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

Joachim von Below; José A. Lubary; Joachim von Below; José A. Lubary

doi:10.3934/nhm.2018031

Networks and Heterogeneous Media

2018, Volume 13, Issue 4: 691-717. doi: 10.3934/nhm.2018031

Previous Article Next Article

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

Joachim von Below ^{1
,
,},
José A. Lubary ^{2
,}

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

Received: 01 December 2017 Revised: 01 August 2018
Primary: 35K57, 35B35, 35B41, 35R02, 35J25

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} $
- Reaction-diffusion-equations,
- metric graphs,
- quantum graphs,
- networks,
- attractors,
- stability
Citation: Joachim von Below, José A. Lubary. Stability implies constancy for fully autonomous reaction-diffusion-equations on finite metric graphs[J]. Networks and Heterogeneous Media, 2018, 13(4): 691-717. doi: 10.3934/nhm.2018031

Related Papers:

Abstract

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} $

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]	Regular solutions of transmission and interaction problems for wave equations. Math. Meth. Appl. Sci. (1989) 11: 665-685.
[3]	Nonlinear interaction problems. Nonlinear Analysis, Theory, Methods & Applications (1993) 20: 27-61.
[4]	H. Amann, Ordinary Differential Equations, de Gruyter, Berlin, 1990. doi: 10.1515/9783110853698
[5]	Classical solvability of linear parabolic equations on networks. J. Differential Equ. (1988) 72: 316-337.
[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]	Eigenvalue asymptotics for second order elliptic operators on networks. Asymptotic Analysis (2012) 77: 147-167.
[9]	Instability of stationary solutions of reaction-diffusion-equations on graphs. Results in Math. (2015) 68: 171-201.
[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]	Stability of local minima and stable nonconstant equilibria. J. Differential Equ (1999) 157: 61-81.
[14]	Stationary states in gas networks. Networks and Heterogeneous Media (2015) 10: 295-320.
[15]	Boundary feedback stabilization of the Schlöl system. Automatica (2015) 51: 192-199.
[16]	Multiplicity of solutions of second order linear differential equations on networks. Lin. Alg. Appl. (1998) 274: 301-315.
[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]	Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. Res. Inst. Math. Sci. Kyoto Univ., (1979) 15: 401-451.
[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]	Stability of nonconstant steady states in reaction-diffusion systems on graphs. Japan J. Indust. Appl. Math. (2001) 18: 25-42.

Reader Comments

Your name:*

Email:*
© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)