Instability of Stationary Solutions of Reaction–Diffusion–Equations on Graphs

Abstract

The nonexistence of stable stationary nonconstant solutions of reaction–diffusion–equations \({\partial_t u_j = \partial_j \left(a_j (x_j)\,\partial_j u_{j} \right) + f_j (u_j)}\) on the edges of a finite (topological) graph is investigated under continuity and consistent Kirchhoff flow conditions at all vertices of the graph. In particular, it is shown that in the balanced autonomous case \({f(u) = u - u^3}\), no such stable stationary solution can exist on any finite graph. Finally, the balanced autonomous case is discussed on the two-sided unbounded path with equal edge lengths.