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Blow-up analysis for a system of heat equations coupled through a nonlinear boundary condition. (English) Zbl 0980.35075

The authors deal with the blow-up properties of the positive solutions to the system of heat equations with nonlinear boundary conditions: \[ \begin{cases} u_{it}=\Delta_x u_i,\quad i= 1,\dots, k,\quad u_{k+1}:= u_1,\quad x\in\Omega,\quad t\in J,\\ {\partial u_i\over\partial\eta}= u^{p_i}_{i+ 1},\quad x\in\partial\Omega,\quad t\in J,\\ u_i(x,0)= u_{i,0}(x),\quad x\in \Omega,\end{cases}\tag{1} \] where \(J= (0,T)\), \(\Omega\subset \mathbb{R}^N\) is a bounded domain with smooth boundary, \(\eta\) is the unit outward normal vector, \(u_{i,0}(x)\) are nonnegative functions satisfying appropriate compatibility conditions. The main goal of this note is to derive the blow-up rate estimates. The authors give the criteria for the solution to have a finite time blow-up. Moreover, the blow-up set is discussed as well.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
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[1] Fila, M., Boundedness of global solutions for the heat equation with nonlinear boundary condition, Comment. math. univ. carolinane, 30, 479-484, (1989) · Zbl 0702.35141
[2] Fila, M.; Quittner, P., The blowup rate for the heat equation with nonlinear boundary condition, Math. methods appl. sci., 14, 197-205, (1991) · Zbl 0735.35014
[3] Hu, B., Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition, Differential and integral equations, 9, 891-901, (1996) · Zbl 0852.35072
[4] Hu, B.; Yin, H.M., The profile near blowup time for solution of the heat equation with a nonlinear boundary condition, Transaction amer. soc., 346, 117-135, (1994) · Zbl 0823.35020
[5] Deng, K., Blow-up rates for parabolic systems, Z angew math. phys., 47, 132-143, (1996) · Zbl 0854.35054
[6] Deng, K., Global existence and blow-up for a system of heat equations with nonlinear boundary condition, Math. methods appl. sci., 18, 307-315, (1995) · Zbl 0822.35074
[7] Lin, Z.G.; Xie, C.H., The blow-up rate for a system of heat equations with nonlinear boundary condition, Nonlinear analysis, TMA, 34, 767-778, (1998) · Zbl 0941.35008
[8] Fujita, H., On the blowing-up of solutions of the Cauchy problem for ut = δu + u1+α, J. fac. sci. univ. Tokyo, sect. I, 13, 109-124, (1966) · Zbl 0163.34002
[9] Friedman, A.; Mcleod, J.B., Blow-up of positive solutions of semilinear heat equation, Indiana univ. math. J., 34, 425-4471, (1985) · Zbl 0576.35068
[10] Giga, Y.; Kohn, R.V., Asymptotically self-similar blow-up of semilinear heat equation, Comm. pure appl. math., 38, 297-319, (1985) · Zbl 0585.35051
[11] Herrero, M.A.; Velazquez, J.J.L., Some results on blow-up for semilinear parabolic problem, (), 105-125 · Zbl 0828.35056
[12] Velazquez, J.J.L., Blow up for semilinear parabolic equations, research in applied mathematics, (), 131-145 · Zbl 0813.35007
[13] Deng, K.; Fila, K.; Levine, H.A., On critical exponents for a system of the heat equation coupled in the boundary conditions, Acta math. univ. comenianne, 63, 169-192, (1994) · Zbl 0824.35048
[14] Fila, M.; Levine, H.A., On critical exponents for a semilinear parabolic system coupled in an equation and a boundary condition, J. math. anal. appl., 204, 494-521, (1996) · Zbl 0870.35049
[15] Levine, H.A., A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations, J. appl. math. phys., 42, 408-430, (1991) · Zbl 0786.35075
[16] K. Deng and H.A. Levine, The role of critical exponents in blowup theorems: The sequel, J. Math. Anal. Appl. (to appear). · Zbl 0942.35025
[17] Renclawowicz, J., Global existence and blow up of solution for a completely coupled Fujita type system of reaction-diffusion equations, Appl. math., 25, 313-326, (1998), Warsaw · Zbl 1002.35070
[18] Fila, M.; Quittner, P., The blowup rate for a semilinear parabolic system, J. math. anal. appl., 238, 468-476, (1999) · Zbl 0934.35062
[19] Wang, M.X.; Wang, Y.M., Reaction diffusion systems with nonlinear boundary conditions, Science in China, ser. A, 39, 834-840, (1996) · Zbl 0861.35043
[20] Pao, C.V., Nonlinear parabolic and elliptic equations, (1992), Plenum Press New York · Zbl 0780.35044
[21] Friedman, A., Partial differential equations of parabolic type, (1964), Prentice-Hall Englewood Cliffs, NJ · Zbl 0144.34903
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