On a parabolic-elliptic chemotactic model with coupled boundary conditions

On a parabolic-elliptic chemotactic model with coupled boundary conditions 1 Manuel Delgado1, Cristian Morales-Rodrigo1, Antonio Su´ arez1and J. Ignacio Tello2 1. Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Fac. de Matem´aticas, Univ. de Sevilla Calle Tarfia s/n, 41012-Sevilla, Spain 2. Departamento de Matem´atica Aplicada, Escuela de Inform´atica, Universidad Polit´ecnica de Madrid, Carretera de Valencia Km 7. Campus Sur. 28031-Madrid, Spain E-mail addresses: [email protected], [email protected], [email protected], [email protected] Abstract This paper deals with a nonlinear system of parabolic-elliptic type with a logistic source term and coupled boundary conditions related to pattern formation. We prove the existence of a unique positive global in time classical solution. We analyze also the stationary problem associated. Moreover it is proved, under the assumption of sufficiently strong logistic dumping, that there is only one nonzero homogeneous equilibrium, and all the solutions to the non-stationary tend to this steady-state for large times. AMS Classification. 35K45, 35K57, 92C17. Keywords. Chemotaxis, Global existence, Asymptotic behavior, Coexistence states. 1 Introduction In many areas of research, from the biology of the embryonic development to the tumoral growth, the models of the cell movement play a fundamental role. Broadly speaking, the continuous models fall in two main classes: the mechanochemical models and the chemotactic models. In the former, cells exert a traction on the extracellular matrix (ECM), which carries out a key role; in the latter, cells secrete a chemical substance, which attracts or repels, and move towards or away from the gradient of this chemical. There exists a large number of examples where both types of models are applied to describe different biological phenomena. Numerical simulations are used to compare the experimental data with the mathematical results and to justify these mathematical models (see, [7], [12]). In any case, the study of these systems have interest by itself (see, for instance, [13]). On the other hand, we must not forget the influence of the boundary conditions on the behavior of the solutions. Recently, nonlinear boundary conditions have been incorporated 1MD and AS have been supported by the Spanish Ministry of Science and Technology under Grant MTM2006-07932 and JIT by Spanish Ministry of Sciences and Technology under grant MTM2009-13655. April 26, 2014 M. Delgado, C. Morales-Rodrigo, A. Su´arez, J. I. Tello permitting a more accurate consideration of several situations (see, for instance, [6] and references therein). In [10], a generalized chemotactic model is considered whose origin was the spatial pattern formation in embryology. It includes a generalized boundary condition which permits to cover a number of them varying the control parameters. We think it is not difficult to apply these models in other fields, for example, in angiogenesis, the cell movement linked to the tumoral growth where the endothelial cells (ECM) move following the gradient of TAF, the chemicals secreted by the tumor cells; for this reason, we think it is interesting to study it. The nondimensional model is                  ut=D∆u−χ∇ · (u∇v) + µsu(1 −u) in Ω ×(0, T), vt= ∆v−sv +su γ+uin Ω ×(0, T), ρ1D∂u ∂n −χu ∂v ∂n= (1 −ρ1)(ρ2−u), ρ3 ∂v ∂n = (1 −ρ3)ρ2 1 + γ−von ∂Ω×(0, T), u(x, 0) = u0(x) in Ω, (1) where Ω ⊂IRdis a bounded domain with regular boundary, uis the cell density, vis the concentration of chemoattractant, D > 0 is the diffusion rate, χ > 0 is the chemotactic coefficient of the motile cells, µis the linear growth rate of the cell population, γis a constant governing the rate of chemoattractant production and degradation