The existence, uniqueness, and instability of spherically symmetric solutions of a system of reaction-diffusion equations
Existence, uniqueness and stability of (steady-state) radially symmetric solutions of the reaction-diffusion system \(\partial x/\partial t=\Delta x+F(x,y),\quad \partial y/\partial t=G(x,y)\) are studied under suitable assumptions on the nonlinear terms F and G. In particular, as is shown in Appendix A, they include the Fitzhugh-Nagumo system for nerve conduction and the Field-Noyes model for the Belousov-Zhabotinskij reaction. These results generalize, under slightly different hypotheses, previous work by C. Jones is the case of one single equation. Under the above mentioned assumptions, the authors prove that the problem (for space dimension \(n>1)\) has a unique radially symmetric solution which is linearly unstable. The proofs are quite complicated from the technical point of view and use nontrivial plane phase arguments, in particular ''backwards shooting'' and center manifolds.
scientific article; zbMATH DE number 3867830