Some non-local population models with non-linear diffusion

Some non-local population models with non-linear diffusion Francisco Julio S.A. Corrˆ ea1, Manuel Delgado2and Antonio Su´ arez2,3 1. Universidade Federal de Campina Grande Centro de Ciˆencias e Tecnologia Unidade Acadˆemica de Matem´atica e Estat´ıstica CEP:58.109-970, Campina Grande - PB - Brazil 2. Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Fac. de Matem´aticas, Univ. de Sevilla Calle Tarfia s/n, 41012-Sevilla, Spain E-mail addresses: [email protected], [email protected], [email protected] Abstract In this paper we present some theoretical results concerning to a non-local elliptic equation with non-linear diffusion arising from population dynamics. Key Words. Population dynamics, non-local terms, non-linear diffusion. AMS Classification. 35K57, 35B32, 92B05. 1 Introduction There is a lot of phenomena that can be modelled by reaction-diffusion PDE of the general form ut−∆u=f(x, u(x, t)) 3Corresponding author, Phone: (+34) 95 455 68 34, Fax:(+34) 95 455 28 98. The authors have been supported by the Spanish Ministry of Science and Technology under Grant MTM2006-07932 and the Spanish Ministry of Science and Innovation under Grant MTM2009-12367. The first author was supported by CNPq/Brazil, under grant 301603/2007-3. 2F. J. Corrˆea, M. Delgado, and A. Su´arez joint with initial and boundary conditions, x∈Ω, Ω is a bounded and regular domain of IRN,N≥1; t≥0, fa regular function and the unknown function u: Ω ×IR+7→ IR. In this kind of equation the relation between uand its derivatives is local, that is, all the functions are taken at the same point x. There are some phenomena where a non-local spatial term has to be included in this model. In this case, the equation has the form ut−∆u=f(x, u, B(u)) where Bis a non-local operator, for instance B(u) = ZΩ g(y, u(y, t))dy, see [18] for a general survey of these equations. In this paper we are interested in the stationary problem associated with the above problem. In fact, due to these motivations, we study the existence, uniqueness or multiplicity, and stability of positive solutions of the equation        −∆u=f(x, u, B(u)) in Ω, u= 0 on ∂Ω, (1.1) where fis a regular function specified later and B(u) = ZΩ b(x)uβ(x)dx, β > 0. Roughly speaking, there are several difficulties that appear when one introduces a nonlocal term in (1.1). Let us point some of them: a) In general, (1.1) has not a variational structure and so we can not apply the powerful tool of “variational methods” to attack (1.1). See [11] where a problem with a nonlocal term has a variational structure. b) In general, the equation (1.1) does not satisfy a maximum principle, and as main consequences, we can not apply directly some classical methods as sub-supersolutions, see Section 4 for more details. c) In general, the linearized operator of (1.1) at a stationary solution is an integraldifferential operator and it will not be self-adjoint, see Section 2 for more details. Some non-local population models with non-linear diffusion 3 Specifically, in this paper we study the following equation, arising in some cases from the population dynamics, of the form          −∆wm=wf x, ZΩ wrin Ω, w= 0 on ∂Ω, (1.2) with r > 0, fis a regular function and m≥1. Here, we are assuming that Ω is fully surrounded by an inhospitable area, since the population density is subject to homogeneous Dirichlet boundary conditions. The real parameter mrepresents the velocity of diffusion, the rate of movement of the species from high-density regions to low-density ones. In this context, m > 1 means that the diffusion is slower than in the linear case (m= 1), which seems to give more realistic models, see [22]. The term m > 1 was introduced in [22], see also [25], by describing the dynamics of biological population whose mobility depends upon their density. Finally, fdenotes the crowding effect. Observe that this term includes a non-local term. Non-local terms have been introduced at least to our knowledge, in population dynamic models in [21]. The presence of the nonlocal terms in (1.2), from the biological point of view means that the crowding effect depends not only on their own point in space but also depends on the entire population. The change wm=utransforms the problem (1.2) into          −∆u=uqfx, ZΩ upin Ω, u= 0 on ∂Ω, (1.3) with 0 < q < 1, p > 0. Specifically, in this note, we are concerned with the the nonlocal elliptic problem                  −∆u=uqλ+a(x)ZΩ b(x)upin Ω, u > 0 in Ω, u= 0 on ∂Ω, (1.4) where Ω is a bounded and