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

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• 作者：Joachim von Below ; José A. Lubary
• 关键词：35K57 ; 35B35 ; 35B41 ; 35R02 ; 35J25 ; Reaction–diffusion–equations ; metric graphs ; networks ; attractors ; stability ; double ; well potential
• 刊名：Results in Mathematics
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：68
• 期：1-2
• 页码：171-201
• 全文大小：771 KB
• 参考文献：1.Amman H.: Ordinary Differential Equations. de Gruyter, Berlin (1990)CrossRef
  2.Below, J. von: Sturm–Liouville eigenvalue problems on networks. Math. Meth. Appl. Sci 10, 383-95 (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. 37-5. 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. 274-83. Longman, Harlow (1992)
  5.Below, J. von: Parabolic Network Equations, 2nd edn. Universit?tsverlag, 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, 251-63 (1975)MathSciNet CrossRef MATH
  10.Lady?enskaya, 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)CrossRef MATH
  12.Wilson R.J.: Introduction to Graph Theory. Oliver & Boyd, Edinburgh (1972)MATH
  13.Yanagida E.: Stability of nonconstant steady states in reaction–diffusion systems on graphs. Japan J. Indust. Appl. Math. 18, 25-2 (2001)MathSciNet CrossRef MATH
  
• 作者单位：Joachim von Below (1)
  José A. Lubary (2)
  
  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
  2. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Campus Nord, Edifici Ω, Jordi Girona, 1-, 08034, Barcelona, Spain
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1420-9012

文摘

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. Keywords Reaction–diffusion–equations metric graphs networks attractors stability double-well potential