Continuity of the blow-up profile with respect to initial data and to the blow-up point for a semilinear heat equation

Khenissy, S.; Rébaï, Y.; Zaag, H.

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 1, pp. 1-26

We consider blow-up solutions for semilinear heat equations with Sobolev subcritical power nonlinearity. Given a blow-up point $\hat{a}$, 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.

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 $\hat{a}$, 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.

[1] S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Birkhäuser Boston Inc., Boston (1995) | MR | Zbl

[2] S. Alinhac, 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) | MR | EuDML | Numdam

[3] J.M. Ball, 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] J. Bricmont, A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity 7 no. 2 (1994), 539-575 | MR | Zbl

[5] C. Fermanian Kammerer, H. Zaag, 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] C. Fermanian Kammerer, F. Merle, H. Zaag, 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] S. Filippas, R.V. Kohn, Refined asymptotics for the blow-up of $u_t-\Delta u=u^p$, Comm. Pure Appl. Math. XLV (1992), 821-869 | MR | Zbl

[8] S. Filippas, W. Liu, On the blowup of multidimensional semilinear heat equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 no. 3 (1993), 313-344 | MR | EuDML | Zbl | Numdam

[9] H. Fujita, On the blowing-up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 13 (1966), 109-124 | MR | Zbl

[10] Y. Giga, R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 no. 3 (1985), 297-319 | MR | Zbl

[11] Y. Giga, R.V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 no. 1 (1987), 1-40 | MR | Zbl

[12] Y. Giga, R.V. Kohn, Nondegeneracy of blow-up for semilinear heat equations, Comm. Pure Appl. Math. 42 no. 6 (1989), 845-884 | MR | Zbl

[13] Y. Giga, S. Matsui, S. Sasayama, Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J. 53 no. 2 (2004), 483-514 | MR | Zbl

[14] M.A. Herrero, J.J.L. Velázquez, Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations 5 (1992), 973-997 | MR | Zbl

[15] M.A. Herrero, J.J.L. Velázquez, 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] M.A. Herrero, J.J.L. Velázquez, Generic behaviour of one-dimensional blow-up patterns, Ann. Sc. Norm. Super. Pisa Cl. Sci. 19 no. 3 (1992), 381-450 | MR | EuDML | Zbl | Numdam

[17] M.A. Herrero, J.J.L. Velázquez, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré 10 no. 2 (1993), 131-189 | MR | EuDML | Zbl | Numdam

[18] M.A. Herrero, J.J.L. Velázquez, Generic behaviour near blow-up points for a N-dimensional semilinear heat equation, in preparation.

[19] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1995) | MR | Zbl

[20] H.A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $P{u}_t=-Au+F\left(u\right)$, Arch. Ration. Mech. Anal. 51 (1973), 371-386 | MR | Zbl

[21] H. Lindblad, C.D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 no. 2 (1995), 357-426 | MR | Zbl

[22] H. Matano, F. Merle, 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] F. Merle, 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] F. Merle, H. Zaag, Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 no. 2 (1998), 139-196 | MR | Zbl

[25] F. Merle, H. Zaag, A Liouville theorem for vector-valued nonlinear heat equations and applications, Math. Ann. 316 no. 1 (2000), 103-137 | MR | Zbl

[26] F. Merle, H. Zaag, 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] N. Nouaili, ${𝒞}^{1,\alpha }$ 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] J.J.L. Velázquez, Higher dimensional blow-up for semilinear parabolic equations, Comm. Partial Differential Equations 17 no. 9–10 (1992), 1567-1596 | MR | Zbl

[29] J.J.L. Velázquez, Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc. 338 no. 1 (1993), 441-464 | MR | Zbl

[30] 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 no. 2 (1993), 445-476 | MR | Zbl

[31] J.J.L. Velázquez, 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] F.B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equations 55 no. 2 (1984), 204-224 | MR | Zbl

[33] H. Zaag, 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 | MR | EuDML | Zbl | Numdam

[34] H. Zaag, One dimensional behavior of singular N dimensional solutions of semilinear heat equations, Comm. Math. Phys. 225 no. 3 (2002), 523-549 | MR | Zbl

[35] H. Zaag, 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] H. Zaag, 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