On the regularity of the blow-up set for semilinear heat equations

Zaag, Hatem

Zaag, Hatem

Annales de l'I.H.P. Analyse non linéaire, Tome 19 (2002) no. 5, pp. 505-542.

MR Zbl | 5 citations dans Numdam

@article{AIHPC_2002__19_5_505_0,
     author = {Zaag, Hatem},
     title = {On the regularity of the blow-up set for semilinear heat equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {505--542},
     publisher = {Elsevier},
     volume = {19},
     number = {5},
     year = {2002},
     mrnumber = {1922468},
     zbl = {1012.35039},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2002__19_5_505_0/}
}

TY  - JOUR
AU  - Zaag, Hatem
TI  - On the regularity of the blow-up set for semilinear heat equations
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2002
SP  - 505
EP  - 542
VL  - 19
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/item/AIHPC_2002__19_5_505_0/
LA  - en
ID  - AIHPC_2002__19_5_505_0
ER  -

%0 Journal Article
%A Zaag, Hatem
%T On the regularity of the blow-up set for semilinear heat equations
%J Annales de l'I.H.P. Analyse non linéaire
%D 2002
%P 505-542
%V 19
%N 5
%I Elsevier
%U http://www.numdam.org/item/AIHPC_2002__19_5_505_0/
%G en
%F AIHPC_2002__19_5_505_0

Zaag, Hatem. On the regularity of the blow-up set for semilinear heat equations. Annales de l'I.H.P. Analyse non linéaire, Tome 19 (2002) no. 5, pp. 505-542. http://www.numdam.org/item/AIHPC_2002__19_5_505_0/

Bibliographie
Cité par

[1] Ball J.M., Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford (Ser. 2) 28 (112) (1977) 473-486. | MR | Zbl

[2] Bernoff A.J., Bertozzi A.L., Witelski T.P., Axisymmetric surface diffusion: dynamics and stability of self-similar pinchoff, J. Statist. Phys. 93 (3-4) (1998) 725-776. | MR | Zbl

[3] M.D. Betterton, M.P. Brenner, Collapsing bacterial cylinders, Preprint.

[4] Brenner M.P., Constantin P., Kadanoff L.P., Schenkel A., Venkataramani S.C., Diffusion, attraction and collapse, Nonlinearity 12 (4) (1999) 1071-1098. | MR | Zbl

[5] Bricmont J., Kupiainen A., Universality in blow-up for nonlinear heat equations, Nonlinearity 7 (2) (1994) 539-575. | MR | Zbl

[6] Chapman S.J., Hunton B.J., Ockendon J.R., Vortices and boundaries, Quart. Appl. Math. 56 (3) (1998) 507-519. | MR | Zbl

[7] Deng K., Levine H.A., The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl. (2000). | MR | Zbl

[8] Fermanian Kammerer C., Merle F., Zaag H., Stability of the blow-up profile of non-linear heat equations from the dynamical system point of view, Math. Annalen 317 (2) (2000) 195-237. | MR | Zbl

[9] Fermanian Kammerer C., Zaag H., Boundedness up to blow-up of the difference between two solutions to a semilinear heat equation, Nonlinearity 13 (4) (2000) 1189-1216. | MR | Zbl

[10] Filippas S., Kohn R.V., Refined asymptotics for the blowup of u_t−Δu=u^p, Comm. Pure Appl. Math. 45 (7) (1992) 821-869. | Zbl

[11] Filippas S., Liu W.X., On the blowup of multidimensional semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (3) (1993) 313-344. | Numdam | MR | Zbl

[12] Fujita H., On the blowing up of solutions of the Cauchy problem for u_t=Δu+u^1+α, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966) 109-124. | Zbl

[13] Giga Y., Kohn R.V., Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (6) (1989) 845-884. | MR | Zbl

[14] Herrero M.A., Velázquez J.J.L., Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (2) (1993) 131-189. | Numdam | MR | Zbl

[15] Kato T., Perturbation Theory for Linear Operators, Springer, Berlin, 1995, Reprint of the 1980 edition. | MR | Zbl

[16] Levine H.A., Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Pu_t=−Au+F(u), Arch. Rational Mech. Anal. 51 (1973) 371-386. | Zbl

[17] Merle F., Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math. 45 (3) (1992) 263-300. | MR | Zbl

[18] Merle F., Zaag H., Reconnection of vortex with the boundary and finite time quenching, Nonlinearity 10 (6) (1997) 1497-1550. | MR | Zbl

[19] Merle F., Zaag H., Stability of the blow-up profile for equations of the type u_t=Δu+|u|^p−1u, Duke Math. J. 86 (1) (1997) 143-195. | Zbl

[20] Merle F., Zaag H., Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 (2) (1998) 139-196. | MR | Zbl

[21] Merle F., Zaag H., A Liouville theorem for vector-valued nonlinear heat equations and applications, Math. Annalen 316 (1) (2000) 103-137. | MR | Zbl

[22] F. Oustry, M.L. Overton, Variational analysis of the total projection for symmetric matrices, 2000.

[23] Soner H.M., Souganidis P.E., Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature, Comm. Partial Differential Equations 18 (5-6) (1993) 859-894. | MR | Zbl

[24] Velázquez J.J.L., Higher-dimensional blow up for semilinear parabolic equations, Comm. Partial Differential Equations 17 (9-10) (1992) 1567-1596. | MR | Zbl

[25] Velázquez J.J.L., Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc. 338 (1) (1993) 441-464. | MR | Zbl

[26] Velázquez J.J.L., Estimates on the (n−1)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J. 42 (2) (1993) 445-476. | Zbl

[27] H. Zaag, One-dimensional behavior of singular N-dimensional solutions of semilinear heat equations, Preprint, 2001. | MR