American Institute of Mathematical Sciences

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

 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.
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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