Nous considérons des solutions explosives de l'équation semilinéaire de la chaleur avec une nonlinéarité sous-critique au sens de Sobolev. Etant donné un point d'explosion , grâce à des travaux antérieurs, on connaît le comportement asymptotique des solutions en variables auto-similaires. Notre objectif est de discuter la stabilité de ce comportement, par rapport à des perturbations du point d'explosion et de la donnée initiale. Introduisant la notion de « l'ordre du profil », nous montrons qu'il est semi-continu supérieurement, et continu uniquement aux points où il est un minimum local.
We consider blow-up solutions for semilinear heat equations with Sobolev subcritical power nonlinearity. Given a blow-up point , we have from earlier literature, the asymptotic behavior in similarity variables. Our aim is to discuss the stability of that behavior, with respect to perturbations in the blow-up point and in initial data. Introducing the notion of “profile order”, we show that it is upper semicontinuous, and continuous only at points where it is a local minimum.
@article{AIHPC_2011__28_1_1_0, author = {Khenissy, S. and R\'eba{\"\i}, Y. and Zaag, H.}, title = {Continuity of the blow-up profile with respect to initial data and to the blow-up point for a semilinear heat equation}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1--26}, publisher = {Elsevier}, volume = {28}, number = {1}, year = {2011}, doi = {10.1016/j.anihpc.2010.09.006}, zbl = {1215.35090}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.006/} }
TY - JOUR AU - Khenissy, S. AU - Rébaï, Y. AU - Zaag, H. TI - Continuity of the blow-up profile with respect to initial data and to the blow-up point for a semilinear heat equation JO - Annales de l'I.H.P. Analyse non linéaire PY - 2011 SP - 1 EP - 26 VL - 28 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.006/ DO - 10.1016/j.anihpc.2010.09.006 LA - en ID - AIHPC_2011__28_1_1_0 ER -
%0 Journal Article %A Khenissy, S. %A Rébaï, Y. %A Zaag, H. %T Continuity of the blow-up profile with respect to initial data and to the blow-up point for a semilinear heat equation %J Annales de l'I.H.P. Analyse non linéaire %D 2011 %P 1-26 %V 28 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.006/ %R 10.1016/j.anihpc.2010.09.006 %G en %F AIHPC_2011__28_1_1_0
Khenissy, S.; Rébaï, Y.; Zaag, H. Continuity of the blow-up profile with respect to initial data and to the blow-up point for a semilinear heat equation. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 1, pp. 1-26. doi : 10.1016/j.anihpc.2010.09.006. http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.006/
[1] Blowup for Nonlinear Hyperbolic Equations, Birkhäuser Boston Inc., Boston (1995) | MR | Zbl
,[2] A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations, Journées “Équations aux Dérivées Partielles”, Forges-Les-Eaux, 2002, Univ. Nantes, Nantes (2002) | EuDML | Numdam | MR
,[3] Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser. (2) 28 no. 112 (1977), 473-486 | MR | Zbl
,[4] Universality in blow-up for nonlinear heat equations, Nonlinearity 7 no. 2 (1994), 539-575 | MR | Zbl
, ,[5] Boundedness up to blow-up of the difference between two solutions to a semilinear heat equation, Nonlinearity 13 no. 4 (2000), 1189-1216 | MR | Zbl
, ,[6] Stability of the blow-up profile of nonlinear heat equations from the dynamical system point of view, Math. Ann. 317 (2000), 347-387 | MR | Zbl
, , ,[7] Refined asymptotics for the blow-up of , Comm. Pure Appl. Math. XLV (1992), 821-869 | MR | Zbl
, ,[8] On the blowup of multidimensional semilinear heat equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 no. 3 (1993), 313-344 | EuDML | Numdam | MR | Zbl
, ,[9] On the blowing-up of solutions of the Cauchy problem for , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 13 (1966), 109-124 | MR | Zbl
,[10] Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 no. 3 (1985), 297-319 | MR | Zbl
, ,[11] Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 no. 1 (1987), 1-40 | MR | Zbl
, ,[12] Nondegeneracy of blow-up for semilinear heat equations, Comm. Pure Appl. Math. 42 no. 6 (1989), 845-884 | MR | Zbl
, ,[13] Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J. 53 no. 2 (2004), 483-514 | MR | Zbl
, , ,[14] Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations 5 (1992), 973-997 | MR | Zbl
, ,[15] Comportement générique au voisinage d'un point d'explosion pour des solutions d'équations paraboliques unidimensionnelles, C. R. Math. Acad. Sci. Paris 314 no. I (1992), 201-203 | Zbl
, ,[16] Generic behaviour of one-dimensional blow-up patterns, Ann. Sc. Norm. Super. Pisa Cl. Sci. 19 no. 3 (1992), 381-450 | EuDML | Numdam | MR | Zbl
, ,[17] Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré 10 no. 2 (1993), 131-189 | EuDML | Numdam | MR | Zbl
, ,[18] M.A. Herrero, J.J.L. Velázquez, Generic behaviour near blow-up points for a N-dimensional semilinear heat equation, in preparation.
[19] Perturbation Theory for Linear Operators, Springer, Berlin (1995) | MR | Zbl
,[20] Some nonexistence and instability theorems for solutions of formally parabolic equations of the form , Arch. Ration. Mech. Anal. 51 (1973), 371-386 | MR | Zbl
,[21] On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 no. 2 (1995), 357-426 | MR | Zbl
, ,[22] On nonexistence of type II blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math. 57 no. 11 (2004), 1494-1541 | MR | Zbl
, ,[23] Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math. 45 no. 3 (1992), 263-300 | MR | Zbl
,[24] Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 no. 2 (1998), 139-196 | MR | Zbl
, ,[25] A Liouville theorem for vector-valued nonlinear heat equations and applications, Math. Ann. 316 no. 1 (2000), 103-137 | MR | Zbl
, ,[26] Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension, J. Funct. Anal. 253 no. 1 (2007), 43-121 | MR | Zbl
, ,[27] regularity of the blow-up curve at non characteristic points for the one dimensional semilinear wave equation, Comm. Partial Differential Equations 33 (2008), 1540-1548 | MR | Zbl
,[28] Higher dimensional blow-up for semilinear parabolic equations, Comm. Partial Differential Equations 17 no. 9–10 (1992), 1567-1596 | MR | Zbl
,[29] Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc. 338 no. 1 (1993), 441-464 | MR | Zbl
,[30] Estimates on the ()-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J. 42 no. 2 (1993), 445-476 | MR | Zbl
,[31] Blow up for semilinear parabolic equations, Recent Advances in Partial Differential Equations, El Escorial, 1992, RAM Res. Appl. Math. vol. 30, Masson, Paris (1994), 131-145
,[32] Single point blow-up for a semilinear initial value problem, J. Differential Equations 55 no. 2 (1984), 204-224 | MR | Zbl
,[33] On the regularity of the blow-up set for semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 no. 5 (2002), 505-542 | EuDML | Numdam | MR | Zbl
,[34] One dimensional behavior of singular N dimensional solutions of semilinear heat equations, Comm. Math. Phys. 225 no. 3 (2002), 523-549 | MR | Zbl
,[35] Regularity of the blow-up set and singular behavior for semilinear heat equations, Mathematics & Mathematics Education, Bethlehem, 2000, World Sci. Publishing, River Edge, NJ (2002), 337-347 | MR | Zbl
,[36] Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation, Duke Math. J. 133 no. 3 (2006), 499-525 | MR | Zbl
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