Torsion numbers, a tool for the examination of symmetric reaction-diffusion systems related to oscillation numbers
Let us put \(w(t,x)= (u(t,x), v(t,x))\), \((t,x)\in (0,\infty)\times[0, 1]\). This interesting and well-written paper deals with the following symmetric reaction-diffusion equation \[ w_t= \lambda w_{xx}+ wg(t, x,w,w_x,w_{xx},\|w\|),\quad w(t,0)= w(t,1)= 0,\tag{1} \] where \(g: (0,\infty)\times [0,1]\times \mathbb{R}^6\times [0,\infty)\to \mathbb{R}\) is continuous and continuously differentiable with respect to all variables except \(t\). For a solution of (1) the author introduces a nonnegative number called torsion number which vanishes iff the solution is planar, where we call the solution \(w\) planar, if the curves \(\gamma_t: [0,1]\ni x\mapsto w(t,x)\in \mathbb{R}^2\), \(t>0\), are contained in a space \(\{k(\cos\alpha, \sin\alpha): k\in\mathbb{R}\}\subset \mathbb{R}^2\), for some \(\alpha\in [0,2\pi)\) and all \(t> 0\). Loosely speaking, the torsion number measures the torsion of the curve \([0,1]\ni x\mapsto (x,w(t,x))\in \mathbb{R}^3\). Torsion numbers are designed to play a role in the investigation of reaction-diffusion systems. Their role is comparable to the role of oscillation numbers which are a useful tool for the examination of solutions of a single equation.
scientific article; zbMATH DE number 1492550