Stability implies constancy for fully autonomous reaction-diffusion-equations on finite metric graphs
Joachim von Below, ,LMPA Joseph Liouville ULCO, FR CNRS Math. 2956, Universités Lille Nord de France, 50, rue F. Buisson, CS 80699, F-62228 Calais, France, José A. Lubary, ,Departament de Matemàtiques, Universitat Politècnica de Catalunya, Campus Nord, Edifici Ω, ordi Girona, 1-3, 08034 Barcelona, Spain · Networks and Heterogeneous Media · 2018
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} $