Travelling waves for reaction--diffusion equations with time depending nonlinearities
This paper concerns the parabolic equation \[ v_t =\Delta v-f(v,\alpha \cdot x-ct),\tag{1} \] where \(v=(v^1,\dots ,v^k):\mathbb{R}^l\times \mathbb{R}\to \mathbb{R}^k\), \(x,\alpha\in \mathbb{R}^l\), \(|\alpha|=1\), \(t\in \mathbb{R}\), \(c>0\) and \(f:\mathbb{R}^k\times \mathbb{R}\to \mathbb{R}^k\) is a continuous function. Under appropriate assumptions the author proves that equation (1) admits a travelling wave solution with end points \(w_{\pm}\), i.e., a solution \(v\) of the form \(v(x,t)=w(\alpha\cdot x-ct)\) and such that \(\lim_{x\to\pm\infty}w(x)=w_{\pm}\in \mathbb{R}^k\). The proof is obtained by reduction of the equation (1) to an ordinary one and by using the Schauder fixed point theorem and a suitable diagonalization procedure.
scientific article; zbMATH DE number 1956852