Multiple Spike Layers in the Shadow Gierer- Meinhardt System: Existence of Equilibria and the Quasi-invariant Manifold
نویسنده
چکیده
where ε is a small parameter, ⊂ R is a bounded domain with C3 boundary, and f ∈ C1,1(R) satisfies f ≡ 0 in (−∞,0), f (u) = O(u) as u → ∞, and 1 < p < (d+2)/(d−2) (if d = 2, then 1< p <∞). Additionally, we assume that there exists q > 1 such that for u > 0 the function f (u)/u is nondecreasing. A typical example of the nonlinearity f satisfying the above conditions is f (u) = u2 for u > 0 and f (u) ≡ 0 otherwise. We should point out here that our results apply for a slightly larger class of equations.We refer the reader to Section 2 for more detailed information about f . Equation (1.1), which is of interest in itself, also plays an important role in the analysis of steady-state solutions to systems of reaction-diffusion equations arising in biology. One well-known example is the Keller-Segel system in chemotaxis (see [28] and the references therein), whose equilibria are obtained by solving (1.1). In this paper we explore the connection between (1.1) and the activator-inhibitor system, which models biological pattern formation and was proposed by Gierer andMeinhardt in [20] and [30]. This aspect of our work is described in Section 1.2. Problem (1.1) has been studied extensively in recent years. Most of the work has been focused on establishing the existence of special solutions to (1.1), which are known as spike-layer solutions. Intuitively, a solution u of (1.1) is called the spikelayer solution if u = o(1) as ε → 0 except at an isolated point (a single spike layer) or points (multiple spike layers). At those points (peaks), u ≥ μ > 0. Single spike layers were studied by Ni and Takagi in a series of remarkable papers
منابع مشابه
On Single Interior Spike Solutions of Gierer-meinhardt System: Uniqueness and Spectrum Estimates
We study the interior spike solutions to a steady state problem of the shadow system of the Gierer-Meinhardt system arising from biological pattern formation. We rst show that at a nondegenerate peak point the interior spike solution is locally unique and then we establish the spectrum estimates of the associated linearized operator. We also prove that the corresponding solution to the shadow s...
متن کاملBifurcation of Spike Equilibria in the Near-Shadow Gierer-Meinhardt Model
In the limit of small activator diffusivity ε, and in a bounded domain in R with N = 1 or N = 2 under homogeneous Neumann boundary conditions, the bifurcation behavior of an equilibrium onespike solution to the Gierer-Meinhardt activator-inhibitor system is analyzed for different ranges of the inhibitor diffusivity D. When D = ∞, and under certain conditions on the exponents in the nonlinear te...
متن کاملAsymmetri Spike Patterns for the One-Dimensional Gierer-Meinhardt Model: Equilibria and Stability
متن کامل
Local Stability of Spike Steady States in a Simplified Gierer-meinhardt System
In this paper we study the stability of the single internal spike solution of a simplified Gierer-Meinhardt’ system of equations in one space dimension. The linearization around this spike consists of a selfadjoint differential operator plus a non-local term, which is a non-selfadjoint compact integral operator. We find the asymptotic behaviour of the small eigenvalues and we prove stability of...
متن کاملHopf Bifurcation of Spike Solutions for the Shadow Gierer-Meinhardt Model
In the limit of small activator diffusivity, the stability of a one-spike solution to the shadow Gierer-Meinhardt activator-inhibitor system is studied for various ranges of the reaction-time constant τ associated with the inhibitor field dynamics. By analyzing the spectrum of the eigenvalue problem associated with the linearization around a one-spike solution, it is proved, for a certain param...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1999