we deduce

ddt

∫Ω

u2 + 2∫Ω

u2 ≤ −2∫Ω

|∇u|2 + C∫Ω

2|∇u| |∇v| + C∫Γ2

|µ|uv1+ v

+ 2C |Ω|. (16)

The boundary term is bounded by the following expressions

C |µ|∫Γ2

uv1+ v

≤ C |µ|∫Γ2

u ≤ ε∫Γ2

u2 + C(ε),

using that for every ε > 0 there exists C(ε) > 0 such that s ≤ εs2 + C(ε) for every s ∈ R.Moreover,

ε

∫Γ2

u2 ≤ C(ε

∫Ω

u2 + ε∫Ω

|∇u|2).

Taking account of the previous estimates in (16) we get

ddt

∫Ω

u2 + (2− Cε)∫Ω

u2 ≤ (Cε − 2)∫Ω

|∇u|2 + C∫Ω

2|∇u| |∇v| + C(ε).

Then,ddt

∫Ω

u2 + (2− Cε)∫Ω

u2 ≤ ((C + 1)ε − 2)∫Ω

|∇u|2 + C(ε)∫Ω

|∇v|2 + C(ε).

Multiplying this inequality by e(2−Cε)t we have

ddt

(e(2−Cε)t

∫Ω

u2)≤ C(ε)e(2−Cε)t

∫Ω

|∇v|2 + Ce(2−Cε)t ,

and integrating in (0, t)we obtain

e(2−Cε)t∫Ω

u2 −∫Ω

u20 ≤ C(ε)∫ t

0

(e(2−Cε)s

∫Ω

|∇v|2)ds+ C

∫ t

0e(2−Cε)sds.

At this point, thanks to (15),∫Ω

u2 ≤ ‖u0‖22e−(2−Cε)t

+ C(‖v0‖2)+ C

and the Lemma is easily concluded.

Lemma 3.4. Let γ ∈ (1,∞), and t ∈ [t0, Tmax) for t0 > 0 small enough. If ‖u(t)‖γ ≤ C, then

‖v(t)‖W1,p(γ ,N) ≤ C,

where

p(γ ,N) = γN

N − 1+ εγ∀ε > 0.

Proof. System (10) has a local classical solution (u(t), v(t)) defined in (0, Tmax). So, we pose the nonhomogeneous linearproblem

vt −1v + v = f (t) := −cu(t)v(t) inΩ × (0, Tmax)∂v

∂n= 0 on Γ1 × (0, Tmax),

∂v

∂n= g(t) := µ

v(t)1+ v(t)

on Γ2 × (0, Tmax),

v(x, 0) = v0(x), inΩ ,

(17)


which we simply denote asvt −A0v = f (t) inΩ × (0, Tmax)B0v = g(t) on Γ × (0, Tmax),v(x, 0) = v0(x), inΩ ,

(18)

(A0,B0) generates an analytical semigroup whose generator is A0 ∈ L(Lγ (Ω)) with dom A0 = kerB0. The operatorB0cKerA0 : KerA0 → W 1−1/γ ,γ (∂Ω) is an homeomorphism and we denote Bc0 its inverse operator. The generalizedvariation-of-constants formula gives, for 2α ∈ (1/γ , 1+ 1/γ ),

v(t) = e−tAα−1v0 +∫ t

0e−(t−τ)Aα−1(f (τ )+ Aα−1Bc0g(τ )) dτ , (19)

and (19) is well defined for (f , g) ∈ C((0, Tmax),W 2α−2γ ,B0× ∂W 2αγ ) ([18], pg. 63). Note that it follows from (7.8) of [18] and

(7) that for−2 ≤ s < 0,

Lγ (Ω) → W s,γB0. (20)

We choose β := 1+ 1γ− ε < 2α < 1+ 1

γ. Owing to Theorem 2.2, there exists some 0 < θ < 1 such that

(W 2α−2,γB0,W 2α,γB0

)θ,γ = Wβ,γB0= Wβ,γ .

So, by (5), it holds that

∀w ∈ W 2α−2,γB0∩W 2α,γB0

= W 2α,γB0, ‖w‖Wβ,γ ≤ C ‖w‖

θ

W2α,γB0

‖w‖1−θW2α−2,γB0

.

If we recall (8), W 2α,γB0= Eα , W

2α−2,γB0

= Eα−1, Aα−1 : Eα → Eα−1 and the semigroup e−tAα−1 : Eα−1 → Eα−1. So, forf ∈ W 2α−2,γB0

= Eα−1,

‖e−(t−τ)Aα−1 f ‖Wβ,γ ≤ C‖e−(t−τ)Aα−1 f ‖θ

W2α,γB0

‖e−(t−τ)Aα−1 f ‖1−θ,γW2α−2,γB0

.

Since Aα−1 is the generator of an analytic semigroup then it holds that ([18], Remark 8.6.(c))

‖w‖W2α,γB0≤ C‖Aα−1w‖W2α−2,γB0

∀w ∈ Eα,

due to [0,+∞) ⊂ ρ(−A0) (ρ is the the resolvent set); this claim holds following Theorem 8.5 and (3.1) of [18]. So

‖e−(t−τ)Aα−1 f ‖Wβ,γ ≤ C‖Aα−1e−(t−τ)Aα−1 f ‖θ

W2α−2,γB0

‖e−(t−τ)Aα−1 f ‖1−θW2α−2,γB0

.

