Skip to main content
SpringerLink
Log in

A bound for global solutions of semilinear heat equations

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We show that global positive solutions of the initial-boundary value problem foru t =Δu+u p are bounded, provided thatp>1 is subcritical. Our bound depends only on sup norm of the initial data and is useful to classify initial data by the asymptotic behavior of the solutions as time tend to infinity.

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.

Similar content being viewed by others

References

  1. Cazenave, T., Lions, P. L.: Solutions globales d'équation de la chaleur semilinéaires. Commun. Partial Differ. Equations9, 955–978 (1984)

    Google Scholar 

  2. Gidas, B. and Spruck, J.: A priori bounds for positive solutions of nonlinear elliptic equations. Commun. Partial Differ. Equations6, 883–901 (1981)

    Google Scholar 

  3. Ladyzhenskaya, O. A., Solonnikov, V. A. Ural'ceva, N. N.: Linear and quasilinear equations of parabolic types. Transl. Math. Monogr. vol.23, Providence, R.I.: Am. Math. Soc. 1968

    Google Scholar 

  4. Lions, P. L.: Asymptotic behavior of some nonlinear heat equations. Nonlinear phenomena. Physica5D. 293–306 (1982)

    Google Scholar 

  5. Ni, W. M., Sacks, P. E., Tavantzis, J.: On the asymptotic behavior of solutions of certain quasilinear equation of parabolic type. J. Differ. Equations54, 97–120 (1984)

    Google Scholar 

  6. Weissler, F.:L p-energy and blow-up for a semilinear heat equation, Proc. AMS Summer Inst. 1983 (to appear)

  7. Weissler, F.: Local existence and nonexistence for semilinear parabolic equations inL p. Indiana Univ. Math. J.29, 79–102 (1980)

    Google Scholar 

Download references

Author information

Author notes

  1. Yoshikazu Giga

    Present address: Department of Mathematics, Hokkaido University, 060, Sapporo, Japan

Authors and Affiliations

  1. Institute for Mathematics and its Applications, University of Minnesota, 514 Vincent Hall, 206 Church St. S.E., 55455, Minneapolis, MN, USA

    Yoshikazu Giga

Authors
  1. Yoshikazu Giga
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by L. Nirenberg

Partially supported by the Sakkokai Foundation

Rights and permissions

Reprints and permissions

About this article

Cite this article

Giga, Y. A bound for global solutions of semilinear heat equations. Commun.Math. Phys. 103, 415–421 (1986). https://doi.org/10.1007/BF01211756

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01211756

Keywords

Search

Navigation