Existence of periodic travelling wave solutions of reaction-diffusion systems with fast-slow dynamics
The paper is concerned with the problem to prove (or disprove) the existence of periodic travelling wave solutions of reaction-diffusion systems \[ (1)\quad u_ t=Su_{xx}+f(u,v),\quad v_ t=Tv_{xx}+\mu g(u,v), \] \(x\in {\mathbb{R}}\), \(u(x,t)\in {\mathbb{R}}^ m\), \(v(x,t)\in {\mathbb{R}}^ n\); S,T are diffusion matrices; \(\mu\) is a small parameter; f,g are smooth functions (representing chemical kinetics). In order to overcome the difficulty of this problem a perturbation problem around \(\mu =0\) is considered, so that a spatially homogeneous solution \(v(x,t)=\alpha\) is obtained, where \(\alpha \in {\mathbb{R}}^ n\) is an arbitrary constant vector. Then the paper assumes that an L-periodic travelling solution or stationary wave solution is obtained for the equation \(u_ t=Su_{xx}+f(u,\alpha)\) and four theorems on the existence (or non existence) of L-periodic travelling solutions of the system (1) are proved by a perturbation method for sufficiently small \(\mu\).
existence of periodic travelling wave solutions
scientific article; zbMATH DE number 4155023