On Extension of Asymptotic Comparison Principle for Time Periodic Reaction-Diffusion-Advection Systems with Boundary and Internal Layers

  • Nikolay NefedovEmail author
  • Aleksei Yagremtsev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)


In this paper we present a further development of our asymptotic comparison principle, applying it for some new important classes of initial boundary value problem for the nonlinear singularly perturbed time periodic parabolic equations, which are called in applications as reaction-diffusion-advection equations. We illustrate our approach for the new problem with balanced nonlinearity. The theorems, which states the existence of the periodic solution with internal layer, gives it’s asymptotic approximation and state their Lyapunov stability are proved.


Singularly perturbed problems Moving fronts Time periodic reaction-diffusion-advection equations 



This work is supported by RFBR, pr. N 13-01-91333.


  1. 1.
    Vasilieva, A.B., Butuzov, V.F., Nefedov, N.N.: Contrast structures in singularly perturbed problems. Fundamentalnaja i prikladnala matematika 4(3), 799–851 (1998)Google Scholar
  2. 2.
    Nefedov, N.: Comparison principle for reaction-diffusion-advection problems with boundary and internal layers. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) NAA 2012. LNCS, vol. 8236, pp. 62–72. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  3. 3.
    Nefedov, N.N., Recke, L., Schnieder, K.R.: Existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations. J. Math. Anal. Appl. 405, 90–103 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Hess, P.: Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Math. Series 247. Longman Scientific and Technical, Harlow (1991)Google Scholar
  5. 5.
    Zabrejko, P.P., Koshelev, A.I., et al.: Integral Equations. M.Nauka, Moscow (1968) Google Scholar
  6. 6.
    Nefedov, N.N.: The method of differential inequalities for some classes of nonlinear singularly perturbed problems with internal layers. Differ. Uravn. 31(7), 1142–1149 (1995)MathSciNetGoogle Scholar
  7. 7.
    Volkov, V., Nefedov, N.: Asymptotic-numerical investigation of generation and motion of fronts in phase transition models. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) NAA 2012. LNCS, vol. 8236, pp. 524–531. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  8. 8.
    Franz, S., Roos, H.-G.: The capriciousness of numerical methods for singular perturbations. SIAM rev. 53, 157–173 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Franz, S., Kopteva, N.: Green’s function estimates for a singularly perturbed convection-diffusion problem. J. Diff. Eq. 252, 1521–1545 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Kopteva, N.: Numerical analysis of a 2d singularly perturbed semilinear reaction-diffusion problem. In: Margenov, S., Vulkov, L.G., Waśniewski, J. (eds.) NAA 2008. LNCS, vol. 5434, pp. 80–91. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  11. 11.
    Vasileva, A.B., Butuzov, V.F., Nefedov, N.N.: Singularly perturbed problems with boundary and internal layers. Proc. Steklov Inst. Math. 268, 258–273 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of PhysicsLomonosov Moscow State UniversityMoscowRussia

Personalised recommendations