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

Chin-Chin Wu; Zhengce Zhang

doi:10.3934/dcdsb.2013.18.2203

Article Contents

2013, Volume 18, Issue 8: 2203-2210. Doi: 10.3934/dcdsb.2013.18.2203

This issue Previous Article Fractional diffusion with Neumann boundary conditions: The logistic equation Next Article

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
2.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049

Received: August 31, 2011

Revised: August 31, 2012

Published: September 2013

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35K20, 35K55; Secondary: 35B40.

Citation:

Related Papers

Cited by

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.