and sis a parameter which controls spatial and temporal scale. In [10], it is presented numerical solutions of this model in one spatial dimension and is discussed the behavior of the model in two dimensions. The authors consider also the possibility to give different values to the parameters ρ1, ρ2, ρ3in different parts of the boundary. During this work, we consider the following system of equations                  ut= ∆u−χ∇ · (u∇v) + µu(1 −u) in Ω ×(0, T), 0 = ∆v−v+u 1 + uin Ω ×(0, T), ∂u ∂n −χu ∂v ∂n =r(θ−u),∂v ∂n =r0θ 2−von ∂Ω×(0, T), u(x, 0) = u0(x) in Ω, (2) where µ, r, r0, θ, χ denote non-negative constants. As we can see, it is the model resulting from (1) for D=s=γ= 1, ρ2=θ,r=1−ρ1 ρ1 ,r0=1−ρ3 ρ3 and supposing that the temporal scale of chemical diffusion is much larger than the scale of diffusion of cells and consequently we can take vt= 0. We will suppose that θ, r and r0are nonnegative constants on ∂Ω. Since we are interested only in non-negative solutions we assume that u0(x)≥0 in Ω. Our purpose is the theoretical study of (2) and its associated stationary problem associated. Our main results can be summarized as follows: a) The parabolic problem has existence and uniqueness of global solution ∀r, r0, θ, µ ≥ 0, for suitable initial data. b) With respect to the elliptic problem, we can summarize the situation as follows: (a) If θ > 0, 2 On a parabolic-elliptic chemotactic model... April 26, 2014 •there exists at least a coexistence state ∀r, r0, µ ≥0, •there exists the trivial solution if, and only if, r=r0= 0; which is unstable for µ > 0, •there exists one semitrivial solution (0, v) if r= 0, r0>0; which is unstable for µ > 0. In particular, if θ= 1, the system has the homogeneous solution (u, v) = (1,1/2). Then, we prove that •if r, r0>0, then (1,1/2) is globally stable when µ≥0, •if r0= 0, r ≥0, then (1,1/2) is globally stable when µis big enough. (b) If θ= 0, there exists the trivial solution and there is no semitrivial solution (0, V ). We can find three curves, µ=h1(r), µ =h2(r), µ =h3(r), being h1(r)≥h2(r)≥h3(r), such that •if µ>h1(r), then there exists a coexistence state, •if µ<h3(r), then there exists no coexistence state, •if µ<h2(r), then the trivial solution is stable. It is worthy to be remarked that we use a fixed point argument to study the evolutive problem due to the fact that our system has one parabolic equation and one elliptic equation and the general theory for parabolic equations (which is used, for example, in [8]) has to be applied in a non-local framework, less constructive than the method we follow. On the other hand, the method of bifurcation to study the stationary problem (see also [8]) needs the knowledge of some nonnegative solution of the system to begin the branch of positive solutions; this solution is usually one of the semitrivial solutions, but in our case the system does not admit any semitrivial solution when r, r0,and θare positive and, for this reason, we use a decoupling method. The paper is organized as follows. In Sections 2 and 3, we study the existence and uniqueness of the global solution for the parabolic problem (2). This problem has only one possible constant coexistence state for θ= 1, (u, v) = (1,1 2); we study in this case the asymptotic behavior of the solutions of (2) in Section 4. In Section 5, we consider the steady problem associated to our system, by a decoupling method and a sub-supersolution method for nonlocal problems. We will obtain some results of existence of solutions and some results of stability of the semitrivial solutions when they exist. 