Instability of Stationary Solutions of Evolution Equations on Graphs Under Dynamical Node Transition

Abstract

The nonexistence of stable stationary nonconstant solutions of reaction–diffusion-equations $\partial _{t}u_{j} = \partial _{j}\left (a_{j}(x_{j})\,\partial _{j}u_{j}\right ) + f(u_{j})$ on the edges of a finite metric graph is investigated under continuity and dynamical consistent Kirchhoff flow conditions at all vertices v _i of the graph:

$$\displaystyle{\sum _{j}d_{\mathit{ij}}a_{j}(v_{i})\partial _{j}u_{j}(v_{i}) +\sigma _{i}\partial _{t}u(v_{i}) = 0.}$$

Various instability criteria are presented, in particular, for some classes of polynomial reaction terms f.