Marius Ghergu, Vicentiu Radulescu
We study the existence or the nonexistence of classical solutions to a singular Gierer-Meinhardt system with Dirichlet boundary condition. The main feature of our model is that the activator and the inhibitor have different sources given by general nonlinearities. Additional regularity and uniqueness results are established for the one dimensional case.
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We establish the existence of multiple solutions for Gierer-Meinhardt system involving Neumann boundary conditions. The approach combines the methods of sub-supersolution and Leray-Schauder topological degree.
Three solutions with precise sign properties for Gierer-Meinhardt type system
该研究针对带Neumann边界条件的符号耦合Gierer-Meinhardt型系统，结合上下解方法与含摄动论证的Leray-Schauder拓扑度理论，证明了存在两个异号常号解和一个分量符号同步的变号解共三个精确符号性质的解，丰富了该类系统的解的存在性结论。
The global analysis of the shadow Gierer-Meinhardt system with multiplicative white noise and general linear boundary conditions is investigated in this paper. For this reaction-diffusion system, we employ a fixed point argument to prove local existence and uniqueness. Our results on global existence are based on a priori estimates of solutions.
The Gierer-Meinhardt system occurs in morphogenesis, where the development of an organism from a single cell is modelled. One of the steps in the development, is the formation of spatial patterns of the cell structure, starting from an almost homogeneous cell distribution. Turing proposed in his pioneering work different activator-inhibitor systems with different diffusion rates, which could trigger the emergence of such cell structures. Mathematically, one describes these activator-inhibitor systems as a coupled systems of reaction-diffusion equations with hugely different diffusion coefficie…
Nodal solutions with synchronous sign changing components and Constant sign solutions for singular Gierer-Meinhardt type system
本文研究了奇异Gierer-Meinhardt型系统，通过结合上下解法与Leray-Schauder拓扑度证明了三个解的存在性，包括两个定号解和一个同步变号结点解。
We consider a nonlocal parabolic PDE, which may be regarded as the standard semilinear heat equation with power nonlinearity, where the nonlinear term is divided by some Sobolev norm of the solution. In this paper, we are interested in constructing blowup solutions under some critical regimes. We had a complete new phenomenon of the blowup speed with a log correction term.
We consider the Gierer-Meinhardt system with small inhibitor diffusivity and very small activator diffusivity in a bounded and smooth two-dimensional domain. For any given positive integer $k$ we construct a spike cluster consisting of $k$ boundary spikes which all approach the same nondegenerate local maximum point of the boundary curvature. We show that this spike cluster is linearly stable. The main idea underpinning these stable spike clusters is the following: due to the small inhibitor diffusivity the interaction between spikes is repulsive and the spikes are attracted towards a nondegen…
The purpose of the current paper is to contribute to the comprehension of the dynamics of the shadow system of an activator-inhibitor system known as a Gierer-Meinhardt model. Shadow systems are intended to work as an intermediate step between single equations and reaction-diffusion systems. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the Gierer-Meinhardt model is reduced to a single though non-local equation whose dynamics will be investigated. We mainly focus on the derivation of blow-up results for this non-local equation wh…
The main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular Gierer-Meinhardt model on an isotropically evolving domain. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the Gierer-Meinhardt model is reduced to a single though non-local equation whose dynamics is thoroughly investigated throughout the manuscript. The main focus is on the derivation of blow-up results for this non-local equation, which can be interpreted as instability patterns of the shadow syste…
We consider the Gierer-Meinhardt system with small inhibitor diffusivity, very small activator diffusivity and a precursor inhomogeneity. For any given positive integer k we construct a spike cluster consisting of $k$ spikes which all approach the same nondegenerate local minimum point of the precursor inhomogeneity. We show that this spike cluster can be linearly stable. In particular, we show the existence of spike clusters for spikes located at the vertices of a polygon with or without centre. Further, the cluster without centre is stable for up to three spikes, whereas the cluster with cen…
In this paper we study ground-states of the fractional Gierer-Meinhardt system on the line, namely the solutions of the problem \begin{equation*} \left\{\begin{array}{ll} (-Δ)^su+u-\frac{u^2}{v}=0,\quad &\mathrm{in}~\mathbb{R},\\ (-Δ)^sv+\varepsilon^{2s}v-u^2=0,\quad &\mathrm{in}~\mathbb{R},\\ u,v>0,\quad u,v\rightarrow0~&\mathrm{as}~|x|\rightarrow+\infty. \end{array}\right. \end{equation*} We prove that given any positive integer $k,$ there exists a solution to this problem for $s\in[\frac12,1)$ exhibiting exactly $k$ bumps in its $u-$component, separated from each other at a distance $O(\var…
This paper deals with an Gierer-Meinhardt model, with three substances, formed Reaction-Diffusion system with fractional reaction. To prove global existence for solutions of this system presents difficulties at the boundednees of fractionar term. The purpose of this paper is to prove the existence of a global solution using a boundary functionel. Our technique is based on the construction of Lyapunov functionel.
