Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-07-02T06:56:04.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence of blow-up solutions of nonlinear heat equations in the supercritical case

Published online by Cambridge University Press:  14 November 2011

J. Matos
J. Matos
Affiliation:
Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper, we study the blow-up behaviour of the radially symmetric non-negative solutions u of the semilinear heat equation with supercritical power nonlinearity up (that is, (N – 2)p> N + 2). We prove the existence of non-trivial self-similar blow-up patterns of u around the blow-up point x = 0. This result follows from a convergence theorem for a nonlinear parabolic equation associated to the initial one after rescaling by similarity variables.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 6 , 1999 , pp. 1197 - 1227
Copyright
Copyright © Royal Society of Edinburgh 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Angenent, S.. The zero set of a solution of a parabolic equation. J. Reine Angew Math. 390 (1988), 79–96.Google Scholar
2Bebernes, J. and Eberly, D.. A description of self-similar blow-up for dimension N ≧ 3. Ann. Inst. H. Poincaré Analyse Non Linéare 5 (1988), 1–21.CrossRefGoogle Scholar
3Berger, M. and Kohn, R. V.. A rescaling algorithm for the numerical calculation of blowing up solutions. Commun. Pure Appl. Math. 41 (1988), 841–863.CrossRefGoogle Scholar
4Bricher, S.. Blow-up behaviour for nonlinearly perturbed semilinear parabolic problems. Proc. R. Soc. Edinb. A 124 (1994), 947–969.CrossRefGoogle Scholar
5Budd, C. J. and Qi, Y.-W.. The existence of bounded solutions of a semilinear elliptic equation. J. Dig. Eqns 82 (1989), 207–218.CrossRefGoogle Scholar
6Chen, X. Y. and Matano, H.. Convergence, asymptotic periodicity and finite point blow-up in one-dimensional semilinear heat equations. J. Dig. Eqns 78 (1989), 160–190.CrossRefGoogle Scholar
7Chen, X. Y. and Poláčik, P.. Asymptotic periodicity of positive solutions of reaction-diffusion equations on a ball. J. Reine Angew Math. 472 (1996), 17–51.Google Scholar
8Coulhon, T. and Lamberton, D.. Régularité Lp pour les équations d'évolution. J. Am. Math. Soc. 1 (1988), 413–439.Google Scholar
9Filippas, S. and Kohn, R.. Refined asymptotics for the blow up of . Commun. Pure Appl. Math. 45 (1992), 821–869.CrossRefGoogle Scholar
10Friedman, A. and McLeod, B.. Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J. 34 (1985), 425–447.CrossRefGoogle Scholar
11Galaktionov, V. A. and Posaskhov, S. A.. Application of new comparison theorems in the investigation of unbounded solutions of nonlinear parabolic equations. Diff. Urav. 22 (1986), 1165–1173. (In Russian.) (Translation: Dig. Eqns 22 (1986), 809–815.)Google Scholar
12Galaktionov, V. A. and Vazquez, J. L.Continuation of blow-up solutions of nonlinear heat equations in several space dimensions. Commun. Pure Appl. Math. 50 (1997), 1–67.3.0.CO;2-H>CrossRefGoogle Scholar
13Galaktionov, V. A., Kurdyumov, S. P. and Samarskii, A. A.. Asymptotic stability of invariant solutions of nonlinear heat-conduction equation with sources. Dig. Urav. 20 (1984), 614–632. (Translation: Dig. Eqns 20 (1984), 461–476.)Google Scholar
14Giga, Y. and Kohn, R.. Asymptotically self-similar blow-up of semilinear heat equations. Commun. Pure Appl. Math. 38 (1985), 297–319.CrossRefGoogle Scholar
15Giga, Y. and Kohn, R.. Characterizing blowup using similarity variables. Indiana Univ. Math. J. 36 (1987), 1–40.CrossRefGoogle Scholar
16Giga, Y. and Kohn, R.. Nondegeneracy of blowup for semilinear heat equations. Commun. Pure Appl. Math. 42 (1989), 845–884.CrossRefGoogle Scholar
17Giga, Y. and Kohn, R.. Removability of blowup points for semilinear heat equations. Lecture Notes in Pure and Applied Mathematics, vol. 118, pp. 257–264 (New York: Dekker, 1989).Google Scholar
18Haraux, A.. Systèmes dynamiques dissipatifs et applications. Recherches en Mathématiques Appliquées, vol. 17 (Paris: Masson, 1991).Google Scholar
19Herrero, M. A. and Veláazquez, J. J. L.. Flat blow up in one dimensional semilinear heat equations. Dig. Int. Eqns 5 (1992), 973–997.Google Scholar
