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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

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