Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition

This paper is concerned with the following Gierer-Meinhardt type systems subject to Dirichlet boundary conditions    ∆u− αu + u p vq + ρ(x) = 0, u > 0 in Ω, ∆v − βv + u r vs = 0, v > 0 in Ω, u = 0, v = 0 on ∂Ω, where Ω ⊂ R (N ≥ 1) is a smooth bounded domain, ρ(x) ≥ 0 in Ω and α, β ≥ 0. We are mainly interested in the case of different source terms, that is, (p, q) 6= (r, s). Under appropriate conditions on the exponents p, q, r and s we establish various results of existence, regularity and boundary behavior. In the one dimensional case a uniqueness result is also presented. AMS classification scheme numbers: Primary 35J55; Secondary 35B40, 35J60 Steady-state solutions for Gierer-Meinhardt systems 2 In 1972 Gierer and Meinhardt [8] proposed a mathematical model for pattern formation of spatial tissue structures in morphogenesis, a biological phenomenon discovered by Trembley [24] in 1744. The mechanism behind the Gierer-Meinhardt’s model is based on the existence of two chemical substances: a slowly diffusing activator and a rapidly diffusing inhibitor. The ratio of their diffusion rates is assumed to be small. The model introduced by Gierer and Meinhardt reads as    ut = d1∆u− αu + cρ p vq + ρ0ρ in Ω× (0, T ), vt = d2∆v − βv + c′ρ′ r vs in Ω× (0, T ), (0.1) subject to Neumann boundary conditions in a smooth bounded domain Ω. Here the unknowns u and v stand for the concentration of activator and inhibitor with the source distributions ρ and ρ′ respectively. In system (0.1), d1, d2 are the diffusion coefficients and α, β, c, c′, ρ0 are positive constants. The exponents p, q, r, s > 0 verify the relation qr > (p− 1)(s + 1) > 0. The model introduced by Gierer and Meinhardt has been used with satisfactory quantitative results for modelling the head regeneration process of hydra, an animal of few millimeters in length, consisting of 100,000 cells of about 15 different types and having a polar structure. The Gierer-Meinhardt system originates in the Turing’s one [23] introduced in 1952 as a mathematical model for the development of complex organisms from a single cell. It has been emphasized that localized peaks in concentration of chemical substances, known as inducers or morphogens, could be responsible for a group of cells developing differently from the surrounding cells. Turing discovered through linear analysis that a large difference in relative size of diffusivities for activating and inhibiting substances carries instability of the homogeneous, constant steady state, thus leading to the presence of nontrivial, possibly stable stationary configurations. A global existence result for a more general system than (0.1) is given in the recent paper of Jiang [10]. It has also been shown that the dynamics of the system (0.1) exhibit various interesting behaviors such as periodic solutions, unbounded oscillating global solutions, and finite time blow-up solutions. We refer the reader to Ni, Suzuki and Takagi [18] for the entire description of dynamics concerning the system (0.1). Many works have been devoted to the study of the steady-state solutions of (0.1), that is, solutions of the stationary system    d1∆u− αu + cρ p vq + ρ0ρ = 0 in Ω, d2∆v − βv + c′ρ′ r vs = 0 in Ω, (0.2) subject to Neumann boundary conditions. The main difficulty in the treatment of (0.2) is the lack of variational structure. Another direction of research is to consider the shadow system associated to (0.2), an idea due to Keener [11]. This system is obtained dividing by d2 in the second equation and then letting d2 → ∞. It has been shown Steady-state solutions for Gierer-Meinhardt systems 3 that nonconstant solutions of the shadow system associated to (0.2) exhibit interior or boundary concentrating points. Among the large number of works in this direction we refer the interested reader to [19], [20], [21], [25], [26] as well as to the survey papers of Ni [16], [17]. In this paper new features of Gierer-Meinhardt type systems are emphasized. More exactly, we shall be concerned with systems of the following type    ∆u− αu + u p vq + ρ(x) = 0, u > 0 in Ω, ∆v − βv + u r vs = 0, v > 0 in Ω, u = 0, v = 0 on ∂Ω, (0.3) in a smooth bounded domain Ω ⊂ R (N ≥ 1). Here u and v represent the concentration of the activator and inhibitor and ρ ∈ C(Ω) (0 < γ < 1) represents the source distribution of the activator. We assume that ρ ≥ 0 in Ω, ρ 6≡ 0 and α, β are nonnegative real numbers. The case ρ ≡ 0 is more delicate and involves a more careful analysis of the Gierer-Meinhardt system. This situation has been analyzed in the recent works [1], [2], [18], [21], [25], [26]. We are mainly interested in this paper in the case where the activator and inhibitor have different source terms, that is, (p, q) 6= (r, s). Let us notice that the homogeneous Dirichlet boundary condition in (0.3) (instead of Neumann’s one as in (0.2)) turns the system singular in the sense that the nonlinearities up vq and u r vs become unbounded around the boundary. The existent results in the literature for (0.3) concern the case of common sources of the concentrations, that is, (p, q) = (r, s). If p = q = r = s = 1 and ρ ≡ 0, the system (0.3) was studied in Choi and McKenna [1]. In Kim [12], [13] it is studied the system (0.3) with p = r and q = s. In the case of common sources, a decouplization of system is suitable in order to provide a priori estimates for the unknowns u and v. More precisely, if p = r and q = s then, subtracting the two equations in (0.3) and letting w = u − v we get the following equivalent form    ∆w − αw + (β − α)wv + ρ(x) = 0, in Ω, ∆v − βv + (v + w) p vq = 0 in Ω, v = w = 0 on ∂Ω. (0.4) Thus, the study of system (0.3) amounts to the study of (0.4) in which the first equation is linear. This is more suitable to derive upper and lower barriers for u and v (see [1], [12], [13]). For more applications of decouplization method in the context of elliptic systems we refer the reader to [14]. We also mention here the paper of Choi and McKenna [2] where the existence of radially symmetric solutions in the case p = r > 1, q = 1, s = 0 and Ω = B1 ⊂ R is discussed. In [2], a priori bounds for concentrations u and v are obtained through sharp estimates for the associated Green’s function. In our case, such a decouplization is not possible due to the fact that (p, q) 6= (r, s). In order to overcome this lack, we shall exploit the boundary behavior of solutions of Steady-state solutions for Gierer-Meinhardt systems 4 single singular equations associated to system (0.3). In turn, this approach requires uniqueness or suitable comparison principles for single singular equations that come from our system. These features are usually associated with nonlinearities having a sublinear growth and that is why we restrict our attention to the case p < 1. Our results extend those presented in [5], [6] and give precise answers to some questions raised in Choi and McKenna [1], [2] and Kim [12], [13]. Also the approach we give in this paper enables us to deal with various type of exponents. For instance, we shall consider the case p < 0 (see Theorems 1.4 and 1.5) which means that the nonlinearity in the first equation of (0.3) is singular in both its variables u and v. Furthermore, these results can be successfully applied to treat the case −1 < s ≤ 0 (see Remark ??).