2 Preliminaries First we observe that χ∇ · (u∇v) = χ(∇u· ∇v+u∆v) = χ∇u· ∇v+uv−u 1 + u. So, we may rewrite the system (2) as follows                    ut= ∆u−χ∇u· ∇v+χu u 1 + u−v+µu(1 −u) in Ω ×(0, T), 0=∆v−v+u 1 + uin Ω ×(0, T), ∂u ∂n =χur0θ 2−v+r(θ−u),∂v ∂n =r0θ 2−von ∂Ω×(0, T), u(x, 0) = u0(x) in Ω. (3) 3 April 26, 2014 M. Delgado, C. Morales-Rodrigo, A. Su´arez, J. I. Tello Next, in order to avoid the singularity of u u+ 1 we define the penalized function h(u) := u+ u++ 1 and we introduce the system                  ut= ∆u−χ∇u· ∇v+χu h(u)−v+µ χ(1 −u)in Ω ×(0, T), 0=∆v−v+h(u) in Ω ×(0, T), ∂u ∂n =χur0θ 2−v+r(θ−u),∂v ∂n =r0θ 2−von ∂Ω×(0, T), u(x, 0) = u0(x) in Ω. (4) Observe that we may consider the equation for uas a linear equation Ut−∆U=−χa(x, t)· ∇U+χb(x, t)Uin Ω, ∂U ∂n =χr0c(x, t)U+r(θ−U) on ∂Ω. where a(x, t) := ∇v, b(x, t) := h(u)−v+µ χ(1 −u), c(x, t) := θ 2−v. Thus, if b(x, t)∈L∞(Ω×(0, Tmax)), then, by the maximum principle, U(x, t) = u(x, t)≥0 in Ω ×[0, T) and a solution to (4) is a solution to (3) and viceversa. Now, having in mind that for positive solutions the systems (3) and (4) are equivalent if u, v ∈L∞(Ω×(0, Tmax)), we will show the local existence theorem for (4). Previously we prove the following result where the notation is taken from [2], Sections 6 and 7: Lemma 2.1 Let p > 1,1< β < 2α < 1 + 1 pand (W2α−2,p B, Aα−1)an element of the interpolation-extrapolation scale generated by A0:= −∆ + Iand the real interpolation functor, then ke−tAα−1ukWβ,p ≤Ce−νtt−κkukW2α−2,p B,(5) with κ(β)∈(0,1) and ν∈(0,1). Proof. By Theorem 7.2 of [2], we have (W2α,p B, W2α−2,p B)κ,p =Wβ,p B, therefore, using this reference, we know that there exists κ∈(0,1) such that ke−tAα−1ukWβ,p B ≤ ke−tAα−1ukκ W2α,p B ke−tAα−1uk1−κ W2α−2,p B . Next, we apply [2, Theorem 8.5] together with [2, (3.1)] and we get ke−tAα−1ukWβ,p B ≤ k(I+Aα−1)e−tAα−1ukκ W2α−2,p B ke−tAα−1uk1−κ W2α−2,p B . 4 On a parabolic-elliptic chemotactic model... April 26, 2014 Taking into account that I+Aα−1and Aα−1are sectorial operators with Reσ(I+Aα−1) = 1 + Reσ(Aα−1) = 2 then we invoke [9, Theorem 1.3.4]. Thus, there exist ν∈(0,1), α∈(0,2) such that ke−tAα−1ukWβ,p B ≤ k(I+Aα−1)e−t(I+Aα−1)etIukκ W2α−2,p B ke−tAα−1uk1−κ W2α−2,p B =eκtk(I+Aα−1)e−t(I+Aα−1)ukκ W2α−2,p B ke−tAα−1uk1−κ W2α−2,p B ≤e(κ(1−α)−ν(1−κ))tt−κkukW2α−2,p B. Finally, we pick 1 −α=−νand use the fact that Wβ,p B=Wβ,p to conclude the proof. Lemma 2.2 Let 1< β < 2α < 1+ 1 pthen there exists κ < 1such that Xκ,→Wβ,p, where Xκ:= D((I+Aα−1)κ) Proof. Arguing in the same manner as we did in the previous lemma we have that there exists θ < 1 such that kukWβ,p ≤Ck(I+Aα−1)ukκ W2α−2,p kuk1−κ W2α−2,p . Finally the lemma can be concluded with the use of [9, pg. 28, Exer. 11]. 3 Global existence in time Theorem 3.1 Let p > d and consider the initial data u0∈W1,p(Ω) with u0≥0. Then there exists τ(ku0kW1,p )such that the system (4) has a unique positive local in time solution (u, v)∈C([0, τ]; W1,p(Ω)) ∩ C1((0, τ); C2+α(Ω))2, and u(x, t), v(x, t)≥0for (x, t)∈Ω×[0, τ]. Moreover, the solution depends continuously on the initial data, i.e. if u(u0)and u(u0)denote the solutions to (4) with inial data u0 and u0respectively then ku(u0)−u(u0)k(C([0,τ];W1,p))2≤Cku0−u0kW1,p . Proof. The proof of the Theorem is based on a standard fixed point argument. Let XT:= C([0, T]; W1,p(Ω)). For each f∈XTwe consider the operator S:XT→ C([0, T]; W2,p(Ω)) f7→ S(f) = v, where vis the unique solution to      −∆v+v=h(f) in Ω ×(0, T), ∂v ∂n =r0θ 2−von ∂Ω×(0, T). 5 April 26, 2014 M. Delgado, C. Morales-Rodrigo, A. Su´arez, J. I. Tello Moreover for each t∈[0, T], thanks to [1], the following estimate is satisfied kv(t)kW2,p ≤C(t)kh(f)kp+kθ/2r0kW1−1/p,p(∂Ω).