Finally, thanks to [21], Theor 1.3.4,

‖Aα−1e−(t−τ)Aα−1 f ‖θW2α−2,γB0

≤ C (t − τ)−θe−δ(t−τ)θ ‖f ‖θW2α−2,γB0

,

‖e−(t−τ)Aα−1 f ‖1−θW2α−2,γB0

≤ C e−δ(t−τ)(1−θ) ‖f ‖1−θW2α−2,γB0

,

and it results in

‖e−(t−τ)Aα−1 f ‖Wβ,γ ≤ C e−δ(t−τ)(t − τ)−θ ‖f ‖W2α−2,γB0

, (21)

for each δ ∈ (0, 1). Taking norm ‖ · ‖Wβ,γ on both sides of (19) and using (21), we get

‖v(t)‖Wβ,γ ≤ C t−θ‖v0‖W2α−2,γB0

+

∫ t

0e−δ(t−τ)(t − τ)−θ (‖f (τ )‖W2α−2,γB0

+ ‖Aα−1Bc0g(τ )‖W2α−2,γB0).

Taking into account Lemma 3.2, (20) and the Sobolev embeddingWβ,γ (Ω) → W 1,p(N,γ )(Ω), we deduce the Lemma.

Lemma 3.5. Let α ∈ [1,+∞). There exists C > 0 such that u(t) ∈ Lα(Ω) and

‖u(t)‖α ≤ C ∀t ∈ [t0, Tmax).


Proof. Fix α > 2. Multiplying the u-equation by αuα−1 and integrating inΩ , we obtain

ddt

∫Ω

uα =−4(α − 1)

α

∫Ω

|∇(uα/2)|2 + α(α − 1)∫Ω

V (u)uα−2∇v · ∇u

−α

∫Γ2

V (u)µv

1+ vuα−1 + λα

∫Ω

uα − α∫Ω

uα+1. (22)

We add∫Ωuα on both sides of the equality. Besides, we estimate the boundary term as follows

−α

∫Γ2

V (u)µv

1+ vuα−1 ≤ C

∫Γ2

uα−1.

At this point, we use the following inequality. Given ε > 0, there exists C(ε) > 0 such that sα−1 ≤ εsα+C(ε) ∀s ∈ R. Then,∫Γ2

uα−1 ≤ ε∫Γ2

uα + C(ε)|Γ2|

≤ C(ε

∫Ω

(uα/2)2 + ε∫Ω

|∇(uα/2)|2)+ C .

Using the Sobolev trace embedding and taking into account the above estimate, in (22), we have

ddt

∫Ω

uα +∫Ω

uα ≤−4(α − 1)

α

∫Ω

|∇(uα/2)|2 + α(α − 1)∫Ω

V (u)uα−2∇v · ∇u

+

∫Ω

((αλ+ 1)uα − αuα+1)+ Cε∫Ω

uα + Cε∫Ω

|∇(uα/2)|2 + C .

By the estimate (αλ+ 1)sα − αsα+1 ≤ C = C(λ, α) for every s ≥ 0 we get,

ddt

∫Ω

uα + (1− Cε)∫Ω

uα ≤(−4(α − 1)

α+ Cε

)∫Ω

|∇(uα/2)|2 + α(α − 1)∫Ω

V (u)uα−2∇v · ∇u+ C .

An easy computation gives

α(α − 1)∫Ω

V (u)uα−2∇v · ∇u = 2(α − 1)∫Ω

uα2−1V (u)∇v · ∇(uα/2)

and replacing it in the previous inequality, we obtain

ddt

∫Ω

uα + (1− ε)∫Ω

uα ≤(−4(α − 1)

α+ ε

)∫Ω

|∇(uα/2)|2 + 2(α − 1)∫Ω

uα2−1V (u)∇v · ∇(uα/2)+ C . (23)

Now, we deal with the second term on the right-hand side,

‖uα2−1V (u)∇v · ∇(uα/2)‖1 ≤ C‖u

α2−1‖θ(p)‖∇v‖p‖∇(uα/2)‖2

≤ C(ε)‖uα2−1‖2θ(p)‖∇v‖

2p + ε‖∇(u

α/2)‖22, (24)

where

θ(p) =2pp− 2

, p > 2.

Replacing (24) into (23), and considering ε small enough, we obtain

ddt‖uα/2‖22 + (1− Cε)‖u

α/2‖22 ≤ C(ε)‖u

α2−1‖2θ(p)‖∇v‖

2p + C . (25)

Now, we begin a recursive algorithm. Taking

γ0 := 2,

by Lemma 3.3 u(t) ∈ L2(Ω), ‖u(t)‖2 ≤ C . By Lemma 3.4 we have that v(t) ∈ W 1,p(γ0,N)(Ω), so, ‖∇v‖2p is finite. Choosingα ≤ 2+ 4

θ(p) we assure that uα2−1 ∈ Lθ (Ω). So, for 2 < α ≤ 2+ 2γ0

θ(p(γ0,N))we have

ddt‖uα/2(t)‖22 + (1− ε)‖u

α/2(t)‖22 ≤ C .


