Explicitly solvable nonlocal eigenvalue problems and the stability of localized stripes in reaction-diffusion systems (Q2793206)

The authors present their theory in the context of the Gierer-Meinhardt system (S) NEWLINE\[NEWLINEv_t=\epsilon^2\Delta v-v+\frac{v^3}{u^q},\quad \tau u_t=\Delta u-u+\frac{v^3}{\epsilon u^s}, \quad \text{in } \Omega\subset\mathbb R^2,NEWLINE\]NEWLINE with homogeneous Neumann boundary conditions on \(\partial\Omega,\) when \(\Omega\) is either the rectangle \(\{-l<x_1<l,0<x_2<d\}\) (section 3), or the ring \(\{(r,\theta)| 0\leq r\leq l,~0\leq \theta<2\pi\}\) (section 4). They assume: \(q>0,~s\geq 0, \frac{3q}{2}-(s+1)>0\). They show, in both cases, that the associated non-local eigenvalue problem is explicitely solvable, in the sense that any unstable eigenvalue of this problem satisfies a simple and explicit transcendental equation. Numerical experiments are presented. Section 5 is devoted to the urban crime model and, in section 6, some open problems are given. The authors point out that their theory can be applied to a more general class of reaction-diffusion systems.