Stability of Equilibrium Solutions of a Nonlinear Reaction-Diffusion Equation

• Autores: César Adolfo Hernández Melo, Luiz Felipe Demetrio
• Localización: Boletín de matemáticas, ISSN 0120-0380, ISSN-e 2357-6529, Vol. 27, Nº. 1 (Versión preliminar), 2020, págs. 1-14
• Idioma: inglés
• Títulos paralelos:
  • Estabilidad de Soluciones de Equilibrio de una Ecuación de Reacción-Difusión no Lineal
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    En el presente trabajo, se analiza la existencia y estabilidad de soluciones de equilibrio de la siguiente ecuación de reacción-difusión no lineal:

    ut = auxx + wu + k ln(u2)u.

    Se proporcionan fórmulas explícitas para una familia de soluciones de equilibrio de la ecuación anterior que decaen a cero en infinito. La inestabilidad de esas soluciones se obtienen mediante el análisis espectral detallado del operador lineal que aproxima las soluciones de la ecuación alrededor de las soluciones de equilibrio. También se establece un resultado sobre la inestabilidad de cualquier solución de equilibrio no trivial de la ecuacióan.

  • English

    In the present work, it is analyzed existence and stability of equilibrium solutions of the following nonlinear reaction-diusion equation:

    ut = auxx + wu + k ln(u2)u.

    Explicit formulas for a family of equilibrium solutions to the former equation which decay to zero at innity are provided. The instability of those solutions is obtained by detailed spectral analysis of the linear operator which approximates the solutions of the equation around the equilibrium solutions. A result about the instability of any non-trivial equilibrium solution of the equation is also established.

• Referencias bibliográficas
  • M. Alfaro and R. Carles, Superexponential growth or decay in the heat equation with a logarithmic nonlinearity, Dynamics of Partial Differential...
  • R. A. Fisher, The wave of advance of advantageous genes, Ann. Eug. 7 (1937), 355{369.
  • V. A. Galaktionov and J. L. Vazquez, The Problem of Blow-Up in Nonlinear Parabolic Equations, Discrete Contin. Dyn. Systems 8 (2002), 399-433.
  • A. Ghazaryan, Y. Latushkin, and S. Schecter, Stability of traveling waves in partly parabolic systems, Math. Model. Nat. Phenom 8 (2013),...
  • B. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction, Volume 60 of Progress in Nonlinear Differential Equations...
  • D. Henry, Geometric theory of semilinear parabolic equations, Springer-Verlag, 1981.
  • P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory, Springer-Verlag. Section 11.2 110-112, 1996.
  • T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Springer, 2013.
  • Y. Liu, Z. Yu, and J. Xia, Exponential stability of traveling waves for non-monotone delayed reaction-diffusion equations, Electronic Journal...
  • C. A. Hernández M., Existence and stability of equilibrium solutions of a nonlinear heat equation, Applied Mathematics and Computation, 1025-1036,...
  • C. A. Hernández M. and E. Y. Mayorga L., Stability of equilibrium solutions of a double power reaction diffusion equation with a Dirac interaction,...
  • A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38-2 (1969), 279-303.
  • L. A. Segel, Distant sidewalls cause slow amplitude modulations of cellular convection, J. Fluid Mech. 38-2 (1969), 203-224.
  • Y. B. Zeldovich and D. A. Frank-Kamenetsky, phys. chem., Moscow 12, 100, 1938.