20Herrero, M. A. and Velázquez, J. J. L.. Blow up profiles in one-dimensional semilinear parabolic problems. Commun. Partial Diff. Eqns 17 (1992), 205–219.CrossRefGoogle Scholar
21Herrero, M. A. and Velázquez, J. J. L.. Blow up behaviour of one dimensional semilinear parabolic equations. Ann. Inst. H. Poincaré, Analyse Nonlin. 10 (1993), 131–189.CrossRefGoogle Scholar
22Herrero, M. A. and Velázquez, J. J. L.. Explosion de solutions des équations paraboliques semilinéaires supercritiques. C. R. Acad. Sd. Paris 319 (1994), 141–145.Google Scholar
23Joseph, D. D. and Lungdren, T. S.. Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Analysis 49 (1973), 241–269.CrossRefGoogle Scholar
24Kurtz, J. C.. Weighted Sobolev spaces with applications to singular nonlinear boundary value problems. J. Diff. Eqns 49 (1983), 105–123.CrossRefGoogle Scholar
25Lamberton, D.. Équations d'évolution linéaires associées à des semi-groups de contractions sur les espaces Lp. J. Fund. Analysis 72 (1987), 252–262.CrossRefGoogle Scholar
26Lee, T.-Y. and Ni, W.-M.. Global existence, large time behaviour and life span of solutions of a semilinear parabolic Cauchy problem. Trans. Am. Math. Soc. 333 (1992), 365–378.CrossRefGoogle Scholar
27Lepin, L. A.. Countable spectrum of eigenfunctions of a nonlinear heat-conduction equation with distributed parameters. Diff. Urav. 24 (1988), 1226–1234. (Translation: Diff. Eqns 24 (1988), 799–805.)Google Scholar
28Lepin, L. A.. Self-similar solutions of a semilinear heat equation. Mathematical Modelling 2 (1990), 63–74. (In Russian.)Google Scholar
29Louzar, T.. Comportement à l'infini des solutions radiales d'une équation de la chaleur semi-linéaire. Thése, Université Paris 6 (1984).Google Scholar
30Matano, H.. Convergence of solutions of one-dimensional parabolic equations. J. Math. Kyoto Univ. 18 (1978), 221–227.Google Scholar
31Matano, H.. Nonincrease of the lap number of a solution for a one-dimensional parabolic equation. J. Fac. Sd. Univ. Tokyo, Sec. 29 (1982), 401–441.Google Scholar
32Matano, H.. Asymptotic behaviour of solutions of semilinear heat equations on S1, Nonlinear Diffusion Equations and their Equilibrium States, pp. 139–162 (Springer, 1988).CrossRefGoogle Scholar
33Matos, J.. Self-similar blow up patterns for supercritical semilinear heat equations. Commun. Appl. Analysis. (In the press.)Google Scholar
34Matos, J.. Blow up of critical and subcritical norms in semilinear heat equations. Adv. Diff. Eqns 3 (1998), 497–532.Google Scholar
35Merle, F. and Zaag, H.. Optimal estimates for blow up rate and behaviour for nonlinear heat equations. Commun. Pure Appl. Math. 51 (1998), 139–196.3.0.CO;2-C>CrossRefGoogle Scholar
36Merle, F. and Zaag, H.. Refined uniform estimates at blow up and applications for nonlinear heat equations. Geom. Fund. Analysis. (In the press.)Google Scholar
37Mueller, C. E. and Weissler, F. B.. Single point blow-up for a general semilinear heat equation. Indiana Univ. Math. J. 34 (1985), 881–913.CrossRefGoogle Scholar
38Protter, M. and Weinberger, H.. Maximum principles in differential equations (Englewood Cliffs, NJ: Prentice-Hall, 1967).Google Scholar
39Souplet, P. and Weissler, F. B.. Self-similar subsolutions and blow-up for nonlinear parabolic equations. J. Math. Analysis Applic. 212 (1997), 60–74.CrossRefGoogle Scholar
40Souplet, P., Tayachi, S. and Weissler, F. B.. Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term. Indiana Univ. Math. J. 45 (1996), 655–682.CrossRefGoogle Scholar
41Troy, W. C.. The existence of bounded solutions of a semilinear heat equation. SIAM J. Math. Analysis 18 (1987), 332–336.CrossRefGoogle Scholar
42Velázquez, J. J. L.. Higher dimensional blow up for semilinear parabolic equations. Commun. Partial Diff. Eqns 17 (1992), 1567–1696.CrossRefGoogle Scholar
43Velázquez, J. J. L.. Classification of singularities for blowing up solutions in higher dimensions. Trans. Am. Math. Soc. 338 (1993), 441–464.CrossRefGoogle Scholar
44Wang, X.. On the Cauchy problem for reaction-diffusion equations. Trans. Am. Math. Soc. 337 (1993), 549–590.CrossRefGoogle Scholar