Skip to main content
SpringerLink
Log in

Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put=−Au+ℱ(u)

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

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

References

  1. Chen, P. J., & M. E. Gurtin, On a theory of heat conduction involving two temperatures. Z. Angew. Math. Phys. 19, 614–627 (1968).

    Google Scholar 

  2. Friedman, A., Partial Differential Equations. New York: Holt, Rinehart, Winston 1969.

    Google Scholar 

  3. Friedman, A., Remarks on nonlinear parabolic equations. Proc. Symp. in Appl. Math. AMS 13, 3–23 (1965).

    CAS  PubMed  Google Scholar 

  4. Fujita, H., On the blowing up of solutions to the Cauchy problem for 385-02. J. Faculty Science, U. of Tokyo 13, 109–124 (1966).

    Google Scholar 

  5. Kaplan, S., On the growth of solutions of quasilinear parabolic equations. Comm. Pure Appl. Math. 16, 305–330 (1963).

    Google Scholar 

  6. Knops, R. J., & L. E. Payne, Growth estimate for solutions of evolutionary equations in Hubert space with applications in elastodynamics. Arch. Rational Mech. Anal. 41, 363–398 (1971).

    Google Scholar 

  7. Levine, H. A., Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu tt =-Au+F(u). Trans. Am. Math. Soc. (in press).

  8. Protter, M. H., Properties of parabolic equations and inequalities. Can. J. Math. 13, 331–345 (1961).

    Google Scholar 

  9. Protter, M. H., & H. F. Weinberger, Maximum Principles in Differential Equations. Englewood Cliffs, N.J.: Prentice Hall 1967.

    Google Scholar 

  10. Strauss, W. A., The Energy Method in Nonlinear Partial Differential Equations. Notas de Mathematica No 41. Instituto de Mathematica Pura e Aplicada, Rio de Janeiro (1969).

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Institute of Technology University of Minnesota, Minneapolis

    Howard A. Levine

Authors
  1. Howard A. Levine
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by M. M. Schiffer

This research was supported in part by the Science Research Council of Great Britain at the University of Dundee, Dundee, Scotland as part of the activities of the North British Symposium on Partial Differential Equations and their Applications held during the academic year 1971–1972 and in part by the Battelle Institute, Advanced Studies Center, Geneva, Switzerland.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Levine, H.A. Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put=−Au+ℱ(u). Arch. Rational Mech. Anal. 51, 371–386 (1973). https://doi.org/10.1007/BF00263041

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00263041

Keywords

Search

Navigation