Journal of Mathematical Sciences

, Volume 244, Issue 2, pp 254–266 | Cite as

On The Stabilization of Solutions of Nonlinear Parabolic Equations with Lower-Order Derivatives

  • A. A. Kon’kovEmail author


For parabolic equations of the form

$$ \frac{\partial u}{\partial t}-\sum \limits_{i,j=1}^n{a}_{ij}\left(x,u\right)\frac{\partial^2u}{\partial {x}_i\partial {x}_j}+f\left(x,u, Du\right)=0\kern0.5em \mathrm{in}\kern0.5em {\mathbb{R}}_{+}^{n+1}, $$

where \( {\displaystyle \begin{array}{cc}{\mathbb{R}}_{+}^{n+1}={\mathbb{R}}^n\times \left(0,\infty \right),& n\ge 1,D=\Big(\partial \end{array}}/\partial {x}_1,...,\partial /\partial {x}_n\Big), \) and f satisfies some constraints, we obtain conditions that ensure the convergence of any its solution to zero as t→∞.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Friedman, “Convergence of solutions of parabolic equations to a steady state,” J. Math. Mech., 8, 57–76 (1959).MathSciNetzbMATHGoogle Scholar
  2. 2.
    V. A. Galaktionov and L. A. Peletier, “Asymptotic behavior near finite-time extinction for the fast diffusion equation,” Arch. Rational Mech. Anal., 139, 83–98 (1997).MathSciNetCrossRefGoogle Scholar
  3. 3.
    A. L. Gladkov, “Self-similar blow-up solutions of the KPZ equation,” Int. J. Differ. Equ., 1–4 (2015).MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. L. Gladkov, M. Guedda, and R. Kersner, “A KPZ growth model with possibly unbounded data: correctness and blow-up,” Nonlinear Anal., 68, No. 7, 2079–2091 (2008).MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Gmira and L. Veron, “Large time behaviour of the solutions of a semilinear parabolic equation in Rn,” J. Differ. Equ., 53, 258–276 (1984).CrossRefGoogle Scholar
  6. 6.
    E. M. Landis, Second order equations of elliptic and parabolic type, Amer. Math. Soc., Providence (1998).zbMATHGoogle Scholar
  7. 7.
    V. A. Kondratiev, “On asymptotic properties of solutions of semilinear second order elliptic equations in cylindrical domains,” Tr. Semin. Petrovskogo, 25, 98–111 (2006).Google Scholar
  8. 8.
    V. A. Kondratiev and L. Veron, “Asymptotic behavior of solutions of some nonlinear parabolic or elliptic equations,” Asymptotic Anal., 14, 117–156 (1997).MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. A. Kon’kov, “Blow-up of solutions for a class of nondivergence elliptic inequalities,” Comput. Math. Math. Phys., 57, No. 3, 453–463 (2017).MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. A. Kon’kov, “On solutions of non-autonomous ordinary differential equations,” Izv. Math., 65, 285–327 (2001).MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. A. Kon’kov, “On the asymptotic behaviour of solutions of nonlinear parabolic equations,” Proc. Royal Soc. Edinburgh, 136, 365–384 (2006).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

Personalised recommendations