The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system (Q1953035)

The paper deals with the one-dimensional steady-state problem associated to the Gierer-Meinhardt (GM) system which describes the process of hydra regeneration, namely, \[ \begin{cases}\varepsilon^2 a'' -a+\dfrac{a^p}{h^q}=0,\quad &x\in (-1,1),\\ d h''-\mu h+\dfrac{1}{\varepsilon}\dfrac{a^r}{h^s}=0,\quad &x\in (-1,1),\\ a'=h'=0,\end{cases}\leqno(1) \] where the solution \((a,h)\) satisfies some properties. The author studies the stability of the stripe solution \((a,h)\) by investigating the one-dimensional eigenvalue problem \[ \begin{cases}\lambda\phi=\varepsilon^2\phi''-(1+\varepsilon^2l^2)\phi+p\dfrac{a^{p-1}}{h^q}\phi-q\dfrac{a^p}{h^{q+1}}\eta\quad &\text{in }(-1,1),\\ \tau\lambda\eta=d\eta''-(\mu+dl^2)\eta+\dfrac{1}{\varepsilon}\left(r\dfrac{a^{r-1}}{h^s}\phi-s\dfrac{a^r}{h^{s+1}}\eta\right)\quad &\text{in }(-1,1),\\ \phi'(\pm 1)=\eta'(\pm 1)=0.\end{cases}\leqno(P) \] He shows that \((P)\) has exactly one real eigenvalue \(\lambda\) satisfying \(\displaystyle\lim_{\varepsilon\to 0}\lambda>0\). The existence of a single-spike solution of \((1)\) is also proved.