Symmetric and Asymmetric Multiple Clusters In a Reaction-Diffusion System

We consider the Gierer-Meinhardt system in the interval (-1,1) with Neumann boundary conditions for small diffusion constant of the activator and finite diffusion constant of the inhibitor. A cluster is a combination of several spikes concentrating at the same point. In this paper, we rigorously show the existence of symmetric and asymmetric multiple clusters. This result is new for systems and seems not to occur for single equations. We reduce the problem to the computation of two matrices which depend on the coefficient of the inhibitor as well as the number of different clusters and the number of spikes within each cluster

On the stationary Cahn-Hilliard equation: Interior spike solutions

We study solutions of the stationary Cahn-Hilliard equation in a bounded smooth domain which have a spike in the interior. We show that a large class of interior points (the "nondegenerate peak" points) have the following property: there exist such solutions whose spike lies close to a given nondegenerate peak point. Our construction uses among others the methods of viscosity solution, weak convergence of measures and Liapunov-Schmidt reduction

Existence, classification and stability analysis of multiple-peaked solutions for the gierer-meinhardt system in R^1

We consider the Gierer-Meinhardt system in R^1. where the exponents (p, q, r, s) satisfy 1< \frac{ qr}{(s+1)( p-1)} < \infty, 1 <p < +\infty, and where \ep<<1, 0<D<\infty, \tau\geq 0, D and \tau are constants which are independent of \ep. We give a rigorous and unified approach to show that the existence and stability of N-peaked steady-states can be reduced to computing two matrices in terms of the coefficients D, N, p, q, r, s. Moreover, it is shown that N-peaked steady-states are generated by exactly two types of peaks, provided their mutual distance is bounded away from zero

Symmetry of Nodal Solutions for Singularly Perturbed Elliptic Problems on a Ball

In [40], it was shown that the following singularly perturbed Dirichlet problem \ep^2 \Delta u - u+ |u|^{p-1} u=0, \ \mbox{in} \ \Om,\] \[ u=0 \ \mbox{on} \ \partial \Om has a nodal solution u_\ep which has the least energy among all nodal solutions. Moreover, it is shown that u_\ep has exactly one local maximum point P_1^\ep with a positive value and one local minimum point P_2^\ep with a negative value and, as \ep \to 0, \varphi (P_1^\ep, P_2^\ep) \to \max_{ (P_1, P_2) \in \Om \times \Om } \varphi (P_1, P_2), where \varphi (P_1, P_2)= \min (\frac{|P_1-P_2}{2}, d(P_1, \partial \Om), d(P_2, \partial \Om)). The following question naturally arises: where is the {\bf nodal surface} \{ u_\ep (x)=0 \}? In this paper, we give an answer in the case of the unit ball \Om=B_1 (0). In particular, we show that for \epsilon sufficiently small, P_1^\ep, P_2^\ep and the origin must lie on a line. Without loss of generality, we may assume that this line is the x_1-axis. Then u_\ep must be even in x_j, j=2, ..., N, and odd in x_1. As a consequence, we show that \{ u_\ep (x)=0 \} = \{ x \in B_1 (0) | x_1=0 \}. Our proof is divided into two steps: first, by using the method of moving planes, we show that P_1^\ep, P_2^\ep and the origin must lie on the x_1-axis and u_\ep must be even in x_j, j=2, ..., N. Then, using the Liapunov-Schmidt reduction method, we prove the uniqueness of u_\ep (which implies the odd symmetry of u_\ep in x_1). Similar results are also proved for the problem with Neumann boundary conditions

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

Multi-interior-spike solutions for the Cahn-Hilliard equation with arbitrarily many peaks

We study the Cahn-Hilliard equation in a bounded smooth domain without any symmetry assumptions. We prove that for any fixed positive integer K there exist interior $K$--spike solutions whose peaks have maximal possible distance from the boundary and from one another. This implies that for any bounded and smooth domain there exist interior K-peak solutions. The central ingredient of our analysis is the novel derivation and exploitation of a reduction of the energy to finite dimensions (Lemma 5.5) with variables which are closely related to the location of the peaks. We do not assume nondegeneracy of the points of maximal distance to the boundary but can do with a global condition instead which in many cases is weaker

On a Two Dimensional Reaction-Diffusion System with Hypercyclical Structure

We study a hypercyclical reaction-diffusion system which arises in the modeling of catalytic networks and describes the emerging of cluster states. We construct single cluster solutions in full two-dimensional space and then establish their stability or instability in terms of the number N of components. We provide a rigorous analysis around the single cluster solutions, which is new for systems of this kind. Our results show that as N increases, the system becomes unstable

Critical Threshold and Stability of Cluster Solutions for Large Reaction-Diffusion Systems in R

We study a large reaction-diffusion system which arises in the modeling of catalytic networks and describes the emerging of cluster states. We construct single cluster solutions on the real line and then establish their stability or instability in terms of the number N of components and the connection matrix. We provide a rigorous analysis around the single cluster solutions, which is new for systems of this kind. Our results show that for N\leq 4 the hypercycle system is linearly stable while for N\geq 5 the hypercycle system is linearly unstable

Clustered spots in the FitzHugh-Nagumo system

We construct {\bf clustered} spots for the following FitzHugh-Nagumo system: \left\{\begin{array}{l}\ep^2\Delta u +f(u)-\delta v =0\quad \mbox{in} \ \Om,\\[2mm]\Delta v+ u=0 \quad \mbox{in} \ \Om,\\[2mm] u= v =0 \quad\mbox{on} \ \partial \Om, \end{array} \right. where \Om is a smooth and bounded domain in $R^2$. More precisely, we show that for any given integer $K$, there exists an \ep_{K}>0 such that for 0<\ep <\ep_K,\, \ep^{m^{'}} \leq \delta \leq \ep^m for some positive numbers $m^{'}, m$, there exists a solution (u_{\ep},v_{\ep}) to the FitzHugh-Nagumo system with the property that u_{\ep} has $K$ spikes Q_{1}^\ep, ..., Q_K^\ep and the following holds: (i) The center of the cluster \frac{1}{K} \sum_{i=1}^K Q_i^\ep approaches a hotspot point Q_0\in\Om. (ii) Set l^\ep=\min_{i \not = j} |Q_i^\ep -Q_j^\ep| =\frac{1}{\sqrt{a}} \log\left(\frac{1}{\delta \ep^2 }\right) \ep ( 1+o(1)). Then (\frac{1}{l^\ep} Q_1^\ep, ..., \frac{1}{l^\ep} Q_K^\ep) approaches an optimal configuration of the following problem: {\it $(*) \ \ \$ Given $K$ points $Q_1, ..., Q_K \in R^2$ with minimum distance $1$, find out the optimal configuration that minimizes the functional $\sum_{i \not = j} \log |Q_i-Q_j|$}

Asymmetric spotty patterns for the Gray-Scott model in R^2

In this paper, we rigorously prove the existence and stability of asymmetric spotty patterns for the Gray-Scott model in a bounded two dimensional domain. We show that given any two positive integers k_1,\,k_2, there are asymmetric solutions with k_1 large spots (type A) and k_2 small spots (type B). We also give conditions for their location and calculate their heights. Most of these asymmetric solutions are shown to be unstable. However, in a narrow range of parameters, asymmetric solutions may be stable