Therefore,

‖uα/2(t)‖22 ≤ ‖u(t0)α/2‖22 + C .

We have just proved that ‖u(t)‖Lα(Ω) ≤ C for 2 < α ≤ 2+ 2γ0θ(p(γ0,N))

.Now, we define

γ1 := 2+2γ0

θ(p(γ0,N)).

Owing to the previous reasoning, we have that u(t) ∈ Lγ1(Ω). So, by Lemma 3.4 v(t) ∈ W 1,p(γ1,N)(Ω). If we chooseα ≤ 2+ 2γ1

θ(p(γ1,N))wewill assure that uα/2−1(t) ∈ Lθ (Ω), and by (25) we obtain that for 2 < α ≤ 2+ 2γ1

θ(p(γ1,N)), u(t) ∈ Lα(Ω)

and ‖u(t)‖Lα(Ω) ≤ C .By a recursive algorithm we get that u(t) ∈ Lα(Ω) and ‖u(t)‖α ≤ C , for every α with 2 < α ≤ 2+ 2γn

θ(p(γn,N)), and

γn = 2+2γn−1

θ(p(γn−1,N)).

Using that

p(γn−1,N) = γn−1N

N − 1+ εγn−1,

we have that

γn = γn−1

(1−

2εN

)+2N.

The limit of γn is 1ε , so for ε > 0 as small as we want, we have that u(t) ∈ Lα(Ω) and ‖u(t)‖α ≤ C for all α ∈ [1,+∞).

Remark 3.6. Consequently, by Lemmas 3.4 and 3.5, we have obtained that u(t) ∈ Lp(Ω), ∇v ∈ Lp(Ω)N , for every p ∈[1,+∞).

In the following result we obtain a better bound of u, a L∞-bound. Let p > 1 and define

B := −1+ I,

with domain

D(B) :=u ∈ W 2,p(Ω) : ∂u/∂n = 0 on ∂Ω

.

For each β ≥ 0, define the sectorial operator Bβ (see [21]) and

Xβ := D(Bβ) with the norm ‖u‖Xβ := ‖Bβu‖p.

Lemma 3.7. Let 2β < 1, then for t ∈ [t0, Tmax) we have

‖u(t)‖Xβ ≤ C .

Proof. We have that

u(t) = e−tBu0 +∫ t

0e−(t−τ)B(−∇ · (V (u)∇v)+ (λ+ 1)u− u2)dτ ,

and so

‖u(t)‖Xβ ≤ ‖e−tBu0‖Xβ +

∫ t

0‖e−(t−τ)B(−∇ · (V (u)∇v)+ (λ+ 1)u− u2)dτ‖Xβ .

First, by [21, Theorem 1.4.3]

‖e−tBu0‖Xβ ≤ Ct−βe−δt‖u0‖p,

and

‖e−(t−τ)B((λ+ 1)u− u2)‖Xβ ≤ (t − τ)−βe−δ(t−τ)((λ+ 1)‖u‖p + ‖u2‖p)

where δ ∈ (0, 1).


Moreover, by [7, Lemma 2.1] we obtain

‖e−(t−τ)B(−∇ · (V (u)∇v))‖Xβ ≤ C‖e−(t−τ)1(−∇ · (V (u)∇v))‖Xβ

≤ C(ε)(t − τ)−1/2−β−εe−δ(t−τ)‖V (u)∇v‖p

where ε > 0 such that−1/2− β − ε > −1. Then,

‖u(t)‖Xβ ≤ Ct−βe−δt‖u0‖p + C

∫ t

0[(t − τ)−1/2−β−εe−δ(t−τ)‖V (u)∇v‖p

+ (t − τ)−βe−δ(t−τ)((λ+ 1)‖u‖p + ‖u2‖p)]dτ . (26)

Now, observe that

‖V (u)∇v‖p ≤ C‖∇v‖p.

Finally, thanks to Lemmas 3.4 and 3.5 we easily conclude the result from (26).

Theorem 3.8. We have that

‖u(t)‖∞ ≤ C for all t ∈ [0, Tmax),

and consequently we have proved the global existence.

Proof. Let p > N , 2β ∈(Np , 1

). Since 2β > N/pwe have by [21, Theorem 1.6.1] that

Xβ → C(Ω).

Thanks to Lemma 3.7 we have that ‖u(t)‖∞ < C for t > t0 > 0. Moreover, the local existence Theorem yields ‖u(t)‖∞ < Cfor t < t0. Therefore, ‖u(t)‖∞ < C for all t ≥ 0. Then, this result and Lemma 3.2 prove the global existence criterion (see(9)).

4. Steady-states

Consider now the stationary problem

−1u = −div(V (u)∇v)+ λu− u2 inΩ ,−1v = −v − cuv inΩ ,∂u∂n=∂v

∂n= 0 on Γ1,

∂u∂n= 0,

∂v

∂n= µ

v

1+ von Γ2.

(27)

First, we need to introduce some notations. For α ∈ (0, 1)we denote

X1 := w ∈ C2+α(Ω) : ∂w/∂n = 0 on ∂Ω,X2 := w ∈ C2+α(Ω) : ∂w/∂n = 0 on Γ1

and finally

X := X1 × X2.