Single-spike solutions to the 1D shadow Gierer-Meinhardt problem
本文通过引入广义双曲函数的假设，确定了一维影格雷尔-梅因哈特问题的精确径向对称解，适用于1 < p < ∞，并提供了内尖解和边界尖解的分析方法。
Steady-states of the Gierer-Meinhardt system in exterior domains
本文研究了外部区域中稳态 Gierer-Meinhardt 系统解的存在性与不存在性，通过结合不动点论证与障碍函数构造了在无穷远处具有预定渐近增长的解，该方法可推广至其他半线性椭圆系统。
We consider a Gierer-Meinhardt system on a surface coupled with a parabolic PDE in the bulk, the domain confined by this surface. Such a model was recently proposed and analyzed for two-dimensional bulk domains by Gomez, Ward and Wei (SIAM J. Appl. Dyn. Syst. 18, 2019). We prove the well-posedness of the bulk-surface system in arbitrary space dimensions and show that solutions remain uniformly bounded in parabolic Hölder spaces for all times. The proof uses Schauders fixed point theorem and a splitting in a surface and a bulk part. We also solve a reduced system, corresponding to the assumptio…
Global existence, uniform boundedness and asymptotic behavior for Gierer--Meinhardt system with critical exponent $qr=(p-1)(s+1)$
本文针对Gierer--Meinhardt系统在临界指数$qr=(p-1)(s+1)$下，建立了全局存在性、一致有界性与渐近行为，适用于任意正扩散系数对。
Localized Pattern Formation and Oscillatory Instabilities in a Three-component Gierer Meinhardt Model
本文提出半强相互作用区域下的三组分Gierer-Meinhardt模型，通过渐近分析与数值方法揭示尖峰振幅振荡和振荡运动两种不稳定性机制，展现比双组分系统更丰富的尖峰动力学行为。
In the current paper, we provide a thorough investigation of the blowing up behaviour induced via diffusion of the solution of the following non local problem \begin{equation*} \left\{\begin{array}{rcl} \partial_t u &=& Δu - u + \displaystyle{\frac{u^p}{ \left(\mathop{\,\rlap{-}\!\!\int}\nolimits_Ωu^r dr \right)^γ}}\quad\text{in}\quad Ω\times (0,T), \\[0.2cm] \frac{ \partial u}{ \partial ν} & = & 0 \text{ on } Γ= \partial Ω\times (0,T),\\ u(0) & = & u_0, \end{array} \right. \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial Ω;$ such problem is derive…
Steady state solutions for the Gierer-Meinhardt system in the whole space
研究全空间中 Gierer-Meinhardt 系统在无穷远处衰减为零的正解，发现在参数 $\lambda,\mu$ 较大且 $0<\sigma\leq1$ 时存在指数衰减解，并在 $\lambda=\mu=0$ 时导出多种存在性结果及临界指数作用。
The structure, linear stability, and dynamics of localized solutions to singularly perturbed reaction-diffusion equations has been the focus of numerous rigorous, asymptotic, and numerical studies in the last few decades. However, with a few exceptions, these studies have often assumed homogeneous boundary conditions. Motivated by the recent focus on the analysis of bulk-surface coupled problems we consider the effect of inhomogeneous Neumann boundary conditions for the activator in the singularly perturbed one-dimensional Gierer-Meinhardt reaction-diffusion system. We show that these boundary…
The Gierer-Meinhardt system in the entire space with non-local proliferation rates
本文研究全空间中带非局部卷积增殖率的稳态 Gierer-Meinhardt 系统，并在核的多种可积性条件下证明经典正解的存在与不存在性。
We analyze a coupled bulk-membrane PDE model in which a scalar linear 2-D bulk diffusion process is coupled through a linear Robin boundary condition to a two-component 1-D reaction-diffusion (RD) system with Gierer-Meinhardt (nonlinear) reaction kinetics defined on the domain boundary. For this coupled model, in the singularly perturbed limit of a long-range inhibition and short-range activation for the membrane-bound species, asymptotic methods are used to analyze the existence of localized steady-state multi-spike membrane-bound patterns, and to derive a nonlocal eigenvalue problem (NLEP) c…
Localized spot patterns, where one or more solution components concentrates at certain points in the domain, are a common class of localized pattern for reaction-diffusion systems, and they arise in a wide range of modeling scenarios. In an arbitrary bounded 3-D domain, the existence, linear stability, and slow dynamics of localized multi-spot patterns is analyzed for the well-known singularly perturbed Gierer-Meinhardt (GM) activator-inhibitor system in the limit of a small activator diffusivity $\varepsilon^2\ll 1$. Our main focus is to classify the different types of multi-spot patterns, an…
We consider positive solutions of the stationary Gierer-Meinhardt system. Under suitable conditions on the exponents $p,q,r$ and $s$, different types of a priori estimates are obtained, existence and non-existence results of nontrivial solutions are derived, for both positive and zero source production cases.
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