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Instability of Stationary Solutions of Reaction–Diffusion–Equations on Graphs

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Abstract

The nonexistence of stable stationary nonconstant solutions of reaction–diffusion–equations \({\partial_t u_j = \partial_j \left(a_j (x_j)\,\partial_j u_{j} \right) + f_j (u_j)}\) 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 \({f(u) = u - u^3}\), 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.

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References

  1. Amman H.: Ordinary Differential Equations. de Gruyter, Berlin (1990)

    Book  Google Scholar 

  2. Below, J. von: Sturm–Liouville eigenvalue problems on networks. Math. Meth. Appl. Sci 10, 383–395 (1988)

  3. Below, J. von: A maximum principle for semilinear parabolic network equations. In: Goldstein, J.A., Kappel, F., Schappacher, W. (eds.) Differential Equations with Applications in Biology, Physics, and Engineering. Lecture Notes in Pure and Applied Mathematics, vol. 133, pp. 37–45. M. Dekker Inc., New York (1991)

  4. Below, J. von: An existence result for semilinear parabolic network equations with dynamical node conditions. In: Bandle, C., Bemelmans, J., Chipot, M., Grüter, M., Saint Jean Paulin, J. (eds.) Progress in Partial Differential Equations: Elliptic and Parabolic Problems. Pitman Research Notes in Mathematics Series, vol. 266, pp. 274–283. Longman, Harlow (1992)

  5. Below, J. von: Parabolic Network Equations, 2nd edn. Universitätsverlag, Tübingen (1994)

  6. Below, J. von, Lubary, J.A.: Graphs without stable nonconstant stationary solutions, in progress

  7. Below, J. von, Vasseur, B.: Instability of stationary solutions of evolution equations on graphs under dynamical node transition. In: Mugnolo, D. (ed.) Mathematical Technology of Networks. Springer, to appear

  8. Biggs, N.L.: Algebraic graph theory. In: Cambridge Tracts in Mathematics, vol. 67. Cambridge University Press, Cambridge (1967)

  9. Hadeler K.-P., Rothe F.: Traveling fronts in nonlinear diffusion equations. J. Math. Biol. 2, 251–263 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ladyženskaya, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)

  11. Walter W.: Differential and Integral Inequalities. Springer, Berlin (1970)

    Book  MATH  Google Scholar 

  12. Wilson R.J.: Introduction to Graph Theory. Oliver & Boyd, Edinburgh (1972)

    MATH  Google Scholar 

  13. Yanagida E.: Stability of nonconstant steady states in reaction–diffusion systems on graphs. Japan J. Indust. Appl. Math. 18, 25–42 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. 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

    Joachim von Below

  2. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Campus Nord, Edifici Ω, Jordi Girona, 1–3, 08034, Barcelona, Spain

    José A. Lubary

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  1. Joachim von Below
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  2. José A. Lubary
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Correspondence to Joachim von Below.

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Supported by the grants MINECO MTM2011-27739-C04-01 and GENCAT 2009SGR-345.

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von Below, J., Lubary, J.A. Instability of Stationary Solutions of Reaction–Diffusion–Equations on Graphs. Results. Math. 68, 171–201 (2015). https://doi.org/10.1007/s00025-014-0429-8

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  • DOI: https://doi.org/10.1007/s00025-014-0429-8

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