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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. Amann, Ordinary Differential Equations, de Gruyter, Berlin, 1990. doi: 10.1515/9783110853698.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. v. Below, Parabolic Network Equations, 2nd ed. Tübingen Universitätsverlag 1994. Google Scholar

[8]

J. v. Below and J. A. Lubary, Eigenvalue asymptotics for second order elliptic operators on networks, Asymptotic Analysis, 77 (2012), 147-167.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[12]

N. L. Biggs, Algebraic Graph Theory, Cambridge Tracts Math. 67, Cambridge University Press, Cambridge UK, 1967.  Google Scholar

[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.  Google Scholar

[14]

M. GugatF. M. HanteM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. Nicaise, Diffusion sur les espaces ramifié, Ph.D. thesis Université Mons, Belgium, 1986. Google Scholar

[20]

R. J. Wilson, Introduction to Graph Theory, Oliver & Boyd Edinburgh UK, 1972. Google Scholar

[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.  Google Scholar

show all references

References:
[1]

F. Ali Mehmeti, Lokale und globale Löungen linearer und nichtlinearer hyperbolischer Evolutionsgleichungen mit Transmission, Ph.D. thesis Johannes Gutenberg-Universität, Mainz, 1987. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. Amann, Ordinary Differential Equations, de Gruyter, Berlin, 1990. doi: 10.1515/9783110853698.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. v. Below, Parabolic Network Equations, 2nd ed. Tübingen Universitätsverlag 1994. Google Scholar

[8]

J. v. Below and J. A. Lubary, Eigenvalue asymptotics for second order elliptic operators on networks, Asymptotic Analysis, 77 (2012), 147-167.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[12]

N. L. Biggs, Algebraic Graph Theory, Cambridge Tracts Math. 67, Cambridge University Press, Cambridge UK, 1967.  Google Scholar

[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.  Google Scholar

[14]

M. GugatF. M. HanteM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. Nicaise, Diffusion sur les espaces ramifié, Ph.D. thesis Université Mons, Belgium, 1986. Google Scholar

[20]

R. J. Wilson, Introduction to Graph Theory, Oliver & Boyd Edinburgh UK, 1972. Google Scholar

[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.  Google Scholar

Figure 1.  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
Figure 2.  Yanagida's exceptional graphs
Figure 3.  More "exceptional" graphs by Theorem 5.2
Figure 4.  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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