Moreover, given a function c ∈ C(Ω)we denote by

cM := maxΩ

c(x), cL := minΩ

c(x).

We are interested in solutions (u, v) ∈ X of (27) with both components non-negative and non-trivial. Observe that thanksto the strong maximum principle, any component, u or v, of a non-negative and non-trivial solution is in fact positive in thewhole domainΩ .Consider functionsm ∈ Cα(Ω), g ∈ C1+α(Γ2) and the eigenvalue problem

−1φ +mφ = λφ inΩ ,∂φ

∂n= 0 on Γ1,

∂φ

∂n+ gφ = 0 on Γ2.

(28)

We are only interested in the principal eigenvalue of (28), i.e., the eigenvalues which have an associated positiveeigenfunction. In the following result we recall its main properties, see [22,23].


Lemma 4.1. Problem (28) admits a unique principal eigenvalue, which will be denoted by λ1(−1+ m;N ,N + g). Moreover,this eigenvalue is simple, and any positive eigenfunction, φ, verifies φ ∈ C2+α(Ω). In addition, λ1(−1 + m;N ,N + g) isseparately increasing in m and g; if m = Kϕ with ϕ > 0 inΩ then

limK→+∞

λ1(−1+ Kϕ;N ,N + g) = +∞. (29)

Moreover, we are going to consider the following Steklov eigenvalue problem with an eigenvalue on the boundary−1φ +mφ = 0 inΩ ,∂φ

∂n= 0 on Γ1,

∂φ

∂n+ gφ = µφ on Γ2.

(30)

It is well-known that there exists a principal eigenvalue of (30), we denote it by

µ1(−1+m;N ,N + g).

It is clear that µ is a principal eigenvalue of (30) if, and only if,

0 = λ1(−1+m;N ,N + g − µ),

and that thanks to Lemma 4.1

0 > λ1(−1+m;N ,N + g − µ)⇐⇒ µ > µ1(−1+m;N ,N + g), (31)

and analogously,

0 < λ1(−1+m;N ,N + g − µ)⇐⇒ µ < µ1(−1+m;N ,N + g).

Finally, we need to introduce the next notation: for a solution U0 of a nonlinear equation, we say that it is linearlyasymptotically stable (l. a. s.) if the first eigenvalue of the linearization around U0 is positive, and unstable if it is negative.

4.1. Semi-trivial solutions

Apart from the trivial solution (u, v) = (0, 0) of (27), there exist the semi-trivial solutions. It is clear that if v ≡ 0, thenu verifies

−1u = λu− u2 inΩ ,∂u∂n= 0 on ∂Ω ,

that is u ≡ λ. Hence, (λ, 0) is a semi-trivial solution of (27).On the other hand, when u ≡ 0 then v satisfies the equation

−1v + v = 0 inΩ ,∂v

∂n= 0 on Γ1,

∂v

∂n= µ

v

1+ von Γ2.

(32)

This equation was analyzed in [24] with Γ1 = ∅; we include a proof for reader’s convenience and some useful estimates.

Proposition 4.2. There exists a positive solution of (32) if, and only if,

µ > µ1 := µ1(−1+ 1;N ,N ).

Moreover, if the solution exists, it is the unique positive solution, and we denote it by θµ. Furthermore, θµ is l. a. s. for µ > µ1, i.e.,

λ1(−1+ 1;N ,N − µ(1/(1+ θµ))2) > 0. (33)

Finally,

1‖ϕ1‖∞

(µ

µ1− 1

)ϕ1 ≤ θµ ≤

1(ϕ1)L

(µ

µ1− 1

)ϕ1, inΩ , (34)

where ϕ1 is a positive eigenfunction associated to µ1.


Proof. Observe that if v is a positive solution of (32) we get

0 = λ1

(−1+ 1;N ,N − µ

11+ v

),

and so µ > 0. Moreover,

0 = λ1

(−1+ 1;N ,N − µ

11+ v

)> λ1(−1+ 1;N ,N − µ)

and then by (31), µ > µ1.To prove the existence of a solution, we apply the sub-supersolutionmethod. Take ϕ1 a positive eigenfunction associated

to µ1. Then (v, v) = (εϕ1,Mϕ1) is a sub-supersolution of (32) if

ε =

µ

µ1− 1

‖ϕ1‖∞and K ≥

µ

µ1− 1

(ϕ1)L.

The uniqueness follows by a standard argument. Indeed, observe that the map s 7→ µs/s(1+ s) = µ/(1+ s) is decreasing.To prove the stability, linearizing (32) around θµ, we need to prove that

λ1(−1+ 1;N ,N − µ(1/(1+ θµ))2) > 0.

For that, observe that v = θµ is a strict-supersolution of the following problem−1v + v = 0 inΩ ,∂v

∂n= 0 on Γ1,

∂v

∂n− µ

(1

1+ θµ

)2v = 0 on Γ2.

The next result provides us with a priori bounds of the solutions of (27) and bounds in the space X .