(6) Next, we consider the operator H:XT→XT f7→ H(f) = u, where uis the unique solution to the linear parabolic problem      ut−∆u=−χ∇f· ∇v+χf(h(f)−v) + µf(1 −f) in Ω ×(0, T ), ∂u ∂n =χfr0θ 2−v+r(θ−f) on ∂Ω×(0, T). (7) Let 2α∈1,1 + 1 p. Thanks to the generalized variations of constants formula, see [2] pg. 63, we can rewrite (7) in the following manner u(t) = e−tAα−1u0+Zt 0 e−(t−τ)Aα−1(F(f, v) + Aα−1Bcγg(f, v))dτ, (8) where γ:W1,p(Ω) →W1−1/p,p(∂Ω) denotes the trace operator and F(f, v) := −χ∇f· ∇v+χf(h(f)−v) + µf(1 −f), g(f, v) := χfr0θ 2−v+r(θ−f). Let us point out that, since Aα−1Bc∈ L(W2α−1−1/p,p(∂Ω), W2α−2,p B) together with the embedding W1−1/p,p(∂Ω) ,→W2α−1−1/p,p(∂Ω), then we can assert that Aα−1Bcγis well defined for g(f, v)∈W1,p(Ω). Next, we define the closed set BT R:= {f∈ C([0, T]; W1,p(Ω)) : kfkXT≤R}. Now, we have to verify that the conditions of the Banach fixed point Theorem are satisfied for the map H. Step 1. There exist R, T > 0 such that for any f∈BT R, it holds that H(f)∈BT R. From (8), thanks to (5) and the embedding Wβ,p(Ω) ,→W1,p(Ω) we get ku(t)kW1,p ≤Cku0kW1,p +CZt 0 e−ν(t−τ)(t−τ)−κ(kF(f, v)kW2α−2,p B+ +kAα−1Bcγg(f, v)kW2α−2,p B)dτ. Taking into account the embedding W1−1/p,p(∂Ω) ,→W2α−1−1/p,p(∂Ω) we have kAα−1Bcγg(f, v)kW2α−2,p(∂Ω) ≤Ckγg(f, v)kW2α−1−1/p,p(∂Ω) ≤Ckγg(f, v)kW1−1/p,p(∂Ω) ≤Ckg(f, v)kW1,p also, having in mind [2, (7.5),(7.8)], we have Lp:= W0,p B,→W2α−2,p B. Thus, we infer ku(t)kW1,p ≤Cku0kW1,p +CZt 0 e−ν(t−τ)(t−τ)−κ(kF(f, v)kp+kg(f, v)kW1,p )dτ. (9) 6 On a parabolic-elliptic chemotactic model... April 26, 2014 On one hand we have kF(f, v)kLp≤χ(k∇f· ∇vkp+kfh(f)kp+kfvkp) + µ(kfkp+kf2kp).(10) Now, we estimate each term of (10) separately k∇f· ∇vkp≤ k∇vk∞k∇fkp ≤CkvkW1,∞kfkW1,p ≤CkvkW2,p kfkW1,p ≤C(kh(f)kp+C(r0))kfkW1,p ≤C(kh(f)k∞+C(r0))kfkW1,p ≤C(kfkW1,p +C(r0))kfkW1,p . The remaining terms of (10) can be estimated in a similar way to obtain kF(f, v)kp≤C(χ, µ, r0,kfkW1,p ),(11) with C(χ, µ, r0,kfkW1,p ) an increasing function on its arguments. On the other hand we have kg(f, v)kW1,p ≤χθr0 2kfkW1,p +χr0kfvkW1,p +krθkW1,p +rkfkW1,p .(12) The term kfvkW1,p is estimated as follows kfvkW1,p ≤CkfkW1,p kvkW1,∞ ≤CkfkW1,p (kh(f)kp+C(r0)) ≤CkfkW1,p (kfkW1,p +C(r0)). So, we obtain kg(f, v)kW1,p ≤C(χ, r, r0, θ, kfkW1,p ),(13) where C(χ, r, r0, θ, kfkW1,p ) is an increasing function on its arguments. Now, we plug (11) and (13) in (9) to obtain ku(t)kW1,p ≤Cku0kW1,p +C(R)Zt 0 Ce−ν(t−τ)(t−τ)−κdτ ≤Cku0kW1,p +C(R)T1−κ. Thus, choosing R > Cku0kW1,p and τ0=T(R) sufficiently small then kukXτ0≤R. Moreover, kukXT≤Rfor all T≤τ0. Now, we fix R > Cku0kW1,p and T≤τ0is free to our disposal. Step 2. H is contractive. Let f1, f2∈BT Rthen, u1=H(f1)∈BT Rand u2=H(f2)∈BT Rand u1(t)−u2(t) = Zt 0 e−(t−τ)Aα−1((F(f1, v1)−F(f2, v2)) + Aα−1Bcγ(g(f1, v1)−g(f2, v2)))dτ. So, we obtain ku1(t)−u2(t)kW1,p ≤Zt 0 e−ν(t−τ)(t−τ)−κ(kF(f1, v1)−F(f2, v2)kp+ +kg(f1, v1)−g(f2, v2)kW1,p )dτ. (14) 7 April 26, 2014 M. Delgado, C. Morales-Rodrigo, A. Su´arez, J. I. Tello On one hand we have kF(f1, v1)−F(f2, v2)kp≤χk∇f2· ∇v2− ∇f1· ∇v1kp+χkf1h(f1)−f2h(f2)kp+ +µkf1−f2kp+µkf2 2−f2 1kp. (15) Now, we estimate each term of (15) separately k∇f2· ∇v2− ∇f1· ∇v1kp≤ k∇(f2−f1)· ∇v2kp+k∇f1· ∇(v2−v1)kp ≤C(kv2kW2,p kf1−f2kW1,p +Rkv1−v2kW2,p ) ≤C(C(R)kf1−f2kW1,p +Rkh(f1)−h(f2)kp). Taking into account that kh(f1)−h(f2)kp≤ k(f1)+−(f2)+k∞≤ kf1−f2k∞we get k∇f2· ∇v2− ∇f1· ∇v1kp≤C(R)kf1−f2kW1,p . For the remaining terms of (15) we can argue in a similar way to obtain kF(f1, v1)−F(f2, v2)kp≤C(R)kf1−f2kW1,p .(16) On the other hand we have kg(f1, v1)−g(f2, v2)kW1,p ≤χθr0 2+rkf1−f2kW1,p +χr0kf2v2−f1v1kW1,p . We deduce kf2v2−f1v1kW1,p ≤ kv2(f2−f1)kW1,p +kf1(v2−v1)kW1,p ≤ kv2kW1,∞kf1−f2kW1,p +kf1kW1,p kv2−v1kW1,∞ ≤C(R)kf1−f2kW1,p . Thus, we get kg(f1, v1)−g(f2, v2)kW1,p ≤C(R)kf1−f2kW1,p .