Journal of Mathematical Sciences

, Volume 239, Issue 5, pp 706–724 | Cite as

On Coercive Solvability of Parabolic Equations with Variable Operators

  • A. R. HanalyevEmail author


In a Banach space E, the Cauchy problem

$$ \upsilon^{\prime }(t)+A(t)\upsilon (t)=f(t)\kern1em \left(0\le t\le 1\right),\kern1em \upsilon (0)={\upsilon}_0, $$

is considered for a differential equation with linear strongly positive operator A(t) such that its domain D = D(A(t)) does not depend on t and is everywhere dense in E and A(t) generates an analytic semigroup exp{−sA(t)}(s ≥ 0). Under natural assumptions on A(t), we prove the coercive solvability of the Cauchy problem in the Banach space \( {C}_0^{\beta, \upgamma} \) (E). We prove a stronger estimate for the solution compared with estimates known earlier, using weaker restrictions on f(t) and v0.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Ashyralyev and A. Khanalyev, “Coercive estimate in Hölder norms for parabolic equations with variable operator,” Modelling of Mining Processes for Gas Deposits and Applied Problems of Theoretical Gas-Hydrodynamics, Ylym, Ashgabat, 154–162 (1998).Google Scholar
  2. 2.
    A. Ashyralyev, A. Khanalyev, and P. E. Sobolevskii, “Coercive solvability of the nonlocal boundary-value problem for parabolic differential equations,” Abstr. Appl. Anal., 6, No. 1, 53–61 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Birkhäuser, Basel–Boston–Berlin (2004).Google Scholar
  4. 4.
    M. A. Krasnosel’skiy, P. P. Zabreyko, E. I. Pustyl’nik, and P. E. Sobolevskiy, Integral Operators in Spaces of Summable Functions [in Russian], Nauka, Moscow (1966).Google Scholar
  5. 5.
    S.G. Kreyn, Linear Differential Equations in Banach Space [in Russian], Nauka, Moscow (1967).Google Scholar
  6. 6.
    S. G. Kreyn and M. I. Khazan, “Differential equations in Banach space,” Mathematical Analysis, 21, 130–264 (1983).Google Scholar
  7. 7.
    V. A. Rudetskiy, “Coercive solvability of parabolic equations in interpolation spaces,” Deposited in VINITI, No. 34-85 (1984).Google Scholar
  8. 8.
    P.E. Sobolevskii, “On equations of parabolic type in a Banach space,” Tr. Mosk. Mat. Obs., 10, 297–350 (1961).MathSciNetGoogle Scholar
  9. 9.
    P. E. Sobolevskii, “Coercivity inequalities for abstract parabolic equations,” Dokl. Akad. Nauk SSSR, 157, No. 1, 52–55 (1964).MathSciNetGoogle Scholar
  10. 10.
    P. E. Sobolevskii, “On fractional norms generated by an unbounded operator in Banach space,” Uspekhi Mat. Nauk, 19, No. 6, 219–222 (1964).Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.RUDN UniversityMoscowRussia

Personalised recommendations