Global existence, uniform boundedness and asymptotic behavior for Gierer--Meinhardt system with critical exponent $qr=(p-1)(s+1)$

Abstract

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.