Advertisement

Journal of Mathematical Sciences

, Volume 244, Issue 2, pp 198–215 | Cite as

Stabilization of Solutions of Parabolic Equations with Growing Leading Coefficients

  • V. N. DenisovEmail author
Article

Abstract

Precise sufficient conditions are obtained for the coefficients of a second-order parabolic equation to ensure that the solutions of the Cauchy problem with polynomially growing initial functions stabilize to zero on compact sets. It is shown, by means of an example, that these sufficient conditions cannot be improved. In the case of bounded initial functions, we find conditions on the coefficients that guarantee that the solutions of the Cauchy problem stabilize to zero at a power rate and this stabilization is uniform in the spatial variables on compact sets.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. M. Il’yin, A. S. Kalashnikov, and O. A. Oleinik, “Linear Second Order Parabolic Equations,” Tr. Semin. Petrovskogo, 21, 9–193 (2001).Google Scholar
  2. 2.
    V. N. Denisov, “Large-time behavior of solutions of parabolic equations,” Uspekhi Mat. Nauk, 60, No. 4, 145–212 (2005).MathSciNetCrossRefGoogle Scholar
  3. 3.
    G. N. Smirnova, “The Cauchy problem for parabolic equations degenerating at infinity,” Mat. Sb., 70, No. 4, 391–604 (1966).MathSciNetGoogle Scholar
  4. 4.
    D. G. Aronson and P. Besala, “Uniqueness of solutions of the Cauchy problem for parabolic equations,” J. Math. Anal. Appl., 13, 516–526 (1966).MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. K. Gushchin, “On the stabilization rate of solutions of a boundary value problem for a parabolic equation,” Sib. Mat. Zh., 10, No. 1, 42–57 (1969).CrossRefGoogle Scholar
  6. 6.
    A. K. Gushchin, “On the stabilization rate of solutions of parabolic equations in unbounded domains,” Differ. Uravn., 6, No. 4, 741–761 (1970).zbMATHGoogle Scholar
  7. 7.
    J. K. Oddson, “On the rate of decay of solutions of parabolic differential equations,” Pacific J. Math., 69, No. 2, 389–396 (1969).MathSciNetCrossRefGoogle Scholar
  8. 8.
    V. N. Denisov, “On the stabilization rate of the solution of the Cauchy problem for a parabolic equation with lower order coefficients in classes of growing initial functions,” Dokl. RAN, 430, No. 5, 586–588 (2010).Google Scholar
  9. 9.
    V. N. Denisov, “Stabilization of solutions of the Cauchy problem for a non-divergent parabolic equation,” Sovrem. Mat. Prilozh., 78, 17–49 (2012).Google Scholar
  10. 10.
    V. N. Denisov, “Conditions of stabilization solutions of the Cauchy problem for a parabolic equation with growing lower order coefficients,” Dokl. RAN, 450, No. 6, 1–3 (2013).Google Scholar
  11. 11.
    V. N. Denisov, “Stabilization of solutions of the Cauchy problem for a nondivergent parabolic equation with growing lower order coefficients,” Differ. Uravn., 49, No. 5, 597–609 (2013).Google Scholar
  12. 12.
    V. N. Denisov, “The stabilization Rate of a solution to the Cauchy problem for a non-divergent parabolic equation,” in: S. V. Pogozin and M. V. Dubatovskay, eds. Analytic Methods of Analysis and Differential Equation: AMADE 2015, Cambridge Sci. Publ. (2016), pp. 49–60.Google Scholar
  13. 13.
    V. N. Denisov, “The stabilization rate of solutions of the Cauchy problem for a parabolic equation with lower order coefficients,” Sovrem. Mat. Fund. Napr., 59, 53–73 (2016).Google Scholar
  14. 14.
    V. N. Denisov, “Stabilization rate of solutions of the Cauchy problem for a parabolic equation with lower order coefficients,” Probl. Mat. Anal., 75, No. 3, 91–97 (2015).MathSciNetzbMATHGoogle Scholar
  15. 15.
    V. N. Denisov, “Large-time asymptotic behavior of solutions of parabolic equations with growing leading coefficients,” Dokl. RAN, 475, No. 1, 10–13 (2017).Google Scholar
  16. 16.
    G. N. Watson, Theory of Bessel Functions, Vol. 1 [Russian translation], Izd. Inostr. Lit., Moscow (1949).Google Scholar
  17. 17.
    G. I. Arkhipov, V. A. Sadovnichii, and V. N. Chubarikov, Lectures on Mathematical Analysis [in Russian], Izd. Mosk. Univ., Moscow (2004).Google Scholar
  18. 18.
    M. I. Fedoryuk, Ordinary Differential Equations [in Russian], Nauka, Moscow (1985).Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

Personalised recommendations