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Single-point blow-up for a semilinear parabolic system. (English) Zbl 1171.35059
The authors study the positive solutions of a semilinear parabolic system coupled by power nonlinearilities in a ball or in the whole space. A single-point blow-up result is proved for a large class of radial decreasing solutions, which answer the open problem posed by {\it A. Friedman} and {\it Y. Giga} [J. Fac. Sci., Univ. Tokyo, Sect. I A 34, 65--79 (1987; Zbl 0648.35042)]. Finally, the author obtain the lower pointwise estimates for the final blow-up profiles.

35K50Systems of parabolic equations, boundary value problems (MSC2000)
35B40Asymptotic behavior of solutions of PDE
35K35Higher order parabolic equations, boundary value problems
35K57Reaction-diffusion equations
Full Text: DOI Link
[1] Andreucci, D., Herrero, M. A., Velázquez, J. J. L.: Liouville theorems and blow up behaviour in semilinear reaction diffusion systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 14, 1-53 (1997) · Zbl 0877.35019 · doi:10.1016/S0294-1449(97)80148-5 · numdam:AIHPC_1997__14_1_1_0 · eudml:78405
[2] Caristi, G., Mitidieri, E.: Blow-up estimates of positive solutions of a parabolic system. J. Differential Equations 113, 265-271 (1994) · Zbl 0807.35066 · doi:10.1006/jdeq.1994.1124
[3] Chen, X.-Y., Matano, H.: Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations. J. Differential Equations 78, 160-190 (1989) · Zbl 0692.35013 · doi:10.1016/0022-0396(89)90081-8
[4] Chlebík, M., Fila, M.: From critical exponents to blow-up rates for parabolic problems. Rend. Mat. Appl. (7) 19, 449-470 (1999) · Zbl 0980.35057
[5] Deng, K.: Blow-up rates for parabolic systems. Z. Angew. Math. Phys. 47, 132-143 (1996) · Zbl 0854.35054 · doi:10.1007/BF00917578