# 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

In memoriam Karl-Peter Hadeler 1936-2017

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:
 [1] F. Ali Mehmeti, Lokale und globale Löungen linearer und nichtlinearer hyperbolischer Evolutionsgleichungen mit Transmission, Ph.D. thesis Johannes Gutenberg-Universität, Mainz, 1987. [2] F. Ali Mehmeti, Regular solutions of transmission and interaction problems for wave equations, Math. Meth. Appl. Sci., 11 (1989), 665-685. doi: 10.1002/mma.1670110507. [3] F. Ali Mehmeti and S. Nicaise, Nonlinear interaction problems, Nonlinear Analysis, Theory, Methods & Applications, 20 (1993), 27-61. doi: 10.1016/0362-546X(93)90183-S. [4] H. Amann, Ordinary Differential Equations, de Gruyter, Berlin, 1990. doi: 10.1515/9783110853698. [5] J. v. Below, Classical solvability of linear parabolic equations on networks, J. Differential Equ., 72 (1988), 316-337. doi: 10.1016/0022-0396(88)90158-1. [6] J. v. Below, A maximum principle for semilinear parabolic network equations, in Differential Equations with Applications in Biology, Physics, and Engineering (eds. J. A. Goldstein, F. Kappel, et W. Schappacher), Lect. Not. Pure and Appl. Math., 133 (1991), 37-45. [7] J. v. Below, Parabolic Network Equations, 2nd ed. Tübingen Universitätsverlag 1994. [8] J. v. Below and J. A. Lubary, Eigenvalue asymptotics for second order elliptic operators on networks, Asymptotic Analysis, 77 (2012), 147-167. [9] J. v. Below and J. A. Lubary, Instability of stationary solutions of reaction-diffusion-equations on graphs, Results in Math., 68 (2015), 171-201. doi: 10.1007/s00025-014-0429-8. [10] J. v. Below and J. A. Lubary, Stability properties of stationary solutions of reaction-diffusion-equations on metric graphs under the anti-Kirchhoff node condition, submitted. [11] J. v. Below and B. Vasseur, Instability of stationary solutions of evolution equations on graphs under dynamical node transition, Mathematical Technology of Networks, (ed. by Delio Mugnolo), Springer Proceedings in Mathematics & Statistics 128 (2015), 13-26. doi: 10.1007/978-3-319-16619-3_2. [12] N. L. Biggs, Algebraic Graph Theory, Cambridge Tracts Math. 67, Cambridge University Press, Cambridge UK, 1967. [13] N. Cònsul and J. de Solà-Morales, Stability of local minima and stable nonconstant equilibria, J. Differential Equ, 157 (1999), 61-81. doi: 10.1006/jdeq.1998.3625. [14] M. Gugat, F. M. Hante, M. Hirsch-Dick and G. Leugering, Stationary states in gas networks, Networks and Heterogeneous Media, 10 (2015), 295-320. doi: 10.3934/nhm.2015.10.295. [15] M. Gugat and F. Trötzsch, Boundary feedback stabilization of the Schlöl system, Automatica, 51 (2015), 192-199. doi: 10.1016/j.automatica.2014.10.106. [16] J. A. Lubary, Multiplicity of solutions of second order linear differential equations on networks, Lin. Alg. Appl., 274 (1998), 301-315. doi: 10.1016/S0024-3795(97)00348-0. [17] J. A. Lubary, On the geometric and algebraic multiplicities for eigenvalue problems on graphs, in Partial Differential Equations on Multistructures (eds. F. Ali Mehmeti, J. v. Below and S. Nicaise) Lecture Notes in Pure and Applied Mathematics 219, Marcel Dekker Inc. New York, (2000), 135-146. [18] H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci. Kyoto Univ.,, 15 (1979), 401-451. doi: 10.2977/prims/1195188180. [19] S. Nicaise, Diffusion sur les espaces ramifié, Ph.D. thesis Université Mons, Belgium, 1986. [20] R. J. Wilson, Introduction to Graph Theory, Oliver & Boyd Edinburgh UK, 1972. [21] E. Yanagida, Stability of nonconstant steady states in reaction-diffusion systems on graphs, Japan J. Indust. Appl. Math., 18 (2001), 25-42. doi: 10.1007/BF03167353.

