Skip to main content
SpringerLink
Log in

On Necessary Conditions for the Existence of Local Solutions to Singular Nonlinear Ordinary Differential Equations and Inequalities

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We obtain conditions for the nonexistence of global solutions and estimates of existence time for local solutions to the problem

$$\frac{{d^k y}}{{dt^k }} \geqslant a_1 (t)\left| y \right|^{q_1 } + a_2 (t)\left| y \right|^{q_2 } + \cdots + a_n (t)\left| y \right|^{q_n } .$$

The proofs are based on the method of trial functions developed by Mitidieri and Pokhozhaev.

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. E. Mitidieri and S. I. Pokhozhaev, "A priori estimates and nonexistence of solutions to nonlinear partial differential equations and inequalities," Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 234 (2001).

  2. J. Hay, "On necessary conditions for the existence of global solutions to nonlinear ordinary differential inequalities of high order" Differentsial' nye Uravneniya [Differential Equations], 38 (2002), no. 3, 1-7.

    Google Scholar 

  3. J. Hay, "On necessary conditions for the existence of global solutions to systems of nonlinear ordinary differential equations and of inequalities of high order" Differentsial' nye Uravneniya [Differential Equations] (to appear).

  4. I. T. Kiguradze and T. A. Chanturiya, Asymptotic Properties of Solutions to Nonautonomous Ordinary Differential Equations [in Russian], Nauka, Moscow, 1990.

    Google Scholar 

  5. I. T. Kiguradze, Initial and Boundary-Value Problems for Systems of Ordinary Differential Equations, I [in Russian], Metsniereba, Tbilisi, 1997.

    Google Scholar 

  6. A. A. Kon'kov, "On solutions of nonautonomous ordinary differential equations," Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 65 (2001), no. 2, 81-125.

    Google Scholar 

  7. E. Mitidieri and S. I. Pokhozhaev, "Nonexistence of global positive solutions to quasilinear elliptic inequalities," Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 359 (1998), no. 4, 456-460.

    Google Scholar 

  8. E. Mitidieri and S. I. Pokhozhaev, "Nonexistence of positive solutions to quasilinear elliptic problems in RN," Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 227 (1999), 192-222.

  9. E. Mitidieri and S. I. Pokhozhaev, "Nonexistence of positive solutions to systems of quasilinear elliptic equations and inequalities in ℝN," Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 366 (1999), no. 1, 13-17.

    Google Scholar 

  10. E. Mitidieri and S. I. Pokhozhaev, "Nonexistence of weak solutions to some degenerate and singular hyperbolic problems in ℝ n+1+ ," Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 232 (2001), 248-267.

    Google Scholar 

  11. H. A. Levine, "The role of critical exponents in blow-up theorems," SIAM Reviews, 32 (1990), 371-386.

    Google Scholar 

  12. G. G. Laptev, "Nonexistence of global positive solutions to systems of semilinear elliptic inequalities in cones," Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 64 (2000), 107-124.

    Google Scholar 

  13. G. G. Laptev, "On the nonexistence of solutions of a class of singular semilinear differential inequalities," Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 232 (2001), 223-235.

    Google Scholar 

  14. G. G. Laptev, "Some nonexistence results for higher-order evolution inequalities in cone-like domains," Electron. Res. Announc. Amer. Math. Soc., 7 (2001), 87-93.

    Google Scholar 

  15. G. G. Laptev, "On the nonexistence of global solutions to parabolic equations of porous medium type," Zh. Vychisl. Mat. i Mat. Fiz. [Comput. Math. and Math. Phys.], 42 (2002), no. 8, 1224-1232.

    Google Scholar 

  16. G. G. Laptev, "Absence of solutions to evolution-type differential inequalities in the exterior of a ball," Russ. J. Math. Phys., 9 (2002), no. 2, 180-187.

    Google Scholar 

  17. G. G. Laptev, "On the nonexistence of solutions of elliptic differential inequalities in conic domains," Mat. Zametki [Math. Notes], 71 (2002), no. 6, 855-866.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. V. A. Steklov Mathematics Institute, Russian Academy of Sciences, Russia

    J. Hay

Authors
  1. J. Hay
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hay, J. On Necessary Conditions for the Existence of Local Solutions to Singular Nonlinear Ordinary Differential Equations and Inequalities. Mathematical Notes 72, 847–857 (2002). https://doi.org/10.1023/A:1021498114996

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021498114996

Search

Navigation