Lemma 4.3. Let (u, v) a coexistence state of (27). Then,

u ≤ λ and v ≤ θµ. (35)

Moreover, consider that (λ, µ) ∈ K ⊂ R2 compact. Then, there exists a constant C (independent of λ and µ) such that for anysolution (u, v) of (27) we have

‖(u, v)‖X ≤ C .

Proof. That v ≤ θµ is clear. On the other hand, observe that the first equation of (27) can be written as

−1u = −V ′(u)∇u · ∇v − V (u)(v + cuv)+ λu− u2.

If we denote by x ∈ Ω such that u(x) = maxΩ u, we have that (see Lemma 2.1 in [25])

u(x) ≤ λ−V (u(x))u(x)

v(x)[1+ cu(x)], (36)

whence we can conclude that u ≤ λ. This completes the proof of (35).Suppose (λ, µ) ∈ K ⊂ R2 compact and let (u, v) be a solution of (27). Then, we have that u and v are bounded in L∞(Ω)

for some constant C not depending on λ or µ. Now, going back to the v-equation and using theW 1,p(Ω)-estimates in [22]we get

‖v‖W1,p ≤ C(‖ − v − cuv‖p + ‖µv/(1+ v)‖p,∂Ω) ≤ C .

On the other hand, sinceW 1,p(Ω) → W 1−1/p,p(∂Ω)we have that

‖µv/(1+ v)‖W1−1/p,p(∂Ω) ≤ C‖µv/(1+ v)‖W1,p ≤ C,

where we have used that the map G(s) = s/(1+ s) is differentiable, G(0) = 0, |G′| ≤ C and the chain rule in Sobolev space.Hence, for p large

‖v‖C1(Ω) ≤ C‖v‖W2,p ≤ C(‖ − v − cuv‖p + ‖µv/(1+ v)‖W1−1/p,p(∂Ω)) ≤ C .

But, the u-equation in (27) can be written as follows

−1u+ V ′(u)∇u · ∇v = λu− u2 − V (u)(v + cuv)


and thus, u is bounded inW 2,p(Ω) for all p > 1, and so in C1(Ω). Now, again using the v-equation and the Schauder Theoryin Hölder spaces (see [26]), v is bounded in X2, and finally u in X1 with constants independent of λ and µ.

As an easy consequence of the above result, we have

Corollary 4.4. If λ ≤ 0 or µ ≤ µ1 then (27) does not possess a positive solution.The following result will be crucial in the existence result:

Proposition 4.5. 1. Assume that V ′(0) > 0 and fix λ > 0. Then, there exists µ0(λ) such that (27) does not possess coexistencestates for µ ≥ µ0(λ).

2. Fix µ > µ1. Then, there exists λ0(µ) such that (27) does not possess coexistence states for λ ≥ λ0(µ).

Proof. 1. Fix λ > 0 and assume that there exists a sequence µn → ∞ and coexistence states (un, vn) of (27). Denote byxn ∈ Ω such that un(xn) = ‖un‖∞. Then, by (36) we have

‖un‖∞ +V (‖un‖∞)‖un‖∞

vn(xn)(1+ c‖un‖∞) ≤ λ. (37)

Moreover, we know that un ≤ λ, and so

−1vn + (1+ cλ)vn ≥ 0,

and so, by a similar argument to the proof of Proposition 4.2, we get that

vn(x) ≥(µn

µ1− 1− cλ

)ϕ1(x),

and so by (37), we have

‖un‖∞ +V (‖un‖∞)‖un‖∞

(1+ c‖un‖∞)(µn

µ1− 1− cλ

)ϕ1(xn) ≤ λ.

Since ϕ1 ≥ δ > 0 inΩ and 1+ c‖un‖∞ ≥ 1, we obtain that V (‖un‖∞)/‖un‖∞ → 0, which is impossible due to V ′(0) > 0.2. Denote by um = minx∈Ω u(x). With a similar argument to the used one to prove (36) we get (see again Lemma 2.1

in [25])

λ ≤ um +V (um)um

v(xm)(1+ cum).

Since v ≤ θµ, if λ→∞ then um →∞. On the other hand, since v is solution of the second equation of (27) we have

0 = λ1

(−1+ 1+ cu;N ,N −

µ

1+ v

)≥ λ1(−1+ 1+ cum;N ,N − µ).

But observe that λ1(−1+ 1+ cum;N ,N − µ)→∞ as λ→∞, a contradiction.

Finally, another eigenvalue problem is analyzed−1φ + div(V ′(0)∇θµφ) = λφ inΩ ,∂φ

∂n= 0 on Γ1 ∪ Γ2.

(38)

Fix µ > µ1, we denote the principal eigenvalue as Λ(µ) and extend Λ(µ) = 0 for µ ≤ µ1. Observe that Λ(µ) = 0 whenV ′(0) = 0. In the following result we show some properties ofΛ(µ).

Proposition 4.6. Assume that V ′(0) > 0. Then

limµ→∞

Λ(µ) = ∞.

Proof. Under a change of variableΦ = eV ′(0)θµ2 φ (see [2] for references about this change of variable), we get

Λ(µ) = λ1

(−1+

V ′(0)2

4|∇θµ|

2+V ′(0)2

θµ;N ,N +V ′(0)2

µθµ

1+ θµ

).

