Fixed points and periodic orbits for a class of symmetric autonomous reaction-diffusion systems (Q2765097)

The author deals with the dynamics of the symmetric autonomous reaction-diffusion system NEWLINE\[NEWLINE{\partial u_k\over\partial t}=\lambda_k {\partial^2 \over\partial x^2}u_k+ u_kg\left( \sqrt{\sum^n_{j=1} u^2_j}\right)NEWLINE\]NEWLINE for \(t \in(0,\infty),\) \(x\in(0,1)\), \(k=1,\dots,n\), with Dirichlet boundary conditions, where the diffusion rates \(\lambda_1,\dots,\lambda_n\) are positive and \(g:[0, \infty)\to\mathbb{R}\) is \(C^1\) and satisfies NEWLINE\[NEWLINE\sup_{x\geq 0}\int^x_0\xi g(\xi)d\xi <+ \infty.NEWLINE\]NEWLINE He considers two different cases: a) the symmetric case, that is all diffusion rates coincide; b) the case of pairwise distinct diffusion rates.