Skip to main content
SpringerLink
Log in

On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term

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

Abstract

We study questions of existence, uniqueness and asymptotic behaviour for the solutions of u(x, t) of the problem

$$\begin{gathered} {\text{ }}u_t - \Delta u = \lambda e^u ,{\text{ }}\lambda {\text{ > 0, }}t > 0,{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ (P){\text{ }}u(x,0) = u_0 (x),{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ {\text{ }}u(x,t) = 0{\text{ }}on{\text{ }}\partial B \times (0,\infty ), \hfill \\ \end{gathered} $$

where B is the unit ball \(\{ x\varepsilon R^N :|x|{\text{ }} \leqq {\text{ }}1\} {\text{ and }}N \geqq 3\). Our interest is focused on the parameter λ 0=2(N−2) for which (P) admits a singular stationary solution of the form

$$S(x) = - 2log|x|$$

.

We study the dynamical stability or instability of S, which depends on the dimension. In particular, there exists a minimal bounded stationary solution u which is stable if \(3 \leqq N \leqq 9\), while S is unstable. For \(N \geqq 10\) there is no bounded minimal solution and S is an attractor from below but not from above. In fact, solutions larger than S cannot exist in any time interval (there is instantaneous blow-up), and this happens for all dimensions.

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. P. Baras & L. Cohen, Complete Blow-Up after T max for the Solution of a Semi-linear Heat Equation, J. Func. Anal. 71 (1987), 142–174.

    Google Scholar 

  2. J. Bebernes, A. Bressan & D. Eberly, A Description of Blowup for the Solid Fuel Ignition Model, Indiana Univ. Math. Jour. 36 (1987), 295–305.

    Google Scholar 

  3. P. Baras & J. Goldstein, The Heat Equation with a Singular Potential, Trans. Amer. Math. Soc. 294 (1984), 121–139.

    Google Scholar 

  4. J. Bebernes & D. Eberly, Mathematical Problems in Combustion Theory, Springer, New York, 1989.

    Google Scholar 

  5. J. Bebernes & D. Eberly, A Description of Selfsimilar Blow-Up for Dimensions n≧3, Ann. Inst. Henri Poincaré, Analyse non linéaire 5 (1989), 1–21.

    Google Scholar 

  6. H. Bellout, On some Singular Solutions of the Equation Δu=−λe u, preprint.

  7. M. F. Bidaut-Véron & L. Véron, Groupe conforme de S 2 et propriétés limites des solutions de −Δu=λe u, C. R. Acad. Sci. Paris 308 (1989), 493–498.

    Google Scholar 

  8. S. Chandrasekar, An Introduction to the Study of Stellar Structure, Dover, New York, 1985.

    Google Scholar 

  9. R. Dautray & J. L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Vols. 1, 2, 3, Masson, Paris, 1984.

    Google Scholar 

  10. C. L. Fefferman, The Uncertainty Principle, Bull. Amer. Math. Soc. 9 (1983), 129–206.

    Google Scholar 

  11. D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press, New York, 1969.

    Google Scholar 

  12. A. Friedman & J. B. McLeod, Blow-up of Positive Solutions of Semilinear Heat Equations, Indiana Univ. Math. Jour. 34 (1985), 425–447.

    Google Scholar 

  13. H. Fujita, On the nonlinear equations Δu+exp u=0 and vt=Δu+exp u, Bull. Amer. Math. Soc. 75 (1969), 132–135.

    Google Scholar 

  14. I. M. Gelfand, Some Problems in the Theory of Quasilinear Equations (see Section 15, due G. I. Barenblatt), Amer. Math. Soc. Transl. (Ser. 2) 29 (1963), 295–381.

    Google Scholar 

  15. Y. Giga & R. Kohn, Nondegeneracy of Blowup for Semilinear Heat Equations, Comm. Pure Appl. Math. 42 (1989), 845–884.

    Google Scholar 

  16. Th. Gallouët, F. Mignot & J. P. Puel, Quelques résultats sur le problème −Δu=exp u, C. R. Acad. Sci. Paris 307 (1988), 289–292.

