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

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 128)

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 vi 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.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.LMPA Joseph Liouville ULCOFR CNRS Math. 2956 Université Lille Nord de FranceCalais CedexFrance

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