And so, using (34) and (29) we obtain that

Λ(µ) > λ1

(−1+

V ′(0)2

µ

µ1− 1

‖ϕ1‖∞ϕ1;N ,N

)→∞

as µ→∞.


||(.,.)|| ||(.,.)||(λ,0) (λ,0)

(0,θμ) (0,θμ)

λλ∞ λ∞Λ(μ) λ

C+ C+

Case (a) Case (b)

Fig. 2. Bifurcation diagrams: Case (a) V ′(0) = 0 and Case (b) V ′(0) > 0.

Now, finally we denote by

F(λ) := µ1(−1+ 1+ cλ;N ,N ).

It is clear that F(0) = µ1, λ 7→ F(λ) is increasing and limλ→+∞ F(λ) = +∞.The main result is:

Theorem 4.7. 1. Assume that V ′(0) > 0. Then there exists at least a coexistence state if(λ−Λ(µ))(µ− F(λ)) > 0. (39)

2. Assume that V ′(0) = 0. Then there exists at least a coexistence state if λ > 0 and

µ > F(λ). (40)

Proof. We fix µ > µ1 and consider λ as bifurcation parameter, see Fig. 2 where we have drawn the two bifurcationdiagrams.First, we apply the Crandall–Rabinowitz theorem [27] in order to find the bifurcation point from the semi-trivial solution

(0, θµ). Consider the map F : R× X1 × X2 7→ Cα(Ω)× Cα(Ω)× Cα(Γ2) defined by

F (λ, u, v) :=(−1u+ div(V (u)∇v)− λu+ u2,−1v + v + cuv,

∂v

∂n− µ

v

1+ v

).

It is clear that F is regular, that F (λ, 0, θµ) = 0 and

D(u,v)F (λ0, u0, v0)(ξη

)=

−1ξ + div(V ′(u0)ξ∇v0 + V (u0)∇η)− λξ + 2u0ξ

−1η + η + cu0η + cv0ξ∂η

∂n− µ

(1

1+ v0

)2η

.Hence, for (u0, v0) = (0, θµ) and λ0 = Λ(µ) and we get that

Ker[D(u,v)F (λ0, 0, θµ)] = span(Φ1,Φ2)

whereΦ1 is an eigenfunction associated toΛ(µ) andΦ2 is the unique solution of(−1+ 1)Φ2 = −cθµΦ1 inΩ,∂Φ2

∂n= 0 on Γ1,

∂Φ2

∂n− µ(1/(1+ θµ))2Φ2 = 0 on Γ2.

Observe thatΦ2 is well-defined by (33). Hence, dim(Ker[D(u,v)F (λ0, 0, θµ)]) = 1.On the other hand, observe that

Dλ(u,v)F (λ0, u0, v0)(ξη

)=

(−ξ00

).

We can show that Dλ(u,v)F (λ0, 0, θµ)(Φ1,Φ2)t 6∈ R(D(u,v)F (λ0, 0, θµ)). Indeed, suppose that there exists (ξ , η) ∈ X suchthat D(u,v)F (λ0, 0, θµ)(ξ , η)t = (−Φ1, 0, 0), and so

−1ξ + V ′(0)div(ξ∇θµ)− λ0ξ = −Φ1 inΩ, ∂ξ/∂n = 0 on ∂Ω .


Under the change of variable ξ = eV′(0)θµς , the above equation is transformed into

−div(eV′(0)θµ∇ς)− λ0eV

′(0)θµς = −Φ1 inΩ ,∂ς

∂n= 0 on Γ1,

∂ς

∂n+ V ′(0)µ

θµ

1+ θµς = 0 on Γ2.

(41)

In a similar way, since Φ1 is an eigenfunction associated to Λ(µ) we can make the change of variable Φ1 = eαθµψ1, and(38) transforms into

−div(eαθµ∇ψ1) = λ0eαθµψ1 inΩ ,∂ψ1

∂n= 0 on Γ1,

∂ψ1

∂n+ V ′(0)µ

θµ

1+ θµψ1 = 0 on Γ2.

(42)

Now, multiplying (41) by ψ1 and (42) by ς , and subtracting we get

0 =∫Ω

Φ1ψ1,

an absurdity.It can be shown that R(D(u,v)F (λ0, 0, θµ)) has co-dimension 1.Hence, the point (λ, u, v) = (Λ(µ), 0, θµ) is a bifurcation point from the semi-trivial solution (0, θµ).Now,we can apply Theorem 4.1 of [28] and conclude the existence of a continuumC+ ⊂ R×X1×X2 of positive solutions

of (27) emanating from the point (λ, u, v) = (Λ(µ), 0, θµ) such that:

(i) C+ is unbounded in R× X1 × X2; or(ii) there exists λ∞ ∈ R such that (λ∞, λ∞, 0) ∈ cl(C+); or(iii) there exists λ ∈ R such that (λ, 0, 0) ∈ cl(C+).