(17) Finally, we put the estimates (16) and (17) in (14) to obtain ku1−u2kXT≤C(R)T1−κkf1−f2kXT. Hence, taking Tsufficiently small we prove that His contractive. Now we deal with the regularity of the solution. Let us fix any t∈(0, τ) then the equation for uhas the abstract representation du dt + (I+Aα−1)u=f(x, t), u(0) = u0. Thus, thanks to [9, Theorem 3.5.2] du dt (t)∈Xκwith κ < 1. In particular, by Lemma 2.2,we have du dt (t)∈Wβ,p for some β > 1, p > d. So, we obtain u∈ C1((0, τ); W1,p(Ω)) and since the v-equation preserves the regularity in time then v∈ C1((0, τ); W1,p(Ω)). We observe that    −∆v(t) + v(t) = h(u)(t) in Ω, ∂v ∂n(t) + r0v(t) = r0θ 2on ∂Ω. 8 On a parabolic-elliptic chemotactic model... April 26, 2014 Taking into account that h(u)(t)∈ Cα(Ω) then the elliptic regularity assures v(t)∈ C2+α(Ω). So, we have proved that v∈ C1((0, τ); C2+α(Ω)). Now we rewrite the u-equation as follows      −∆u(t) + ∇u(t)· ∇v(t) = f(t) in Ω, ∂u ∂n(t) + r−χr0θ 2−vu(t) = rθ on ∂Ω, where f(t) := (uh(u)−uv +µu(1 −u)−ut)(t). Since f(t)∈ Cα(Ω), r−χr0θ 2−v(t)∈ C1+α(∂Ω) and ∇v(t)∈ Cα(Ω) then elliptic regularity entails u(t)∈ C2+α(Ω). Next we observe that the positivity of (u, v) is consequence of the maximum principle for parabolic equations. At the end we show the continuity respect to the initial data, for this purpose we argue in the following manner. Let R > C(ku0kW1,p +ku0kW1,p ). We have u(u0)(t) = e−tAα−1u0+Zt 0 e−(t−τ)Aα−1(F(u(u0), v(u0)) + Aα−1Bcγg(u(u0), v(u0)))dτ =e−tAα−1u0+H(u(u0)) −e−tAα−1u0. Hence, we infer k(u(u0)−u(u0))(t)kW1,p ≤2ke−tAα−1(u0−u0)kW1,p +kH(u(u0)−H(u(u0))kW1,p . Taking supremum on time, thanks to the contractivity of H, we obtain ku(u0)−u(u0)kXT≤Cku0−u0kW1,p +kku(u0)−u(u0)kXT, with k < 1. Also we have k(v(u0)−v(u0))(t)kW2,p ≤C(t)kh(u(u0))(t)−h(u0(u0))(t)kLp ≤C(t)ku(u0)−u(u0)kW1,p .(18) From (18) the proof of the continuity can be easily concluded. Now we deal with the issue of global in time solutions. To this end we have just to show that ku(t)kW1,p ≤C(t) for all t<Tmax where Tmax stands for the maximal interval of existence. We observe that      −∆v+v=u 1 + uin Ω ×(0, Tmax), ∂v ∂n +r0v=r0θ 2on ∂Ω×(0, Tmax). Since    u(t) 1+u(t)  ∞≤1 then kv(t)kW2,p ≤Cfor all t∈[0, Tmax). Next we put the formula of generalized variations of constants to obtain u(t) = e−tAα−1u0+Zt 0 e−(t−τ)Aα−1(F(u, v) + Aα−1Bcg(u, v))dτ. 9 April 26, 2014 M. Delgado, C. Morales-Rodrigo, A. Su´arez, J. I. Tello where in the last inequality we used that f+=−(−f)−. We note that −∆v+v−u 1 + u=U1 (1 + ξ2)2 for some ξ2(x, t)∈(min{u, u},max{u, u}). After multiplying the previous expression by (v−u 1+u)−we get ZΩ ∇ v−u 1 + u−! 2 +ZΩv−u 1 + u2 − =ZΩ U1 (1 + ξ2)2v−u 1 + u− ≤ZΩ U− 1 (1 + ξ2)2v−u 1 + u− ≤1 2ZΩ U2 −+1 2ZΩv−u 1 + u2 − Taking into account that the boundary term is non-positive then, from the above inequality we deduce 1 2ZΩv−u 1 + u2 − ≤1 2ZΩ U2 −.(38) Plugging the estimate (38) into (37) we obtain χZΩ U+u 1 + u−v≤χ−1 0ZΩ U2 ++ZΩ U2 −. The previous inequality provides with the following bound in (36) d 2dt ZΩ U2 +≤ZΩ U2 +g(u, v, u) + χ−1 0ZΩ U2 ++ZΩ U2 −. Since g(u, v, u)≤Cthen d 2dt ZΩ U2 +≤CZΩ U2 ++ZΩ U2 −. In the same fashion we have d 2dt ZΩ U2 −≤CZΩ U2 ++ZΩ U2 −. Adding the above inequalities and taking into account that (U0)+= (U0)−= 0, we may invoke Gronwall’s Lemma to achieve U+=U−= 0 (39) which proves the step. Step 5. Since u≤u≤uand u≤1≤uthen ku(t)−1k∞≤u−u≤ −−1 0ln(0)eγ00t∀t > 0.(40) 16 On a parabolic-elliptic chemotactic model... April 26, 2014 Next we observe that ξ=1 2is the unique solution to the problem      −∆ξ+ξ=1 2in Ω ×(0,+∞), ∂ξ ∂n = 0 on ∂Ω×(0,+∞). Therefore z:= v−ξsatisfies      −∆z+z=u 1 + u−1 2in Ω ×(0,+∞), ∂z ∂n = 0 on ∂Ω×(0,+∞), (41) and elliptic regularity asserts kz(t)kW2,p ≤C    u(t)−1 2(1 + u(t))   ∞ ≤Cku(t)−1k∞, concluding the result. Corollary 4.4 If µ≥2χthen the solution (u, v)to (2) is globally exponentially asymptotically stable and converges to the homogeneous steady-state (1,1 2). Moreover, if µ≥2χ, then the previous homogeneous steady-state is the only positive solution to the steady-state problem associated to (2). Proof. Assume min u0= 0 then, by the strong maximum principle, min u(τ)>0 for arbitrary τ > 0 small as desired. Next, we observe that γ0<2χ−µ≤0 and thanks to Theorem 4.2 we conclude the first part. The second part is a direct consequence of the global stability. Another consequence of Theorem 4.2 is the next Corollary Corollary 4.5 If µ > χ 2then the solution (1,1 2)to (2) is locally exponentially asymptotically stable. 