发表机构
Tianjin University; Beijing University of Technology（天津大学; 北京工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
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。
AI 中文摘要
本文研究如下Gierer--Meinhardt系统：\begin{equation*} \begin{cases} u_t=d_1\Delta u-a_1u+u^p/v^q+\delta_1(x),\quad x\in \Omega, t>0,\\\\ v_t=d_2\Delta v-a_2v+u^r/v^s+\delta_2(x), \quad x\in \Omega, t>0,\\\\ \partial_\nu u=\partial_\nu v=0,\qquad\qquad \qquad\qquad\quad x\in \partial \Omega, t>0,\\\\ u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in \Omega, \end{cases} \end{equation*} 其中$\Omega\subset \mathbb{R}^N$是有界域，$\nu$是光滑边界$\partial \Omega$的外法向量，$d_1,d_2, p,q,r>0$，$s>-1$，$p-1<r$，$qr=(p-1)(s+1)$，$\delta_1(x)$，$\delta_2(x)$，$u_0(x)$和$v_0(x)$是连续函数。对于每一对$d_1,d_2>0$，我们在一定条件下分别建立了临界指数$qr=(p-1)(s+1)$下Gierer--Meinhardt系统的全局存在性、一致有界性与渐近行为。
英文摘要
In this paper, we study the following Gierer--Meinhardt system \begin{equation*} \begin{cases} u_t=d_1Δu-a_1u+u^p/v^q+δ_1(x),\quad x\in Ω, t>0,\\ v_t=d_2Δv-a_2v+u^r/v^s+δ_2(x), \quad x\in Ω, t>0,\\ \partial_νu=\partial_νv=0,\qquad\qquad \qquad\qquad\quad x\in \partial Ω, t>0,\\ u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in Ω, \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^N$ is a bounded domain, $ν$ is the outer norm vector with respect to the smooth boundary $\partial Ω$, $d_1,d_2, p,q,r>0$, $s>-1$, $p-1<r$, $qr=(p-1)(s+1)$, $δ_1(x)$, $δ_2(x)$, $u_0(x)$ and $v_0(x)$ are continuous functions. For every pair $d_1,d_2>0$, we respectively establish the global existence, uniform boundedness and asymptotic behavior for Gierer--Meinhardt system with critical exponent $qr=(p-1)(s+1)$ under certain conditions.
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