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##### References:
 [1] F. Ali Mehmeti, Lokale und globale Löungen linearer und nichtlinearer hyperbolischer Evolutionsgleichungen mit Transmission, Ph.D. thesis Johannes Gutenberg-Universität, Mainz, 1987. [2] F. Ali Mehmeti, Regular solutions of transmission and interaction problems for wave equations, Math. Meth. Appl. Sci., 11 (1989), 665-685. doi: 10.1002/mma.1670110507. [3] F. Ali Mehmeti and S. Nicaise, Nonlinear interaction problems, Nonlinear Analysis, Theory, Methods & Applications, 20 (1993), 27-61. doi: 10.1016/0362-546X(93)90183-S. [4] H. Amann, Ordinary Differential Equations, de Gruyter, Berlin, 1990. doi: 10.1515/9783110853698. [5] J. v. Below, Classical solvability of linear parabolic equations on networks, J. Differential Equ., 72 (1988), 316-337. doi: 10.1016/0022-0396(88)90158-1. [6] J. v. Below, A maximum principle for semilinear parabolic network equations, in Differential Equations with Applications in Biology, Physics, and Engineering (eds. J. A. Goldstein, F. Kappel, et W. Schappacher), Lect. Not. Pure and Appl. Math., 133 (1991), 37-45. [7] J. v. Below, Parabolic Network Equations, 2nd ed. Tübingen Universitätsverlag 1994. [8] J. v. Below and J. A. Lubary, Eigenvalue asymptotics for second order elliptic operators on networks, Asymptotic Analysis, 77 (2012), 147-167. [9] J. v. Below and J. A. Lubary, Instability of stationary solutions of reaction-diffusion-equations on graphs, Results in Math., 68 (2015), 171-201. doi: 10.1007/s00025-014-0429-8. [10] J. v. Below and J. A. Lubary, Stability properties of stationary solutions of reaction-diffusion-equations on metric graphs under the anti-Kirchhoff node condition, submitted. [11] J. v. Below and B. Vasseur, Instability of stationary solutions of evolution equations on graphs under dynamical node transition, Mathematical Technology of Networks, (ed. by Delio Mugnolo), Springer Proceedings in Mathematics & Statistics 128 (2015), 13-26. doi: 10.1007/978-3-319-16619-3_2. [12] N. L. Biggs, Algebraic Graph Theory, Cambridge Tracts Math. 67, Cambridge University Press, Cambridge UK, 1967. [13] N. Cònsul and J. de Solà-Morales, Stability of local minima and stable nonconstant equilibria, J. Differential Equ, 157 (1999), 61-81. doi: 10.1006/jdeq.1998.3625. [14] M. Gugat, F. M. Hante, M. Hirsch-Dick and G. Leugering, Stationary states in gas networks, Networks and Heterogeneous Media, 10 (2015), 295-320. doi: 10.3934/nhm.2015.10.295. [15] M. Gugat and F. Trötzsch, Boundary feedback stabilization of the Schlöl system, Automatica, 51 (2015), 192-199. doi: 10.1016/j.automatica.2014.10.106. [16] J. A. Lubary, Multiplicity of solutions of second order linear differential equations on networks, Lin. Alg. Appl., 274 (1998), 301-315. doi: 10.1016/S0024-3795(97)00348-0. [17] J. A. Lubary, On the geometric and algebraic multiplicities for eigenvalue problems on graphs, in Partial Differential Equations on Multistructures (eds. F. Ali Mehmeti, J. v. Below and S. Nicaise) Lecture Notes in Pure and Applied Mathematics 219, Marcel Dekker Inc. New York, (2000), 135-146. [18] H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci. Kyoto Univ.,, 15 (1979), 401-451. doi: 10.2977/prims/1195188180. [19] S. Nicaise, Diffusion sur les espaces ramifié, Ph.D. thesis Université Mons, Belgium, 1986. [20] R. J. Wilson, Introduction to Graph Theory, Oliver & Boyd Edinburgh UK, 1972. [21] E. Yanagida, Stability of nonconstant steady states in reaction-diffusion systems on graphs, Japan J. Indust. Appl. Math., 18 (2001), 25-42. doi: 10.1007/BF03167353.
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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