5 The stationary problem In this section, we analyze the stationary problem associated to (2), that is                      −∆u=−χ∇ · (u∇v) + µu(1 −u) in Ω, −∆v=−v+u 1 + uin Ω, ∂u ∂n −χu ∂v ∂n =r(θ−u) on ∂Ω, ∂v ∂n =r0θ 2−von ∂Ω. (42) First, in order to present and prove our main result, we need introduce some notation. Given functions a, b ∈ C(Ω) with a≥a0>0, r∈ C(∂Ω), we denote by λ1(a;b;N+r) the principal eigenvalue of the problem    −div(a(x)∇u) = λb(x)uin Ω, ∂u ∂n +r(x)u= 0 on ∂Ω. (43) 17 April 26, 2014 M. Delgado, C. Morales-Rodrigo, A. Su´arez, J. I. Tello Recall that λ1(a;b;N+r) is increasing in aand rand decreasing in b. On the other hand, denote by vr0the unique positive solution of    −∆v+v= 1 in Ω, ∂v ∂n +r0v=r0θ 2on ∂Ω. (44) It is clear that vr0= 1 if r0= 0. Moreover, if r0>0 min{1, θ/2} ≤ vr0≤max{1, θ/2}. Finally, observe that since u/(u+ 1) ≤1 we get v≤vr0in Ω. (45) Now, we are ready to state our main result: Theorem 5.1 a) Assume that θ > 0and r0≥0. Then, there exists at least a positive solution of (42) if µ≥0and r≥0. b) Assume that θ= 0 and r0≥0. Then, there exists at least a positive solution of (42) if µ>λ1(eχvr0; 1; N+r), and (42) does not possess a positive solution if 0≤µ≤λ1(1; eχvr0;N+r). Corollary 5.2 Assume that θ=r0= 0. Then, there exists at least a positive solution of (42) if, and only if, µ>λ1(1; 1; N+r), In order to prove the main result we are going to use a decoupling method and a subsupersolution method for non-local problems. Let us begin showing the validity of the this last method. Consider a continuous map B:C(Ω) 7→ C(Ω) and the non-linear equation    −∆u=f(x, u, B(u)) in Ω, ∂u ∂n +r(x)u=h(x) on ∂Ω, (46) where f: Ω ×IR × C(Ω) 7→ IR is a regular function; r, h ∈ C(∂Ω). Definition 5.3 We say that u, u ∈ C2(Ω) ∩ C(Ω) is a sub-supersolution of (46) if u≤u in Ωand a) −∆u−f(x, u, B(u)) ≤0≤ −∆u−f(x, u, B(u)) in Ω,∀u∈[u, u], b) ∂u ∂n +r(x)u≤h(x)≤∂u ∂n +r(x)uon ∂Ω. 18 On a parabolic-elliptic chemotactic model... April 26, 2014 Proposition 5.4 Assume that there exists a sub-supersolution of (46), u, u, in the sense of Definition 5.3. Then, there exists a solution u∈[u, u]of (46). Proof. Being the proof standard, we outline only it. Take M > 0 large enough such that u7→ f(x, u, ξ) + Mu is increasing for all x∈Ω and ξ∈ C(Ω) and λ1(1; 1; N+r) + M > 0. Consider the map T: [u, u]2→[u, u]2 w7→ u:= T(w), being uthe unique solution of the following problem    −∆u+Mu =f(x, w, B(w)) + Mw in Ω, ∂u ∂n +r(x)u=h(x) on ∂Ω, (47) where [u, u]2:= {u∈L2(Ω) : u≤u≤u}. It is not hard to show that we can apply the Schauder fixed point theorem to Tand conclude the result. To study system (42) we are going to apply the change of variable u=eχvw which transforms the first equation of (42) into    −div(eχv∇w) = µeχvw(1 −eχvw) in Ω, ∂w ∂n +rw =rθe−χv on ∂Ω. (48) With respect to this equation, we get: Proposition 5.5 Fix v∈ C(Ω) and denote by vL:= min x∈Ω v(x), vM:= max x∈Ω v(x). a) Assume that rθ > 0. Then, there exists a unique positive solution, denoted w, of (48) for µ≥0. Moreover, min{1, θ}e−χvM≤w≤max{1, θ}e−χvLif µ > 0, θe−χvM≤w≤θe−χvLif µ= 0.(49) b) Assume that rθ = 0. Then, if µ > 0there exists a unique positive solution of (48) if, and only if, µ>λ1(eχv;eχv;N+r). Moreover, µ−λ1(eχv;eχv;N+r) µeχvMkϕk∞ ϕ≤w≤e−χvL(50) where ϕis a positive eigenfunction associated to λ1(eχv;eχv, N +r). If µ= 0 there exists a positive solution if, and only if, r= 0. In such case, any positive constant is solution. 