Alternative (iii) is not possible. Indeed, if a sequence of positive solutions (λn, un, vn) ∈ cl(C+) such that λn → λ and(un, vn)→ (0, 0) uniformly, then denoting by

Vn =vn

‖vn‖∞,

and using the elliptic regularity, we have that Vn → V ≥ 0 is non-trivial in C2(Ω)with

−1V + V = 0 inΩ ,∂V∂n= 0 on Γ1,

∂V∂n= µV on Γ2,

and so µ = µ1, a contradiction.Fixedµ > µ1, we know by Corollary 4.4 and Proposition 4.5 that (27) does not possess a positive solution if λ ≤ 0 or λ is

large. Moreover, by Lemma 4.3 it follows thatC+ is bounded in X uniformly on compact subintervals of λ. Hence, alternative(i) does not occur. Therefore, alternative (ii) holds. When this alternative occurs, there exists a sequence (λn, un, vn) ofsolutions of (27) such that (λn, un, vn)→ (λ∞, λ∞, 0). Denoting by

Vn =vn

‖vn‖∞,

we obtain that Vn → V in C2(Ω)with

(−1+ 1+ cλ∞)V = 0 inΩ , ∂V/∂n = 0 on Γ1, ∂V/∂n = µV on Γ2,

that is, µ = F(λ∞). So, we can conclude the existence of a coexistence state for

λ ∈ (minΛ(µ), λ∞,maxΛ(µ), λ∞).

Observe that if V ′(0) = 0 thenΛ(µ) = 0. This completes the proof of the theorem.

In Fig. 3 we have drawn the coexistence regions, denoted by A, defined by (39) and (40) in Theorem 4.7 in both cases:Case (a) V ′(0) = 0 and Case (b) V ′(0) > 0.


A A

μ μ

μ1 μ1

λ1

μ=F(λ)

μ=F(λ)

λ=Λ(μ)

λ λ

Case (a) Case (b)

Fig. 3. Coexistence regions: Case (a) V ′(0) = 0 and Case (b) V ′(0) > 0.

4.2. Local stability

We study the local stability of the trivial and semi-trivial solutions.

Proposition 4.8. 1. The trivial solution of (27) is l. a. s. if λ < 0 and µ < µ1 and unstable if λ > 0 or µ > µ1.2. Assume that λ > 0. The semi-trivial solution (λ, 0) is l. a. s. if µ < F(λ) and unstable if µ > F(λ).3. Assume that µ > µ1. The semi-trivial solution (0, θµ) is l. a. s. if λ < Λ(µ) and unstable if λ > Λ(µ).

Proof. We only prove the third paragraph in the result, the other ones are similarly followed. Observe that the stability of(0, θµ) is given by the real parts of the eigenvalues for which the following problem admits a solution (ξ , η) ∈ X \ (0, 0)

−1ξ + div(V ′(0)∇θµξ)− λξ = σξ inΩ ,(−1+ 1+ cθµ)η = ση inΩ ,∂η

∂n= µ

(1

1+ θµ

)2η on Γ2.

(43)

Assume that ξ ≡ 0, then for some j ≥ 1 and we have

σ = λj

(−1+ 1+ cθµ;N ,N − µ

(1

1+ θµ

)2)> λ1

(−1+ 1;N ,N − µ

(1

1+ θµ

)2)> 0.

Now suppose that ξ 6≡ 0, then from the first equation of (43) we get that λ+ σ is a real eigenvalue associated to (38). Sinceλ < Λ(µ) it follows that σ > 0.Assume that λ > Λ(µ). Then,

σ1 := λ1(−1+ div(V ′(0)∇θµ)− λ;N ,N ) < 0.

Denote by ξ a positive eigenfunction associated to σ1, that is

−1ξ + div(V ′(0)∇θµξ)− λξ = σ1ξ inΩ , ∂ξ/∂n = 0 on ∂Ω .

Since σ1 < 0, then

λ1

(−1+ 1+ cθµ − σ1;N ,N − µ

(1

1+ θµ

)2)> 0,

and so there exists η solution of

(−1+ 1+ cθµ)η = σ1η inΩ ,∂η

∂n= µ

(1

1+ θµ

)2η on Γ2.

Then, σ1 < 0 is an eigenvalue of (43) with the eigenfunction associated (ξ , η), so (0, θµ) is unstable.

5. Interpretation

In this paper we have analyzed a problemmodeling the angiogenesis. For that, we have included a nonlinear chemotacticsensitivity, a logistic term to model the growth rate of the ECs and a nonlinear term at the boundary of the tumor. We haveshown the validity of the model proving the existence and the uniqueness of the global positive solution of the model.


Let us interpret some of our results. Fix the growth rate of ECs, that is, fix λ > 0; then F(λ) is also fixed. Hence, we candefine µ1(λ) and µ2(λ) as

µ1(λ) := minµ : λ = Λ(µ), µ2(λ) := maxµ : λ = Λ(µ).

It is clear that µ1(λ) = µ2(λ) = +∞ if V ′(0) = 0. On the other hand, remember that in the case V ′(0) > 0 for µ > µ0(λ)there does not exist any coexistence state, see Proposition 4.5. Our results of local stability, Proposition 4.8, show that whenµ is large enough, Λ(µ) > λ, then the semi-trivial solution (0, θµ) is linearly asymptotically stable, and so, this solutionindicates the long time behavior of the parabolic problem.Taking V ′(0) > 0 but small, we get that F(λ) < µ1(λ).With this notation, we know that:

1. In both cases (V ′(0) = 0 and V ′(0) > 0) for µ small, that is µ < F(λ), the TAF disappears, i.e., if the TAF generated bythe tumor is consumed by the ECs.

