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Blow-up analysis for a nonlinear diffusion equation with nonlinear boundary conditions. (English) Zbl 1056.35087

Summary: The blow-up rate for a nonlinear diffusion equation \( u_t= (u^m)_{xx}+ u^p\), \(0< x< 1\) with a nonlinear boundary condition \((u^m)_x= u^q\) at \(x=0\), \((u^m)_x= 0\) at \(x=1\) is established together with the necessary and sufficient blow-up conditions.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K65 Degenerate parabolic equations
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[1] Levine, H.A., The role of critical exponents in blow-up theorems, SIAM rev., 32, 268-288, (1990)
[2] Lieberman, G.M., Second parabolic differential equations, (1996), World Sci · Zbl 0884.35001
[3] Dibenedetto, E., Degenerate parabolic equations, (1993), Springer New York · Zbl 0794.35090
[4] Peletier, L.A., The porous medium equation, (1981), Pitman London · Zbl 0497.76083
[5] Budd, C.J.; Collins, G.J.; Galaktionov, V.A., An asymptotic and numerical description of self-similar blow-up in quasilinear parabolic equations, J. comp. appl. math., 97, 51-80, (1998) · Zbl 0932.65099
[6] Deng, K.; Levine, H., The role of critical exponents in blow-up theorems: the sequel, J. math. anal. appl., 243, 85-126, (2000) · Zbl 0942.35025
[7] Filo, J., Diffusivity versus absorption through the boundary, J. diff. equ., 99, 281-305, (1992) · Zbl 0761.35048
[8] Friedman, A.; Mcleod, J.B., Blow-up of solutions of nonlinear degenerate parabolic equations, Arch. rat. mech. anal., 96, 55-80, (1987) · Zbl 0619.35060
[9] Friedman, A., Partial differential equations of parabolic type, (1964), Prentice-Hall Englewood Cliffs, NJ · Zbl 0144.34903
[10] Giga, Y.; Kohn, R., Asymptotically self-similar blow-up of semilinear heat equations, Comm. pure appl. math., 38, 297-319, (1985) · Zbl 0585.35051
[11] Acsta, G.; Rossi, J.D., Blow-up vs. global existence for quasilinear parabolic systems with a nonlinear boundary condition, Z. angew. math. phys., 48, 5, 711-724, (1997) · Zbl 0893.35046
[12] Wang, M., Long time behaviors of solutions for some quasilinear parabolic equations with nonlinear boundary conditions, Acta math. sinica, 39, 1, 118-124, (1996) · Zbl 0880.35059
[13] Lin, Z.; Wang, M., 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
[14] Zheng, S.N., Global boundedness of solutions to a reaction-diffusion system, Math. meth. appl. sci., 22, 43-54, (1999) · Zbl 0919.35060
[15] Chipot, M.; Fila, M.; Quittner, P., Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta. math. univ. comen., LX, 1, 35-103, (1991) · Zbl 0743.35038
[16] Hu, B.; Yin, H.M., The profile near blow-up time for the solution of the heat equation with a nonlinear boundary condition, Trans. amer. math. soc., 346, 117-135, (1995)
[17] Rossi, J.D., The blow-up rate for a semilinear parabolic equation with a nonlinear boundary condition, Acta. math. univ. comen., LXVII, 2, 343-350, (1998) · Zbl 0924.35017
[18] Dibenedetto, E., Continuity of weak solutions to a general porous medium equation, Indiana univ. math. J., 32, 83-118, (1983) · Zbl 0526.35042
[19] Gidas, B.; Spruck, J., A priori bounds for positive solutions of nonlinear elliptic equations, Comm. partial differential equations, 6, 883-901, (1981) · Zbl 0462.35041
[20] Lieberman, G.M., Holder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. di math. pura ed appl., 148, 77-99, (1987) · Zbl 0658.35050
[21] Pao, C.V., Nonlinear parabolic and elliptic equations, (1992), Plenum New York · Zbl 0780.35044
[22] Smoller, J., Shock waves and reaction diffusion equations, (1983), Springer-Verlag New York · Zbl 0508.35002
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