19 April 26, 2014 M. Delgado, C. Morales-Rodrigo, A. Su´arez, J. I. Tello Proof. We will apply the sub-supersolution method. a) Assume that rθ > 0. Take w:= ε > 0 and w:= K > 0 with ε, K to be chosen. Then, εand Kmust satisfy 0≤µ(1 −eχvε) in Ω, ε≤θe−χv on ∂Ω, and 0≥µ(1 −eχvK) in Ω, K≥θe−χv on ∂Ω. This proves (49). The uniqueness follows because the map w7→ µ(1 −eχvw) is decreasing when µ > 0, see [5]. In the case µ= 0 the equation (48) is linear, and hence it is clear the uniqueness result. b) Assume rθ = 0. Let wbe a positive solution of (48). Then, −div(eχv∇w)< µeχvwin Ω, ∂w ∂n +rw = 0 on ∂Ω, and so, multiplying by ϕand integrating we get that µ>λ1(eχv;eχv;N+r).For the existence of solution take in this case w:= εϕ with ε=µ−λ1(eχv;eχv;N+r) µeχvMkϕk∞ . Again, the uniqueness follows similarly. Finally, the case µ= 0 follows easily. We fix now µ > 0, the case µ= 0 will be treated separately. Fix v∈ C(Ω) and consider the equation (48). Denote by w(v) := (wvif rθ > 0 or if rθ = 0 and µ>λ1(eχv;eχv;N+r) , 0 in other case, being wvthe unique positive solution of (46), which exists by Proposition 5.5. Lemma 5.6 The operator v∈ C(Ω) 7→ w(v)∈ C(Ω) is continuous. Proof. Consider sequences vn→vin C(Ω) and wn:= w(vn). Observe that wnis solution of −div(an(x)∇wn) + wn=hn,in Ω, an(x)∂wn ∂n =gnon ∂Ω, being an(x) := eχvn, hn:= wn+µwn·(1 −eχvnwn), gn:= an·(−rwn+rθe−χvn). Observe that 0< α ≤an≤β < ∞ and thanks to (49) and (50) we get that khnk∞≤Cand kgnkL∞(∂Ω) ≤C, and then, by Theorem 2.1 in [3], it follows that kwnkCν(Ω) ≤C, 20 On a parabolic-elliptic chemotactic model... April 26, 2014 for ν∈(0,1) and some constants depending on αand β. Hence, for a subsequence, wn→win C(Ω), being wa weak solution of (46). The elliptic regularity proves that wis a classical solution of (46). Now, we can follow the lines of the proof of Lemma 3.1 of [4] to conclude that in fact w=w(v). Proof of Theorem 5.1. Now, we have to study the following non-local and nonlinear equation        −∆v+v=eχvw(v) 1 + eχvw(v)in Ω, ∂v ∂n +r0v=r0θ 2on ∂Ω. (51) To study this equation we apply Proposition 5.4 with v=vr0being vr0the unique positive solution of (44). We take as subsolution v= 0 . Observe that v= 0 is sub-solution and no solution of (51) if θr0≥0 and w(v)≥0∀v∈[0, vr0] and some inequality strict. It is clear that this holds if θr0>0 and paragraph a) follows for µ > 0. Assume now that r0= 0 and θ > 0. If r > 0 then rθ > 0 and by Proposition 5.5 we have that w(v)≥min{1, θ}e−χvM>0 for all µ > 0. If r= 0, by Proposition 5.5 µ−λ1(eχv;eχv;N+r) µeχvMkϕk∞ ϕ≤w(v) if µ>λ1(eχv;eχv;N) = 0. Hence, w(v)>0 if µ > 0. Finally assume that r0≥0 and θ= 0. If r= 0, by a similar reasoning, we need that µ > 0. If r > 0 we need that µ>λ1(eχv;eχv;N+r)∀v∈[0, vr0]. Thanks to (45) we get λ1(eχv;eχv;N+r)≤λ1(eχvr0; 1; N+r), hence it is enough that µ>λ1(eχvr0; 1; N+r). Finally, we show the non-existence result. Assume that rθ = 0. Observe that w(v) = 0 if µ > 0 and µ≤λ1(eχv;eχv;N+r). But λ1(eχv;eχv;N+r)≥λ1(1; eχvr0;N+r) whence we deduce the result. Consider now the case µ= 0. If rθ > 0 then there exists a unique w(v)>0 solution of (48) and the result follows in a similar way. If θ= 0 and r > 0 then w(v) = 0, and then u= 0. However, if r= 0, we obtain that w=Cfor any positive constant C, and then u=eχvC. 21 April 26, 2014 M. Delgado, C. Morales-Rodrigo, A. Su´arez, J. I. Tello It is enough now to study the equation for v −∆v+v=Ceχv 1 + Ceχv in Ω,∂v ∂n +r0v=r0θ 2on ∂Ω. We can argue as in the above case and conclude the result. With respect to the semitrivial and trivial solutions, we get: Proposition 5.7 a) The trivial solution (u, v) = (0,0) exists if and only if rθ =r0θ= 0. In such case, the solution is unstable for µ > λ1(1; 1; N+r)and stable for µ<λ1(1; 1; N+r). b) The semitrivial solution (0, Vr0)exists if and only if r= 0 and r0, θ > 0, being Vr0 the unique solution of    −∆V+V= 0 in Ω, ∂V ∂n +r0V=r0θ 2on ∂Ω.