2. If V ′(0) = 0 and µ > F(λ), then both populations, TAF and ECs, coexist and so angiogenesis occurs.3. If V ′(0) > 0 and µ ∈ (F(λ), µ1(λ)), then both populations again coexist and so angiogenesis occurs. However, when µis much larger, µ > maxµ2(λ), µ0(λ), then the ECs tends to the extinction, and so angiogenesis does not occur.

Hence, forµ large there is a drastic change of behavior between the cases V ′(0) = 0 and V ′(0) > 0. Indeed, when V ′(0) > 0and µ is large, the ECs are sensitive to the TAF, they move towards the tumor and there, as µ is large, the TAF is also large,the two populations compete and they come into contact and the ECs die. However, when V ′(0) = 0, the ECs are nearlyinsensitive to the signal of TAF, and so they do not move quickly towards the tumor and so they do not come into contactwith TAF. In summary, in this model a case of low sensitivity response and a tumor generating a lot of TAF drives the systemto the extinction of ECs in the domain, due to the competition between the TAF and the ECS.

Acknowledgement

Supported by MEC under grants MTM2006-07932.

References

[1] N.V. Mantzaris, S. Webb, H.G. Othmer, Mathematical modeling of tumor-induced angiogenesis, J. Math. Biol. 49 (2004) 111–187.[2] M. Delgado, A. Suárez, Study of an elliptic system arising from angiogenesis with chemotaxis and flux at the boundary, J. Differential Equations 244(2008) 3119–3150.

[3] T. Hillen, K.J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009) 183–217.[4] T. Hillen, K.J. Painter, Global existence for a parabolic chemotaxis model with preventing of overcrowding, Adv. Appl. Math. 26 (2001) 280–301.[5] K.J. Painter, T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart. 10 (2002) 501–543.[6] D. Wrzosek, Global attractor for a chemotaxis model with prevention of overcrowding, Nonlinear Anal. TMA 59 (2004) 1293–1310.[7] D. Horstmann, M. Winkler, Boundedness vs Blow-up in a chemotaxis sytem, J. Differential Equations 215 (2005) 52–107.[8] V. Calvez, J.A. Carrillo, Volume effects in the Keller–Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl. 86 (2006) 155–175.[9] T. Cieslak, C. Morales-Rodrigo, Quasilinear non-uniformly parabolic–elliptic system modelling chemotaxis with volume filling effect. Existence anduniqueness of global-in-time solutions, Topol. Methods Nonlinear Anal. 29 (2007) 361–381.

[10] K. Osaki, T. Tsujikawa, A. Yagi, M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. TMA 51 (2002)119–144.

[11] J.I. Tello, M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations 32 (2007) 849–877.[12] M.Winkler, t Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl. 348 (2008) 708–729.[13] L. Dung, Coexistence with chemotaxis, SIAM J. Math. Anal. 32 (2000) 504–521.[14] L. Dung, H.L. Smith, Steady states of models of microbial growth and competition with chemotaxis, J. Math. Anal. Appl. 229 (1999) 295–318.[15] A. Kettemann, M. Neuss-Radu, Derivation and analysis of a systemmodeling the chemotactic movement of hematopoietic stem cells, J. Math. Biol. 56

(2008) 579–610.[16] R.S. Cantrell, C. Cosner, Spatial ecology via reaction-diffusion equations, in: Wiley Series in Mathematical and Computational Biology, John Wiley &

Sons, Ltd., Chichester, 2003.[17] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992.[18] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in: Schmeisser, Triebel (Eds.), Function Spaces,

Differential Operators and Nonlinear Analysis, in: Teubner Texte zur Mathematik, vol. 133, 1993, pp. 9–126.[19] R. Adams, Sobolev Spaces, Academic Press, New York, 1975.[20] L. Tartar, An introduction to Sobolev spaces and interpolation spaces, in: Lecture Notes of the Unione Matematica Italiana, vol. 3, Springer, Berlin,

2007.[21] D. Henry, Geometric theory of semilinear parabolic equations, in: Lecture Notes Math., vol. 840, Springer, 1981.[22] H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, in: W. Eckhaus (Ed.), New Developments in Differential Equations,

in: Math Studies, vol. 21, North-Holland, Amsterdam, 1976, pp. 43–63.[23] S. Cano-Casanova, J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems,

J. Differential Equations 178 (2002) 123–211.[24] K. Umezu, Nonlinear elliptic boundary value problems suggested by fermentation, NoDEA Nonlinear Differential Equations Appl. 7 (2000) 143–155.[25] Y. Lou, W.M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations 154 (1999) 157–190.[26] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, 1983.[27] M.G. Crandall, P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971) 321–340.[28] J. López-Gómez, Nonlinear eigenvalues and global bifurcation: Application to the search of positive solutions for general Lotka-Volterra reaction-

diffusion systems with two species, Differ. Integral Equ. 7 (1994) 1427–1452.

an angiogenesis model with nonlinear chemotactic response and flux at the tumor boundary

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