Multi-Peak Solutions for a Wide Class of Singular Perturbation Problems

In this paper we are concerned with a wide class of singular perturbation problems arising from such diverse fields as phase transitions, chemotaxis, pattern formation, population dynamics and chemical reaction theory. We study the corresponding elliptic equations in a bounded domain without any symmetry assumptions. We assume that the mean curvature of the boundary has \overline{M} isolated, non-degenerate critical points. Then we show that for any positive integer m\leq \overline{M} there exists a stationary solution with M local peaks which are attained on the boundary and which lie close to these critical points. Our method is based on Liapunov-Schmidt reduction

Solutions for the Cahn-Hilliard Equation With Many Boundary Spike Layers

In this paper we construct new classes of stationary solutions for the Cahn-Hilliard equation by a novel approach. One of the results is as follows: Given a positive integer K and a (not necessarily nondegenerate) local minimum point of the mean curvature of the boundary then there are boundary K-spike solutions whose peaks all approach this point. This implies that for any smooth and bounded domain there exist boundary K-spike solutions. The central ingredient of our analysis is the novel derivation and exploitation of a reduction of the energy to finite dimensions (Lemma 3.5), where the variables are closely related to the peak loations

Stationary solutions for the Cahn-Hilliard equation

We study the Cahn-Hilliard equation in a bounded domain without any symmetry assumptions. We assume that the mean curvature of the boundary has a nongenerate critical point. Then we show that there exists a spike-like stationary solution whose global maximum lies on the boundary. Our method is based on Lyapunov-Schmidt reduction and the Brouwer fixed-point theorem

Asymmetric patterns for the Gierer-Meinhardt system

In this paper, we rigorously prove the existence and stability of K-peaked asymmetric patterns for the Gierer-Meinhardt system in a two dimensional domain which are far from spatial homogeneity. We show that given any positive integers k_1,\,k_2 \geq 1 with k_1+k_2=K, there are asymmetric patterns with k_1 large peaks and k_2 small peaks. Most of these asymmetric patterns are shown to be unstable. However, in a narrow range of parameters, asymmetric patterns may be stable (in contrast to the one-dimensional case)

Higher-Order Energy Expansions and Spike Locations

We consider the following singularly perturbed semilinear elliptic problem: (I)\left\{ \begin{array}{l} \epsilon^{2} \Delta u - u + f(u)=0 \ \ \mbox{in} \ \Omega, \\ u>0 \ \ \mbox{in} \ \ \Omega \ \ \mbox{and} \ \frac{\partial u}{\partial \nu} =0 \ \mbox{on} \ \partial \Omega, \end{array} \right. where \Om is a bounded domain in R^N with smooth boundary \partial \Om, \ep>0 is a small constant and f is some superlinear but subcritical nonlinearity. Associated with (I) is the energy functional J_\ep defined by J_\ep [u]:= \int_\Om \left(\frac{\ep^2}{2} |\nabla u|^2 + \frac{1}{2} u^2- F(u)\right) dx \ \ \ \ \ \mbox{for} \ u \in H^1 (\Om), where F(u)=\int_0^u f(s)ds. Ni and Takagi proved that for a single boundary spike solution u_\ep, the following asymptotic expansion holds: J_\ep [u_\ep] =\ep^{N} \Bigg[ \frac{1}{2} I[w] -c_1 \ep H(P_\ep) + o(\ep)\Bigg], where c_1>0 is a generic constant, P_\ep is the unique local maximum point of u_\ep and H(P_\ep) is the boundary mean curvature function at P_\ep \in \partial \Om. In this paper, we obtain a higher-order expansion of J_\ep [u_\ep]: J_\ep [u_\ep] =\ep^{N} \Bigg[ \frac{1}{2} I[w] -c_1 \ep H(P_\ep) + \ep^2 [c_2 (H(P_\ep))^2 + c_3 R (P_\ep)]+ o(\ep^2)\Bigg] where c_2, c_3 are generic constants and R(P_\ep) is the Ricci scalar curvature at P_\ep. In particular c_3 >0. Some applications of this expansion are given

Mutually exclusive spiky pattern and segmentation modelled by the five-component meinhardt-gierer system

We consider the five-component Meinhardt-Gierer model for mutually exclusive patterns and segmentation. We prove rigorous results on the existence and stability of mutually exclusive spikes which are located in different positions for the two activators. Sufficient conditions for existence and stability are derived, which depend in particular on the relative size of the various diffusion constants. Our main analytical methods are the Liapunov-Schmidt reduction and nonlocal eigenvalue problems. The analytical results are confirmed by numerical simulations

Existence and stability of multiple spot solutions for the gray-scott model in R^2

We study the Gray-Scott model in a bounded two dimensional domain and establish the existence and stability of {\bf symmetric} and {\bf asymmetric} multiple spotty patterns. The Green's function and its derivatives together with two nonlocal eigenvalue problems both play a major role in the analysis. For symmetric spots, we establish a threshold behavior for stability: If a certain inequality for the parameters holds then we get stability, otherwise we get instability of multiple spot solutions. For asymmetric spots, we show that they can be stable within a narrow parameter range

On the Gierer-Meinhardt system with precursors

We consider the Gierer-Meinhardt system with a for the activator. Such an equation exhibits a typical Turing bifurcation of the second kind, i.e., homogeneous uniform steady states do not exist in the system. We establish the existence and stability of N-peaked steady-states in terms of the precursor and the inhibitor diffusivity. It is shown that the precursor plays an essential role for both existence and stability of spiky patterns. In particular, we show that precursors can give rise to instability. This is a new effect which is not present in the homogeneous case

On the Stationary Cahn-Hilliard Equation: Bubble Solutions

We study stationary solutions of the Cahn--Hilliard equation in a bounded smooth domain which have an interior spherical interface (bubbles). We show that a large class of interior points (the ``nondegenerate peak'' points) have the following property: there exists such a solution whose bubble center lies close to a given nondegenerate peak point. Our construction uses among others the Liapunov-Schmidt reduction method and exponential asymptotics

Stationary Multiple Spots for Reaction-Diffusion Systems

In this paper, we review analytical methods for a rigorous study of the existence and stability of stationary, multiple spots for reaction-diffusion systems. We will consider two classes of reaction-diffusion systems: activator-inhibitor systems (such as the Gierer-Meinhardt system) and activator-substrate systems (such as the Gray-Scott system or the Schnakenberg model). The main ideas are presented in the context of the Schnakenberg model, and these results are new to the literature. We will consider the systems in a two-dimensional, bounded and smooth domain for small diffusion constant of the activator. Existence of multi-spots is proved using tools from nonlinear functional analysis such as Liapunov-Schmidt reduction and fixed-point theorems. The amplitudes and positions of spots follow from this analysis. Stability is shown in two parts, for eigenvalues of order one and eigenvalues converging to zero, respectively. Eigenvalues of order one are studied by deriving their leading-order asymptotic behavior and reducing the eigenvalue problem to a nonlocal eigenvalue problem (NLEP). A study of the NLEP reveals a condition for the maximal number of stable spots. Eigenvalues converging to zero are investigated using a projection similar to Liapunov-Schmidt reduction and conditions on the positions for stable spots are derived. The Green's function of the Laplacian plays a central role in the analysis. The results are interpreted in the biological, chemical and ecological contexts. They are confirmed by numerical simulations