Skip to main content
SpringerLink
Log in

Semilinear Heat Models

  • Chapter
  • First Online:
Book cover Methods for Partial Differential Equations

Abstract

In this chapter we consider the semilinear heat model with power nonlinearity

$$\displaystyle {u_t} - \varDelta u =\pm |u|^{p-1}u,\,\,\,u(0,x)=\varphi (x).$$

Here ±|u|p−1 u is an example of a source nonlinearity (positive sign) and of an absorbing nonlinearity (negative sign) (see, for example, [153]).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 99.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. R. Adams, Sobolev Spaces (Academic, New York, 1975)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. H. Brezis, W.A. Strauss, Semilinear second order elliptic equation in L 1. J. Math. Soc. Jpn. 25, 565–590 (1973)

    Article  MATH  Google Scholar 

  4. H. Brezis, L.A. Peletier, D. Terman, A very singular solution of the heat equation with absorption. Arch. Ration. Mech. Anal. 95(3), 185–209 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. T. Cazanave, F. Dickstein, F.B. Weissler, Multi-scale multi-profile global solutions of parabolic equations in \(\mathbb {R}^n\). Discrete Contin. Dyn. Syst. S 5(3), 449–472 (2012)

    Google Scholar 

  6. M.G. Crandall, T.M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces. Am. J. Math. 93, 265–298 (1971)

    Article  MATH  Google Scholar 

  7. M. Escobedo, O. Kavian, Variational problems related to self-similar solutions of the heat equation. Nonlinear Anal. 11(10), 1103–1133 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Escobedo, O. Kavian, Asymptotic behavior of positive solutions of a non-linear heat equation. Houst. J. Math. 14, 39–50 (1988)

    MATH  Google Scholar 

  9. M. Escobedo, O. Kavian, H. Matano, Large time behavior of solutions of a dissipative semilinear heat equation. Commun. Partial Differ. Equ. 20(7–8), 1427–1452 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Fujita, On the blowing-up of solutions of the Cauchy problem for u t  = Δu + u 1+α. J. Fac. Sci. Univ. of Tokyo, Sect. 1 13, 109–124 (1966)

    Google Scholar 

  11. A. Gmira, L. Veron, Large time behavior of the solutions of a semilinear parabolic equation in \(\mathbb {R}^n\). J. Differ. Equ. 53, 258–276 (1984)

    Google Scholar 

  12. A. Haraux, F.B. Weissler, Non-uniqueness for a semilinear initial value problem. Indiana Univ. Math. J. 31(2), 167–189 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. K. Hayakawa, On nonexistence of global solutions of some semi-linear parabolic differential equations. Proc. Jpn. Acad. 49, 503–505 (1973)

    Article  MATH  Google Scholar 

  14. O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation. Annales de l’ I.H.P. Sect. C 4, 423–452 (1987)

    Google Scholar 

  15. K. Kobayashi, T. Sirao, H. Tanaka, On the growing up problem for semi-linear heat equations. J. Math. Soc. Jpn. 29, 407–424 (1977)

    Article  MATH  Google Scholar 

  16. K. Nishihara, Asymptotic behavior of solutions for a system of semilinear heat equations and the corresponding damped wave system. Osaka J. Math. 49, 331–348 (2012)

    MathSciNet  MATH  Google Scholar 

  17. K. Nishihara, Diffusion phenomena of solutions to the Cauchy problem for the damped wave equations. Sugaku Expositions 26(1), 29–47 (2013)

    MathSciNet  MATH  Google Scholar 

  18. F.B. Weissler, Existence and nonexistence of global solutions for a semi-linear heat equation. Isr. J. Math. 38(1–2), 29–40 (1981)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computing and Mathematics, University of São Paulo, Ribeirão Preto, São Paulo, Brazil

    Marcelo R. Ebert

  2. Institute of Applied Analysis, TU Bergakademie Freiberg, Freiberg, Germany

    Michael Reissig

Authors
  1. Marcelo R. Ebert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Reissig
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Ebert, M.R., Reissig, M. (2018). Semilinear Heat Models. In: Methods for Partial Differential Equations. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-66456-9_17

Download citation

Publish with us

Policies and ethics

Search

Navigation