Convergence to steady state solutions of a particular class of fracctional cooperative systems

• Yangari, Miguel ^[1]
  1. [1] Escuela Politécnica Nacional

     Escuela Politécnica Nacional

     Quito, Ecuador

     
• Localización: Revista Politécnica, ISSN-e 2477-8990, Vol. 35, Nº. 2, 2015 (Ejemplar dedicado a: Revista Politécnica), págs. 12-12
• Idioma: español
• Títulos paralelos:
  • Convergence to Steady State Solutions of a Particular Class of Fractional Cooperative Systems.
• Enlaces
  • Texto completo
• Resumen
  • español

    The aim of this paper is to prove that under some appropriate assumptions on the nonlinearity and the initial datum, the solution of the fractional reaction-diffusion cooperative system converge to the smallest positive steady solution. Also, we prove that this convergence is exponential in time and that the exponent of propagation depends on the principal eigenvalue of the derivative of reaction term and on the smallest index of the fractional laplacians.

  • English

    Resumen: El objetivo de este artículo es probar que bajo ciertas hipótesis sobre la nolinearidad y la condición inicial, la solución de un sistema cooperativo de reacción-difusión fraccionario converge a la solución positiva más pequeña de estado estable. Además, probamos que esta convergencia es exponencial en tiempo y que el exponente de propagación depende del primer valor propio de la derivada del término de reacción y del índice más pequeño de los Laplacianos fraccionarios.Abstract: The aim of this paper is to prove that under some appropriate assumptions on the nonlinearity and the initial datum, the solution of the fractional reaction-diffusion cooperative system converge to the smallest positive steady solution. Also, we prove that this convergence is exponential in time and that the exponent of propagation depends on the principal eigenvalue of the derivative of reaction term and on the smallest index of the fractional laplacians. 

• Referencias bibliográficas
  • D.G. Aronson and H.F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math. 30, 1978, pp. 33-76.
  • H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol 51,...
  • H. Berestycki, J.-M. Roquejoffre, and L. Rossi, The periodic patch model for population dynamics with fractional diffusion, Discrete Contin....
  • X. Cabré and J. Roquejoffre. Propagation de fronts dans les équations de Fisher KPP avec diffusion fractionnaire. C. R. Math. Acad. Sci. Paris...
  • X. Cabré and J. Roquejoffre. The influence of fractional diffusion in Fisher-KPP equation. Commun. Math. Phys 320, 2013, pp 679-722.
  • A-C. Coulon and M. Yangari. (2014, Sept.), Exponential propagation for fractional reaction-diffusion cooperative systems with fast decaying
  • initial conditions, 2014, Available:
  • url{http://arxiv.org/pdf/1405.5113v2.pdf}.
  • D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981, pp. 49-62.
  • A.N. Kolmogorov, I.G. Petrovsky and N.S. Piskunov, 'Etude de l'equation de la diffusion avec croissance de la quantit'e de mati'ere...
  • B. Li, H. Weinberger and M. Lewis. Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. 196, 2005, pp. 82-98.
  • R. Mancinelli, D. Vergni, and A. Vulpiani. Front propagation in reactive systems with anomalous diffusion, Phys. D 185, 2003, pp. 175-195.
  • H.F. Weinberger, M. Lewis and B. Li. Analysis of linear determinacy for spread in cooperative models. J. Math. Biol. 45, 2002, pp. 183-218.