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Blowup estimates for a semilinear reaction diffusion system. (English) Zbl 0990.35065
The author deals with the problem of blow-up estimates for positive solutions of the semilinear parabolic system: \[ u_t=\Delta u +u^\alpha v^p,\qquad v_t=\Delta v +u^q v^\beta, \] subject to zero Dirichlet boundary condition on a ball of \(\mathbb{R}^N\) with nonnegative continuous initial data. The results improve previous work of S. N. Zheng [J. Math. Anal. Appl. 232, 293-311 (1999; Zbl 0935.35042)]. The method of proof relies on the maximum principle.

MSC:
35K57 Reaction-diffusion equations
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35B50 Maximum principles in context of PDEs
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[1] Caristi, G.; Mitidieri, E., Blow-up estimates of positive solutions of a parabolic system, J. differential equations, 113, 265-271, (1994) · Zbl 0807.35066
[2] M. Chlebik, and, M. Fila, From critical exponents to blow-up rates for parabolic problems, preprint. · Zbl 0980.35057
[3] Deng, K., Blow-up rates for parabolic systems, Z. angew. math. phys., 47, 132-143, (1996) · Zbl 0854.35054
[4] Deng, K., The blow-up behavior of the heat equation with Neumann boundary conditions, J. math. anal. appl., 188, 641-650, (1994) · Zbl 0809.35006
[5] Deng, K.; Levine, H.A., The role of critical exponents in blow-up theorems: the sequel, J. math. anal. appl., 243, 85-126, (2000) · Zbl 0942.35025
[6] Escobedo, M.; Herrero, M.A., A semilinear parabolic system in a bounded domain, Ann. mat. pura appl., 165, 315-336, (1993) · Zbl 0806.35088
[7] Fila, M.; Quittner, P., The blow-up rate for a semilinear parabolic system, J. math. anal. appl., 238, 468-476, (1999) · Zbl 0934.35062
[8] Friedman, A.; McLeod, B., Blow-up of positive solutions of semilinear heat equations, Indiana univ. math. J., 34, 425-447, (1985) · Zbl 0576.35068
[9] Giga, Y.; Kohn, V., Characterizing blow-up using similarity variables, Indiana univ. math. J., 36, 425-447, (1987)
[10] Hu, B., Remarks on the blowup estimate for solutions of the heat equation with a nonlinear boundary condition, Differential integral equations, 9, 891-901, (1996) · Zbl 0852.35072
[11] Hu, B.; Yin, H.M., The profile near blow-up time for solution of the heat equation with a nonlinear boundary condition, Trans. amer. math. soc., 346, 117-135, (1994) · Zbl 0823.35020
[12] Lin, Z.G.; Wang, M.X., The blow-up properties of solutions to semilinear heat equations with nonlinear boundary conditions, Z. angew. math. phys., 50, 361-374, (1999) · Zbl 0926.35062
[13] Liu, W.X., The blow-up rate of solutions of semilinear heat equations, J. differential equations, 77, 104-122, (1989) · Zbl 0672.35035
[14] Pao, C.V., Nonlinear parabolic and elliptic equations, (1992), Plenum New York/London · Zbl 0780.35044
[15] Samarskii, A.A.; Galaktionov, V.A.; Kurdyumov, S.P.; Mikhailov, A.P., Blow-up in quasilinear parabolic equations, (1995), de Gruyter Berlin · Zbl 1020.35001
[16] Wang, M.X., Global existence and finite time blow up for a reaction-diffusion system, Z. angew. math. phys., 51, 160-167, (2000) · Zbl 0984.35088
[17] M. X. Wang, Blow-up rate estimates for semilinear parabolic systems, J. Differential Equations, to appear. · Zbl 0979.35065
[18] Weissler, F.B., An L∞ blow-up estimate for a nonlinear heat equation, Comm. pure appl. math., 38, 291-295, (1985) · Zbl 0592.35071
[19] Zheng, S.N., Nonexistence of positive solutions to a semilinear elliptic system and blow-up estimates for a reaction-diffusion system, J. math. anal. appl., 232, 293-311, (1999) · Zbl 0935.35042
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