caa a22 4500 606201033 CHVBK 20210128100946.0 cr unu---uuuuu 210128e20150901xx s 000 0 eng 10.1007/s00025-014-0429-8 doi (NATIONALLICENCE)springer-10.1007/s00025-014-0429-8 Instability of Stationary Solutions of Reaction-Diffusion-Equations on Graphs [Elektronische Daten] [Joachim von Below, José Lubary] The nonexistence of stable stationary nonconstant solutions of reaction-diffusion-equations $${\partial_t u_j = \partial_j \left(a_j (x_j)\,\partial_j u_{j} \right) + f_j (u_j)}$$ ∂ t u j = ∂ j a j ( x j ) ∂ j u j + 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 $${f(u) = u - u^3}$$ 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. Springer Basel, 2015 Reaction-diffusion-equations nationallicence metric graphs nationallicence networks nationallicence attractors nationallicence stability nationallicence double-well potential nationallicence von Below Joachim LMPA Joseph Liouville ULCO, FR CNRS Math. 2956, Université Lille Nord de France ULCO, 50, rue F. Buisson, B.P. 699, 62228, Calais Cedex, France aut Lubary José Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Campus Nord, Edifici Ω, Jordi Girona, 1-3, 08034, Barcelona, Spain aut Results in Mathematics Springer Basel 68/1-2(2015-09-01), 171-201 1422-6383 68:1-2<171 2015 68 25 https://doi.org/10.1007/s00025-014-0429-8 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00025-014-0429-8 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- von Below Joachim LMPA Joseph Liouville ULCO, FR CNRS Math. 2956, Université Lille Nord de France ULCO, 50, rue F. Buisson, B.P. 699, 62228, Calais Cedex, France aut NATIONALLICENCE 700 1- Lubary José Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Campus Nord, Edifici Ω, Jordi Girona, 1-3, 08034, Barcelona, Spain aut NATIONALLICENCE 773 0- Results in Mathematics Springer Basel 68/1-2(2015-09-01), 171-201 1422-6383 68:1-2<171 2015 68 25