Blow-up results for vector-valued nonlinear heat equations with no gradient structure
Zaag, Hatem
Annales de l'I.H.P. Analyse non linéaire, Tome 15 (1998), p. 581-622 / Harvested from Numdam
Publié le : 1998-01-01
@article{AIHPC_1998__15_5_581_0,
     author = {Zaag, Hatem},
     title = {Blow-up results for vector-valued nonlinear heat equations with no gradient structure},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {15},
     year = {1998},
     pages = {581-622},
     mrnumber = {1643389},
     zbl = {0902.35050},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_1998__15_5_581_0}
}
Zaag, Hatem. Blow-up results for vector-valued nonlinear heat equations with no gradient structure. Annales de l'I.H.P. Analyse non linéaire, Tome 15 (1998) pp. 581-622. http://gdmltest.u-ga.fr/item/AIHPC_1998__15_5_581_0/

[1] J. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford, Vol. 28, 1977, pp. 473-486. | MR 473484 | Zbl 0377.35037

[2] M. Berger and R. Kohn, A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. Pure Appl. Math., Vol. 41, 1988, pp. 841-863. | MR 948774 | Zbl 0652.65070

[3] J. Bricmont and A. Kupiainen, Renormalization group and nonlinear PDEs, Quantum and non-commutative analysis, past present and future perspectives, Kluwer (Boston), 1993. | MR 1276284 | Zbl 0842.35040

[4] J. Bricmont and A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity, 7, 1994, pp. 539-575. | MR 1267701 | Zbl 0857.35018

[5] S. Filippas and R. Kohn, Refined asymptotics for the blowup of ut - Δu = up, Comm. Pure Appl. Math., Vol. 45, 1992, pp. 821-869 | MR 1164066 | Zbl 0784.35010

[6] S. Filippas and F. Merle, Modulation theory for the blowup of vector-valued nonlinear heat equations, J. Diff. Equations, Vol. 116, 1995, pp. 119-148. | MR 1317705 | Zbl 0814.35043

[7] V.A. Galaktionov, S.P. Kurdyumov and A.A. Samarskii, On approximate self-similar solutions for some class of quasilinear heat equations with sources, Math. USSR-Sb, Vol. 52, 1985, pp. 155-180. | Zbl 0573.35049

[8] V.A. Galaktionov and J.L. Vazquez, Regional blow-up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. Math. Anal., Vol. 24, 1993, pp. 1254-1276. | MR 1234014 | Zbl 0813.35033

[9] Y. Giga and R. Kohn, Asymptotically self-similar blowup of semilinear heat equations, Comm. Pure Appl. Math., Vol. 38, 1985, pp. 297-319. | MR 784476 | Zbl 0585.35051

[10] Y. Giga and R. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J., Vol. 36, 1987, pp. 1-40. | MR 876989 | Zbl 0601.35052

[11] Y. Giga and R. Kohn, Nondegeneracy of blow-up for semilinear heat equations, Comm. Pure Appl. Math., Vol. 42, 1989, pp. 845-884. | MR 1003437 | Zbl 0703.35020

[12] R.S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II, Internat. Press, Cambridge, 1995, pp. 7-136. | MR 1375255 | Zbl 0867.53030

[13] M.A. Herrero and J.J.L. Velazquez, Blow-up behavior of one-dimensional semilinear parabolic equations, Ann. Inst. Henri Poincaré, Vol. 10, 1993, pp. 131-189. | Numdam | MR 1220032 | Zbl 0813.35007

[14] M.A. Herrero and J.J.L. Velazquez, Flat blow-up in one-dimensional semilinear heat equations, Differential and Integral eqns., Vol. 5, 1992, pp. 973-997. | MR 1171974 | Zbl 0767.35036

[15] C.D. Levermore and M. Oliver, The complex Ginzburg-Landau equation as a model problem, Dynamical systems and probabilistic methods in partial differential equations (Berkeley, 1994), Lectures in Appl. Math., Vol. 31, Amer. Math. Soc., Providence, RI, 1996, pp. 141-190. | MR 1363028 | Zbl 0845.35003

[16] H. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put = -Au + F(u), Arch. Rat. Mech. Anal., Vol. 51, 1973, pp. 371-386. | MR 348216 | Zbl 0278.35052

[17] F. Merle, Solution of a nonlinear heat equation with arbitrary given blow-up points, Comm. Pure Appl. Math., Vol. 45, 1992, pp. 263-300. | MR 1151268 | Zbl 0785.35012

[18] F. Merle and H. Zaag, Stability of blow-up profile for equation of the type ut = Δu + |u|p-1u, preprint.

[19] J.J.L. Velazquez, Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc., Vol. 338, 1993, pp. 441-464. | MR 1134760 | Zbl 0803.35015

[20] F. Weissler, Single-point blowup for a semilinear initial value problem, J. Diff. Equations, Vol. 55, 1984, pp. 204-224. | MR 764124 | Zbl 0555.35061