# American Institute of Mathematical Sciences

December  2018, 13(4): 691-717. doi: 10.3934/nhm.2018031

## Stability implies constancy for fully autonomous reaction-diffusion-equations on finite metric graphs

 1 LMPA Joseph Liouville ULCO, FR CNRS Math. 2956, Universités Lille Nord de France, 50, rue F. Buisson, CS 80699, F-62228 Calais, France 2 Departament de Matemàtiques, Universitat Politècnica de Catalunya, Campus Nord, Edifici Ω, ordi Girona, 1-3, 08034 Barcelona, Spain

* Corresponding author

Received  December 2017 Revised  August 2018 Published  November 2018

Fund Project: The second author is supported by MINECO grant MTM2014-52402-C3-1-P.

We show that there are no stable stationary nonconstant solutions of the evolution problem (1) for fully autonomous reaction-diffusion-equations on the edges of a finite metric graph
 $G$
under continuity and Kirchhoff flow transition conditions at the vertices.
 $(1) \ \ \ \ \ \ \ \ \ \ \begin{cases} u∈ \mathcal{C}(G×[0,∞))\cap \mathcal{C}^{2,1}_{K}(G×(0,∞)),\\\partial_t u_j=\partial_j^2u_{j}+f(u_j) & \text{on the edges }k_j,\\ \displaystyle(K)\ \ \ \ \sum\limits_{j=1}^N d_{ij} c_{ij}\partial_ju_{j}(v_i,t)=0 &\text{at the vertices } v_i.\end{cases}$
Citation: Joachim von Below, José A. Lubary. Stability implies constancy for fully autonomous reaction-diffusion-equations on finite metric graphs. Networks & Heterogeneous Media, 2018, 13 (4) : 691-717. doi: 10.3934/nhm.2018031
##### References:

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##### References:
Cutting at $p$ with $\partial u(p) = 0$. The original graph $\Gamma$ is drawn on the left, while the resulting graph $\tilde{\Gamma}$ is drawn on the right
Yanagida's exceptional graphs
More "exceptional" graphs by Theorem 5.2
Proof of Lemma 6.2. The indicated signs are those of the $\Delta_{ij}$. The two thin arrows indicate the nodes $v_m$ and $v_{1}$ fulfilling the assertion
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