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October  2013, 18(8): 2203-2210. doi: 10.3934/dcdsb.2013.18.2203

Dead-core rates for the heat equation with a spatially dependent strong absorption

Chin-Chin Wu 1, and  Zhengce Zhang 2,

1. 

Department of Applied Mathematics, National Chung Hsing University, 250, Kuo Kuang Road, Taichung 402, Taiwan
2. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China

Received  September 2011 Revised  September 2012 Published  July 2013

This work is to study the dead-core behavior for a semilinear heat equation with a spatially dependent strong absorption term. We first give a criterion on the initial data such that the dead-core occurs. Then we prove the temporal dead-core rate is non-self-similar. This is based on the standard limiting process with the uniqueness of the self-similar solutions in a certain class.
Keywords: strong absorption., Dead-core, heat equation.
Mathematics Subject Classification: Primary: 35K20, 35K55; Secondary: 35B4.
Citation: Chin-Chin Wu, Zhengce Zhang. Dead-core rates for the heat equation with a spatially dependent strong absorption. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2203-2210. doi: 10.3934/dcdsb.2013.18.2203
References:
[1]

C. Bandle, T. Nanbu and I. Stakgold, Porous medium equation with absorption, SIAM J. Math. Anal., 29 (1998), 1268-1278. doi: 10.1137/S0036141096311423.

[2]

C. Bandle and I. Stakgold, The formation of the dead core in parabolic reaction-diffusion problems, Trans. Amer. Math. Soc., 286 (1984), 275-293. doi: 10.1090/S0002-9947-1984-0756040-1.

[3]

X. Chen, J.-S. Guo and B. Hu, Dead-core rates for the porous medium equation with a strong absorption, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1761-1774. doi: 10.3934/dcdsb.2012.17.1761.

[4]

Q. Chen and L. Wang, On the dead core behavior for a semilinear heat equation, Math. Appl., 10 (1997), 22-25.

[5]

M. S. Floater, Blow-up at the boundary for degenerate semilinear parabolic equations, Arch. Rational Mech. Anal., 114 (1991), 57-77. doi: 10.1007/BF00375685.

[6]

J.-S. Guo, C.-T. Ling and Ph. Souplet, Non-self-similar dead-core rate for the fast diffusion equation with strong absorption, Nonlinearity, 23 (2010), 657-673. doi: 10.1088/0951-7715/23/3/013.

[7]

J.-S. Guo, H. Matano and C.-C. Wu, An application of braid group theory to the finite time dead-core rate, J. Evol. Equ., 10 (2010), 835-855. doi: 10.1007/s00028-010-0072-0.

[8]

J.-S. Guo and Ph. Souplet, Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up, Math. Ann., 331 (2005), 651-667. doi: 10.1007/s00208-004-0601-7.

[9]

J.-S. Guo and C.-C. Wu, Finite time dead-core rate for the heat equation with a strong absorption, Tohoku Math. J. (2), 60 (2008), 37-70. doi: 10.2748/tmj/1206734406.

[10]

H. Ockendon, Channel flow with temperature-dependent viscosity and internal viscous dissipation, J. Fluid Mech., 93 (1979), 737-746. doi: 10.1017/S0022112079002007.

[11]

Ph. Souplet and F. B. Weissler, Self-similar subsolutions and blowup for nonlinear parabolic equations, J. Math. Anal. Appl., 212 (1997), 60-74. doi: 10.1006/jmaa.1997.5452.

[12]

I. Stakgold, Reaction-diffusion problems in chemical engineering, in "Nonlinear Diffusion Problems" (Montecatini Terme, 1985), Lecture Notes in Math., 1224, Springer, Berlin, (1986), 119-152. doi: 10.1007/BFb0072689.

show all references

References:
[1]

C. Bandle, T. Nanbu and I. Stakgold, Porous medium equation with absorption, SIAM J. Math. Anal., 29 (1998), 1268-1278. doi: 10.1137/S0036141096311423.

[2]

C. Bandle and I. Stakgold, The formation of the dead core in parabolic reaction-diffusion problems, Trans. Amer. Math. Soc., 286 (1984), 275-293. doi: 10.1090/S0002-9947-1984-0756040-1.

[3]

X. Chen, J.-S. Guo and B. Hu, Dead-core rates for the porous medium equation with a strong absorption, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1761-1774. doi: 10.3934/dcdsb.2012.17.1761.

[4]

Q. Chen and L. Wang, On the dead core behavior for a semilinear heat equation, Math. Appl., 10 (1997), 22-25.

[5]

M. S. Floater, Blow-up at the boundary for degenerate semilinear parabolic equations, Arch. Rational Mech. Anal., 114 (1991), 57-77. doi: 10.1007/BF00375685.

[6]

J.-S. Guo, C.-T. Ling and Ph. Souplet, Non-self-similar dead-core rate for the fast diffusion equation with strong absorption, Nonlinearity, 23 (2010), 657-673. doi: 10.1088/0951-7715/23/3/013.

[7]

J.-S. Guo, H. Matano and C.-C. Wu, An application of braid group theory to the finite time dead-core rate, J. Evol. Equ., 10 (2010), 835-855. doi: 10.1007/s00028-010-0072-0.

[8]

J.-S. Guo and Ph. Souplet, Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up, Math. Ann., 331 (2005), 651-667. doi: 10.1007/s00208-004-0601-7.

[9]

J.-S. Guo and C.-C. Wu, Finite time dead-core rate for the heat equation with a strong absorption, Tohoku Math. J. (2), 60 (2008), 37-70. doi: 10.2748/tmj/1206734406.

[10]

H. Ockendon, Channel flow with temperature-dependent viscosity and internal viscous dissipation, J. Fluid Mech., 93 (1979), 737-746. doi: 10.1017/S0022112079002007.

[11]

Ph. Souplet and F. B. Weissler, Self-similar subsolutions and blowup for nonlinear parabolic equations, J. Math. Anal. Appl., 212 (1997), 60-74. doi: 10.1006/jmaa.1997.5452.

[12]

I. Stakgold, Reaction-diffusion problems in chemical engineering, in "Nonlinear Diffusion Problems" (Montecatini Terme, 1985), Lecture Notes in Math., 1224, Springer, Berlin, (1986), 119-152. doi: 10.1007/BFb0072689.

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