Liouville theorems and blow up behaviour in semilinear reaction diffusion systems
Annales de l'I.H.P. Analyse non linéaire, Volume 14 (1997) no. 1, p. 1-53
@article{AIHPC_1997__14_1_1_0,
     author = {Andreucci, D. and Herrero, Miguel A. and Velazquez, Juan J. L.},
     title = {Liouville theorems and blow up behaviour in semilinear reaction diffusion systems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Gauthier-Villars},
     volume = {14},
     number = {1},
     year = {1997},
     pages = {1-53},
     zbl = {0877.35019},
     mrnumber = {1437188},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_1997__14_1_1_0}
}
Andreucci, D.; Herrero, M. A.; Velázquez, J. J. L. Liouville theorems and blow up behaviour in semilinear reaction diffusion systems. Annales de l'I.H.P. Analyse non linéaire, Volume 14 (1997) no. 1, pp. 1-53. http://www.numdam.org/item/AIHPC_1997__14_1_1_0/

[1] D. Amadori, Unstable blow up patterns, to appear in Diff. and Integr. Equations. | MR 1348961 | Zbl 0839.35062

[2] D. Andreucci, New results on the Cauchy problem for parabolic systems and with strongly nonlinear sources, Manuscripta Math., Vol. 77, 1992, pp. 127-159 | MR 1188577 | Zbl 0801.35043

[3] D. Andreucci, Degenerate parabolic equations with initial data measures, to Trans. Amer. Mat. Soc. | MR 1333384 | Zbl 0885.35056

[4] D. Andreucci and E. Dibenedetto, On the Cauchy problem and initial traces for a of evolution equations with strongly nonlinear sources, Ann. Sc. Normale Pisa, V 1991. | Numdam | Zbl 0762.35052

[5] J. Bebernes and A. Lacey, Finite-time blow up for a particular parabolic system, J. Math. Anal., Vol. 21, no. 6, 1990, pp. 1415-1425. | MR 1075585 | Zbl 0721.35009

[6] J. Bebernes and A. Lacey, Finite-time blow up for semilinear reactive-diffusive systems, to appear in J. Diff. Eq.

[7] A. Bressan, On the asymptotic shape of blow up, Indiana Univ. Math. J., Vol. 39, no. 4 1990, pp. 947-960 | MR 1087180 | Zbl 0798.35020

[8] A. Bressan, Stable blow up patterns, J. Diff. Equations, Vol. 98, 1992, pp. 57-75. | MR 1168971 | Zbl 0770.35010

[9] J. Bricmont and A. Kupiainen, Universality in blow up for nonlinear heat equations, preprint, 1993. | MR 1267701

[10] G. Caristi and E. Mitidieri, Blow up estimates of positive solutions of a parabolic system to appear in J. Diff. Eq. | MR 1297658 | Zbl 0807.35066

[11] E. Dibenedetto, Degenerate parabolic equations, Springer-Verlag, New York, 1993. | MR 1230384 | Zbl 0794.35090

[12] M. Escobedo and M.A. Herrero, Boundedness and blow up for a semilinear reaction diffusion system, J. Diff. Eq., Vol. 89, no. 1, 1991, pp. 176-202. | MR 1088342 | Zbl 0735.35013

[13] M. Escobedo and M.A. Herrero, A semilinear parabolic system in a bounded domain Annali Mat. Pura Appl. (IV), CLXV, 1993, pp. 315-336. | MR 1271424 | Zbl 0806.35088

[14] M. Escobedo and H.A. Levine, Critical blow up and global existence numbers for a weakly coupled system of reaction-diffusion equations, to appear in Trans. Amer. Math. Soc. | MR 1328471

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

[16] S. Filippas and F. Merle, Modulation theory for the blow up of vector valued nonlinear heat equations, to appear in J. Diff. Eq. | Zbl 0814.35043

[17] V.A. Galaktionov and S.A. Posashkov, Application of new comparison theorems in the investigation of unbounded solutions of nonlinear parabolic equations, Diff. Urav., Vol. 22, no. 7, 1986, pp. 1165-1173. | MR 853803 | Zbl 0632.35028

[18] B. Gidas and J. Spruck, Global and local behaviour of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., Vol. 34, 1981, pp. 525-598. | MR 615628 | Zbl 0465.35003

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

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

[21] M.A. Herrero and J.J.L. Velázquez, Blow up behaviour of one dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré, Vol. 10, no. 2, 1993, pp. 131-189. | Numdam | MR 1220032 | Zbl 0813.35007

[22] M.A. Herrero and J.J.L. Velázquez, Flat blow up in one dimensional semilinear heat equations, Diff. and Integral Eq., Vol. 5, 1992, pp. 973-998. | MR 1171974 | Zbl 0767.35036

[23] M.A. Herrero and J.J.L. Velázquez, Explosion de solutions d'equations paraboliques semilinéaires supercritiques, C. R. Acad. Sci. Série A, Vol. 319, 1994, pp. 141-145. | MR 1288393 | Zbl 0806.35005

[24] O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Uraltseva, Linear and quasilinear equations of parabolic type, AMS Translations of Math., Monographs, XXIII, Providence RI, 1968.

[25] F. Rothe, Global solutions of reaction-diffusion systems, in Lecture Notes in Mathematics, 1072, Springer-Verlag, New York, 1984. | MR 755878 | Zbl 0546.35003

[26] J.J.L. Velázquez, Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc., Vol. 338, no. 1, 1993, pp. 441-464. | MR 1134760 | Zbl 0803.35015

[27] J.J.L. Velázquez, Higher dimensional blow up for semilinear parabolic equations, Comm. in PDE, Vol. 17, no. 9&10, 1992, pp. 1567-1596. | Zbl 0813.35009

[28] J.J.L. Velázquez, Blow up of semilinear parabolic equations, in Recent advances in partial differential equations, eds. M. A. Herrero and E. Zuazua, Research in Applied Mathematics, Masson & J. Wiley, 1994, pp. 131-145. | Zbl 0798.35072

[29] J.J.L. Velázquez, Curvature blow up in perturbations of minimizing cones evolving by mean curvature flow, Ann. Scuola Normale Sup. Pisa, Serie IV, Vol. XXI, 1994, pp. 595-628. | Numdam | MR 1318773 | Zbl 0926.35023

[30] F.B. Weissler, An L∞ blow up estimate for a nolinear heat equation, Comm. Pure Appl. Math., Vol. 38, 1985, pp. 291-295. | MR 784475 | Zbl 0592.35071