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Instability of stationary solutions of reaction-diffusion-equations on graphs. (English) Zbl 1332.35373

Summary: 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.

MSC:

35R02 PDEs on graphs and networks (ramified or polygonal spaces)
35K57 Reaction-diffusion equations
35B35 Stability in context of PDEs
35B41 Attractors
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