(52) In such case, (0, wr0)is unstable for µ > 0. Proof. a) The existence result is not hard to show. On the other hand, the stability of (0,0) is given by the real parts of the eigenvalues for which the following problem admits a solution (ξ, η)6= (0,0)                  −∆ξ−µξ =σξ in Ω, −∆η+η=ξ+ση in Ω, ∂ξ ∂n +rξ = 0 on ∂Ω, ∂η ∂n +r0η= 0 on ∂Ω. (53) If ξ= 0, then σ=λj(1; 1; N+r0)+1>0. Assume now that ξ6= 0, so σ=λj(1; 1; N+r)−µ≥λ1(1; 1; N+r)−µ. Then, if µ<λ1(1; 1; N+r) we obtain that σ > 0 and (0,0) is stable. Assume now that µ>λ1(1; 1; N+r). Then, σ1:= λ1(1; 1; N+r)−µ < 0. Denote by ξa positive eigenfunction associated to σ1, that is −∆ξ−µξ =σ1ξin Ω, ∂ξ ∂n +rξ = 0 on ∂Ω. Since σ1<0, then λ1(1; 1; N+r0)+1−σ1>0, and so there exists η6= 0 such that −∆η+η−σ1η=ξin Ω,∂η ∂n +r0η= 0 on ∂Ω. 22 On a parabolic-elliptic chemotactic model... April 26, 2014 Then, σ1<0 is an eigenvalue of (53) with associated eigenfunction (ξ, η), so (0,0) is unstable. b) Again the existence result is direct. Now, the linearization around (0, Vr0) is                    −∆ξ=−div(χξ∇Vr0) + µξ +σξ in Ω, −∆η+η=ξ+ση in Ω, ∂ξ ∂n −χξ ∂Vr0 ∂n = 0 on ∂Ω, ∂η ∂n +r0η= 0 on ∂Ω. (54) The first equation, after the change of variable ξ=eχVr0ψ, is transformed into −div(eχVr0∇ψ)=(µ+σ)eχVr0ψin Ω, ∂ψ ∂n = 0 on ∂Ω. If ξ= 0 then, since η6= 0, then σ=λj(1; 1; N+r0) + 1 >0. On the other hand, if ξ6= 0 we get σ=λj(eχVr0;eχVr0;N)−µ≥λ1(eχVr0;eχVr0;N)−µ > 0, since λ1(eχVr0;eχVr0;N) = 0 and µ < 0. Assume now that µ > 0, then σ1:= λ1(eχVr0;eχVr0;N)−µ < 0 and consider ψa positive eigenfunction associated to σ1, that is −div(eχVr0∇ψ) = (µ+σ1)eχVr0ψin Ω, ∂ψ ∂n = 0 on ∂Ω. Again, consider the change of variable ξ=eχVr0ψ, and ηthe solution of −∆η+η=ξ+σ1ηin Ω, ∂η ∂n +r0η= 0 on ∂Ω, which exists because λ1(1; 1; N+r0)+1−σ1>0. Then, σ1<0 is an eigenvalue of (54) with associated eigenfunction (ξ, η), so (0, Vr0) is unstable. References [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Comm. Pure Appl. Math. 12, (1959) 623-727. [2] H. Amann, Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems, Eds. Schmeisser and Triebel, Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Texte zur Mathematik, 133 (1993) 9-126. [3] D. Arcoya and J. D. Rossi, Antimaximum principle for quasilinear problems, Adv. Differential Equations 9(2004), 1185-1200. 23 April 26, 2014 M. Delgado, C. Morales-Rodrigo, A. Su´arez, J. I. Tello [4] J. Blat and K. J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 21-34. [5] H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986) 55–64. [6] R. S. Cantrell, C. Cosner and S. Mart´ınez, Global bifurcation of solutions to diffusive logistic equations on bounded domains subject to nonlinear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), 45-56. [7] M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci. 15 (2005), 1685-1734. [8] M. Delgado, I. Gayte, C. Morales-Rodrigo and A. Su´arez, An angiogenesis model with nonlinear chemotactic response and flux at the tumor boundary, Nonlinear Analysis (2009), doi:10.1016/j.na.2009.06.057 [9] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes Math. 840, Springer 1981. [10] M. R. Myerscough, P. K. Maini and K. J. Painter, Pattern formation in a generalized chemotactic model, Bulletin of Mathematical Biology 60 (1998), 1-26. [11] J. I. Tello and M. Winkler, On a Chemotaxis with logistic source, Communications in Partial Differential Equations, 32 (2007), 849 - 877. [12] M. J. Tindall, P. K. Maini, S. L. Porter and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis. II. Bacterial populations. Bull. Math. Biol. 70 (2008), 1570-1607 [13] M. Winkler, Chemotaxis with logistic source: very weak global solutions and their boundedness properties, J. Math. Anal. Appl. 348 (2008), 708-729. 24