Liouville theorems and blow up behaviour in semilinear reaction diffusion systems

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1. [1] D. Amadori, Unstable blow up patterns, to appear in Diff. and Integr. Equations. Zbl0839.35062MR1348961
2. [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 Zbl0801.35043MR1188577
3. [3] D. Andreucci, Degenerate parabolic equations with initial data measures, to Trans. Amer. Mat. Soc. Zbl0885.35056MR1333384
4. [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, V1991. Zbl0762.35052
5. [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. Zbl0721.35009MR1075585
6. [6] J. Bebernes and A. Lacey, Finite-time blow up for semilinear reactive-diffusive systems, to appear in J. Diff. Eq. Zbl0801.35050
7. [7] A. Bressan, On the asymptotic shape of blow up, Indiana Univ. Math. J., Vol. 39, no. 41990, pp. 947-960 Zbl0798.35020MR1087180
8. [8] A. Bressan, Stable blow up patterns, J. Diff. Equations, Vol. 98, 1992, pp. 57-75. Zbl0770.35010MR1168971
9. [9] J. Bricmont and A. Kupiainen, Universality in blow up for nonlinear heat equations, preprint, 1993. Zbl0857.35018MR1267701
10. [10] G. Caristi and E. Mitidieri, Blow up estimates of positive solutions of a parabolic system to appear in J. Diff. Eq. Zbl0807.35066MR1297658
11. [11] E. Dibenedetto, Degenerate parabolic equations, Springer-Verlag, New York, 1993. Zbl0794.35090MR1230384
12. [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. Zbl0735.35013MR1088342
13. [13] M. Escobedo and M.A. Herrero, A semilinear parabolic system in a bounded domainAnnali Mat. Pura Appl. (IV), CLXV, 1993, pp. 315-336. Zbl0806.35088MR1271424
14. [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. Zbl0822.35068MR1328471
15. [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. Zbl0784.35010MR1164066
16. [16] S. Filippas and F. Merle, Modulation theory for the blow up of vector valued nonlinear heat equations, to appear in J. Diff. Eq. Zbl0814.35043
17. [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. Zbl0632.35028MR853803
18. [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. Zbl0465.35003MR615628
19. [19] Y. Giga and R.V. Kohn, Asymptotically self-similar blow up of semilinear heat equationsComm. Pure Appl. Math., Vol. 38, 1985, pp. 297-319. Zbl0585.35051MR784476
20. [20] Y. Giga, R.V. Kohn, Nondegeneracy of blow up for semilinear heat equations, Comm. Pure Appl. Math., Vol. 42, 1989, pp. 845-884. Zbl0703.35020MR1003437
21. [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. Zbl0813.35007MR1220032
22. [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. Zbl0767.35036MR1171974
23. [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. Zbl0806.35005MR1288393
24. [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. [25] F. Rothe, Global solutions of reaction-diffusion systems, in Lecture Notes in Mathematics, 1072, Springer-Verlag, New York, 1984. Zbl0546.35003MR755878
26. [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. Zbl0803.35015MR1134760
27. [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. Zbl0813.35009
28. [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. Zbl0798.35072
29. [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. Zbl0926.35023MR1318773
30. [30] F.B. Weissler, An L∞ blow up estimate for a nolinear heat equation, Comm. Pure Appl. Math., Vol. 38, 1985, pp. 291-295. Zbl0592.35071MR784475

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