regular domain of IRN,N≥1, a, b ∈C(Ω), b≥0, b6≡ 0, λ∈IR,0< q < 1, p > 0, 4F. J. Corrˆea, M. Delgado, and A. Su´arez and averifies either a > 0 or a < 0. The above equation is a nonlocal counterpart of the well known logistic equation, whose more general version is given by                  −∆u=uq(λ+a(x)up) in Ω, u > 0 in Ω, u= 0 on ∂Ω, (1.5) where λ, p, q and aare as above. Let us point an important fact on equations (1.5) and (1.4). When q= 1, the strong maximum principle implies that any positive solution u(positive means non-negative and non-trivial) is strictly positive (u(x)>0 for all x∈Ω.) This means that there are uniquely two kinds of solutions in this case: the trivial solution (the species is dead) and the strictly positive solution (the species survives in whole domain). However, when q < 1 a new type of solution appears: a non-negative and non-trivial solution ubut vanishing in a part of the domain Ω0⊂Ω, that is u(x) = 0 for x∈Ω0. This set is called dead-core, see [26] and [12] where conditions on the coefficients are given to assure the existence of dead cores. On the other hand, when q < 1 the nonlinear reaction term is not derivable at u= 0. This entails some theoretical problems to linearize: we can not linearize at u≡0 and some singular terms appear when one linearizes at a positive solutions u > 0. With respect to the mathematical analysis of (1.4) we consider two situations: (i) The Homogeneous Case. Here we suppose that ais a constant and we use fixed point to obtain existence results. In this case we are able to describe exactly the set of positive solution of (1.4). (ii) The Non-Homogeneous Case. Here we consider the situation in which adepends on x∈Ω. In this case, bifurcation theory and sub-supersolution method plays a key role. When q=p= 1 and a=−1 in [5] the authors proved the existence, uniqueness and stability of positive solution when λ > λ1,λ1stands for the principal eigenvalue of the operator −∆ in Ω under homogeneous Dirichlet boundary conditions. In this case, the Some non-local population models with non-linear diffusion 5 solution can be explicitly built, it is proportional to a positive eigenfunction associated to λ1. More recently, and again with q=p= 1 but aa function such that a≤a0<0, in [7] the authors proved the existence and uniqueness of positive solution for λ > λ1. In this paper, the authors used bifurcation methods to prove the results. See also [8] for a related problem. We study questions of stability of positive solutions by using heavily the results on nonlocal and singular eigenvalue problems contained in section 2 of this work. In sections 3 and 4 we study the local (1.5) and (1.4) equations, respectively. In the last section we discuss the main results of this paper and the differences between the local and the non-local equations. 2 Non-local eigenvalue problems In this section we study a non-local and singular eigenvalue problem, which appears when one linearizes around a positive solution of (1.4). Specifically, we study the following problem          −∆u+m(x)u−h(x)ZΩ g(x)u=λu in Ω, u= 0 on ∂Ω, (2.1) where m∈C1(Ω), h∈C(Ω) and g∈C1(Ω) and verify: for some α∈(−1,1) and β < 1 (Hm)|∂im|d(x, ∂Ω)2−αare bounded for all x∈Ω and i= 1, ..., N; (Hg) there exists K > 0 such that g(x)≤Kd(x, ∂Ω)−β, where d(x, ∂Ω) := dist(x, ∂Ω). Basically, in (2.1) there is a combination between a differential and an integral operator. Moreover, (2.1) has different difficulties: the existence of singular terms and that the operator is not self-adjoint (in fact it is self-adjoint if and only if hand gare proportional). When the coefficients are bounded, in [17] (see also [19]) the authors proved the existence of a sequences {λi}in the complex plane with finite multiplicity. Since we are interested in the existence of positive solution of (1.4), with respect to (2.1) we want to prove the existence of a principal eigenvalue of (2.1), that is, a real and simple eigenvalue with positive eigenfunction associated to it. Moreover, it is less than 6F. J. Corrˆea, M. Delgado, and A. Su´arez all the real parts of the other eigenvalues. In order to prove the existence of a principal eigenvalue for a non self-adjoint operator, the Krein-Rutman Theorem is a powerful tool, see [1] for a general version of this result. In our setting, this theorem says: consider f≥0, f6= 0, and consider the solution vof the linear integro-differential problem        −∆v+m(x)v−h(x)RΩg(x)v=f(x) in Ω, v= 0 on ∂Ω. If vis strictly positive, then there exists the principal eigenvalue, λ1∈IR. The