Skip to main content
SpringerLink
Log in

Removability of an isolated singularity of solutions of the Neumann problem for quasilinear parabolic equations with absorption that admit double degeneration

  • Published:
Ukrainian Mathematical Journal Aims and scope

We consider the Neumann initial boundary-value problem for the equation

$$ {u_t} = {\text{div}}\left( {{u^{m - 1}}{{\left| {Du} \right|}^{\lambda - 1}}Du} \right) - {u^p} $$

in domains with noncompact boundary and with initial Dirac delta function. In the case of slow diffusion (m + λ − 2 > 0) and critical absorption exponent (p = m + λ − 1 + (λ + 1)/N), we prove that the singularity at the point (0, 0) is removable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Bresis and A. Friedman, “Nonlinear parabolic equations involving measures as initial conditions,” J. Math. Pures Appl., 62, 73–97 (1983).

    MathSciNet  Google Scholar 

  2. A. Gmira, “On quasilinear parabolic equations involving measure data,” Asymptotic Anal., 3, 43–56 (1990).

    MATH  MathSciNet  Google Scholar 

  3. I. I. Skrypnik, “Removability of isolated singularities of solutions of quasilinear parabolic equations with absorption,” Sb. Math., 196, 1693–1713 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  4. V. A. Galaktionov and A. E. Shishkov, “Higher-order quasilinear parabolic equations with singular initial data,” Commun. Cont. Math., 8(3), 331–354 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  5. O. M. Boldovskaya, “Existence and nonexistence of a weak solution of the Neumann problem for degenerate quasilinear parabolic equations in domains with noncompact boundary. Case of slow diffusion,” in: Proceedings of the Institute of Mathematics and Mechanics of the Ukrainian National Academy of Sciences [in Russian], 16 (2008), pp. 33–54.

  6. D. Andreucci and A. F. Tedeev, “A Fujita-type result for degenerate Neumann problem in domains with noncompact boundary,” J. Math. Anal. Appl., 231, 543–567 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  7. V. I. Ushakov, “Stabilization of solutions of the third mixed problem for a parabolic equation of the second order in a noncylindrical domain,” Mat. Sb., 111 (153), 95–115 (1980).

    MathSciNet  Google Scholar 

  8. F. Kh. Mukminov, “Stabilization of a solution of the first mixed problem for a parabolic equation of the second order,” Mat. Sb., 111 (153), 503–521 (1980).

    MathSciNet  Google Scholar 

  9. A. K. Gushchin, “On estimates for solutions of boundary-value problems for a parabolic equation of the second order,” Tr. Mat. Inst. Akad. Nauk SSSR, 126, 5–45 (1973).

    MATH  Google Scholar 

  10. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk, Ukraine

    O. M. Boldovskaya

Authors
  1. O. M. Boldovskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 7, pp. 894–912, July, 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Boldovskaya, O.M. Removability of an isolated singularity of solutions of the Neumann problem for quasilinear parabolic equations with absorption that admit double degeneration. Ukr Math J 62, 1040–1060 (2010). https://doi.org/10.1007/s11253-010-0412-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-010-0412-9

Keywords

Search

Navigation