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2000 | 52 | 1 | 221-226
Tytuł artykułu

Blow-up behavior in nonlocal vs local heat equations

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We present some recent results on the blow-up behavior of solutions of heat equations with nonlocal nonlinearities. These results concern blow-up sets, rates and profiles. We then compare them with the corresponding results in the local case, and we show that the two types of problems exhibit "dual" blow-up behaviors.
Rocznik
Tom
52
Numer
1
Strony
221-226
Opis fizyczny
Daty
wydano
2000
Twórcy
  • Département de Mathématiques, Université de Picardie, INSSET, 02109 St-Quentin, France
Bibliografia
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  • [D] K. Deng, Nonlocal nonlinearity versus global blow-up, Math. Applicata (1995), 124-129.
  • [FM] A. Friedman and J. B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425-447.
  • [GK1] Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 8 (1985), 297-319.
  • [GK2] Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 1-40.
  • [GK3] Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equation, Comm. Pure Appl. Math. 42 (1989), 845-884.
  • [HV] M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Annali Sc. Norm. Sup. Pisa 19, 3 (1992), 381-450.
  • [MZ1] F. Merle and H. Zaag, Stability of blow-up profile for equation of the type $u_t=Δu + |u|^p-1u$, Duke Math. J. 86 (1997), 143-195.
  • [MZ2] F. Merle and H. Zaag, Optimal estimates for blow-up rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 (1998), 139-196.
  • [MW] C. E. Mueller and F. B. Weissler, Single point blow-up for general semilinear heat equation, Indiana Univ. Math. J. 34 (1985), 881-913.
  • [S1] Ph. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal. 29 (1998), 1301-1334.
  • [S2] Ph. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differ. Equations 153 (1999), 374-406.
  • [S3] Ph. Souplet, Some blow-up results for nonlocal reaction-diffusion equations, in: Actes du 3ème Congrès Européen sur les problèmes elliptiques et paraboliques (Pont-à-Mousson, juin 1997), Pitman Research Notes in Mathematics Series, n° 384, Addison Wesley Longman, 1998.
  • [V1] J. J. L. Velázquez, Classification of singularities for blowing-up solutions in higher dimensions, Trans. Amer. Math. Soc. 338 (1993), 441-464.
  • [V2] J. J. L. Velázquez, Estimates on the (n-1)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J. 42 (1993), 445-476.
  • [V3] J. J. L. Velázquez, Blow up for semilinear parabolic equations, in: Recent advances in partial differential equations, M. A. Herrero et al. (eds.), Res. Notes Appl. Math. 30, Masson, Paris, 1993, 131-145.
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  • [W2] F. B. Weissler, An $L^∞$ blow-up estimate for a nonlinear heat equation, Comm. Pure Appl. Math. 38 (1985), 291-295.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv52z1p221bwm
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