following results are consequences of [4]: a) If h > 0, then vis strictly positive, and the existence of a principal eigenvalue follows. b) Given f≥0, f6= 0 there exists h < 0 such that the solution vbecomes negative in some part of Ω. c) There exists h0>0 such that if khk∞∈(0, h0) then vis strictly positive. What happens if khk∞large? In [4] the authors showed an example, with homogeneous Neumann boundary conditions, in which for khk∞large there are several eigenvalues less than an eigenvalue having a positive eigenfunction associated. Using the Krein-Rutman Theorem and results from [23], the next theorems were proved in [9]: Theorem 2.1. Assume that mverifies (Hm),h∈C1(Ω) ∩C(Ω), a non-negative and non-trivial function, g∈C1(Ω) is a non-negative and non-trivial function and verifies (Hg). Then, there exists a principal eigenvalue of (2.1), denoted by λ1(−∆ + m;h;g), which has an associated positive eigenfunction ϕ1∈C2(Ω) ∩C1,δ 0(Ω) for some δ∈(0,1). Moreover, λ1(−∆+m;h;g)is simple, and it is the unique eigenvalue having an associated eigenfunction without change of sign. As we said before, we need to study the sign of the principal eigenvalue in order to know the stability of a positive solution of (1.4). In the following result we give a criteria to ascertain the sign of λ1(−∆ + m;h;g). Some non-local population models with non-linear diffusion 7 Proposition 2.2. Assume that there exists a positive function u∈C2(Ω) ∩C1,δ 0(Ω), δ∈(0,1), such that −∆u+m(x)u−h(x)ZΩ g(x)u > 0in Ω(resp. <0in Ω). Then, λ1(−∆ + m;h;g)>0(resp. λ1(−∆ + m;h;g)<0). 3 The local problem In this section we collect the main results concerning to the local equation (1.5). We employ the following notation σ1(−∆ + m) := λ1(−∆ + m; 0; 0). From the results of [26], [2], [13], [14], [15] and references therein, the results can be summarized in the following way: Theorem 3.1. a) Assume a < 0. Then, there exists a positive solution of (1.5) if and only if λ > 0. When λ > 0the solution is strictly positive, unique and stable. b) Assume a > 0. (a) Assume p+q < 1. Then, there exists a value λ < 0such that there exists a positive solution of (1.5) if and only if λ≥λ. Moreover, if λ≥0the solution is strictly positive, unique and stable. (b) Assume p+q= 1. i. If σ1(−∆−a)>0there exists a positive solution of (1.5) if and only if λ > 0. The solution is strictly positive, unique and stable. ii. If σ1(−∆−a) = 0 there exists a positive solution of (1.5) if and only if λ= 0. Moreover, there exist infinite solutions and any positive solution is neutrally stable. iii. If σ1(−∆−a)<0there exists a positive solution of (1.5) if and only if λ < 0. 8F. J. Corrˆea, M. Delgado, and A. Su´arez (c) Assume 1< p+q < (N+2)/(N−2). Then there exists a value λ > 0such that (1.5) possesses a positive solution if and only if λ≤λ. Moreover, for λ∈(0, λ) the problem (1.5) possesses at least two positive solution, one of them is the minimal solution and this is the unique stable solution. In Figure 1 we have represented the bifurcation diagrams of (1.5). The case a < 0 is shown in Case 1. Assume now that a > 0. Cases 1, 2 and 3 show the case p+q= 1 and σ1(−∆−a)>0, σ1(−∆−a) = 0 and σ1(−∆−a)<0, respectively. In Cases 4 and 5 we have drawn the cases p+q < 1 and 1 < p+q < (N+2)/(N−2). We remark that in Cases 3, 4 and 5 the drawings are simple representations of the set of solutions, for example in Case 3 the solutions need not be unique in spite of the figure. caso1.pdf Figure 1: Bifurcation diagrams. Some non-local population models with non-linear diffusion 9 4 The non-local problem Before we state the main results in this case we need some notations. Consider a∈C(Ω), a > 0 and denote by ωathe unique positive solution of        −∆u=a(x)uqin Ω, u= 0 on ∂Ω, (4.1) and A:= ZΩ b(x)ωp a.(4.2) As we said in the introduction, in the non-local case the results and techniques used depend on the a. We distinguish two cases: 4.1 The homogeneous case Assume that ais constant, that is a∈IR. In this case, we denote by R:= ZΩ b(x)up(x)dx. We study the equation        −∆w=wq(λ+aR) in Ω, w= 0 on ∂Ω. (4.3) It is well-known that (4.3) possesses a unique positive solution, denoted by uR, if and only if λ+aR > 0 and in fact uR= ((λ+aR)+)1/(1−q)ω1. Then, to find positive of (1.4) is equivalent to study the unidimensional equation R= ((λ+aR)+)p/(1−q)ZΩ b(x)ωp 1= ((λ+aR)+)p/(1−q)A. We prove in [9]: Theorem 4.1. a) Assume a < 0. Then, there exists a positive solution of (1.4) if and only if λ > 0. Moreover, the solution, uλ, is unique and lim λ→0kuλk∞= 0,lim λ→+∞ kuλk∞= +∞.