    Google Scholar 

  17. B. Gidas, W.-M. Ni & L. Nirenberg, Symmetry and Related Properties via the Maximum Principle, Comm. Math. Phys. 68 (1979), 209–243.

    Google Scholar 

  18. J. García Azorero, I. Peral Alonso & J. P. Puel, Quasilinear Problems with Exponential Growth in the Reaction Term, Nonlinear Anal. 22 (1994), 481–498.

    Google Scholar 

  19. B. Guerch & L. Véron, Properties of Solutions of some Nonlinear Singular Schrödinger Equations, Revista Matem. Iberoamericana 7 (1991), 65–114.

    Google Scholar 

  20. G. Hardy, J. E. Littlewood & G. Polya, Inequalities, Cambridge University Press, 1934.

  21. D. Joseph & T. S. Lundgren, Quasilinear Dirichlet Problems Driven by Positive Sources, Arch. Rational Mech. Anal. 49 (1973), 241–269.

    Google Scholar 

  22. J. L. Kazdan & F. W. Warner, Curvature Functions for Compact 2-Manifolds, Annals of Mathematics 99 (1974), 14–47.

    Google Scholar 

  23. E. Lieb, Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities, Annals Math. 118 (1983), 349–374.

    Google Scholar 

  24. O. A. Ladyženskaja, V. A. Solonnikov & N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs, Amer. Math. Soc., Vol. 23, 1968.

  25. A. A. Lacey & D. Tzanetis, Global Existence and Convergence to a Singular Steady State far a Semilinear Heat Equation, Proc. Royal Soc. Edinburgh 105A (1987), 289–305.

    Google Scholar 

  26. F. Mignot & J. P. Puel, Solution radiale singulière de −Δu=eu, C. R. Acad. Sci. Paris 307 (1988), 379–382.

    Google Scholar 

  27. P. L. Lions, The Concentration-Compactness Principle in the Calculus of Variations (The Limit Case, Part 1), Revista Mat. Iberoamericana 1, no. 1 (1985), 145–201.

    Google Scholar 

  28. P. L. Lions, The Concentration-Compactness Principle in the Calculus of Variations (The Limit Case, Part 2), Revista Mat. Iberoamericana 1, no. 2 (1985), 45–121.

    Google Scholar 

  29. J. L. Vazquez & C. Yarur, Isolated Singularities of Schrödinger Equations with a Radial Potential, Arch. Rational Mech. Anal. 98 (1987), 251–284.

    Google Scholar 

  30. F. B. Weissler, Local Existence and Nonexistence for Semilinear Parabolic Equations in L p, Indiana Univ. Math. J. 29 (1980), 79–102.

    Google Scholar 

  31. F. B. Weissler, Existence and Nonexistence of Global Solutions for a Semilinear Heat Equation, Israel J. of Math. 38 (1981), 29–40.

    Google Scholar 

  32. F. B. Weissler, L p-Energy and Blow-up for a Semilinear Heat Equation, Proc. Symposia Pure Mathematics 45, Part 2 (1986), 545–551.

    Google Scholar 

  33. Y. B. Zeldovich, G. I. Barenblatt, V. B. Librovich & G. M. Makhviladze, The Mathematical Theory of Combustion and Explosions, Consulatants Bureau, New York and London, 1985.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid

    I. Peral & J. L. Vazquez

Authors
  1. I. Peral
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. L. Vazquez
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by H. Brezis

Rights and permissions

Reprints and permissions

About this article

Cite this article

Peral, I., Vazquez, J.L. On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term. Arch. Rational Mech. Anal. 129, 201–224 (1995). https://doi.org/10.1007/BF00383673

Download citation

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00383673

Keywords

Search

Navigation