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.


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
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.