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 & 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.  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.  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.  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.   Google Scholar

[5]

M. S. Floater, Blow-up at the boundary for degenerate semilinear parabolic equations,, Arch. Rational Mech. Anal., 114 (1991), 57.  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.  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.  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.  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.  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.  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.  doi: 10.1006/jmaa.1997.5452.  Google Scholar

[12]

I. Stakgold, Reaction-diffusion problems in chemical engineering,, in, 1224 (1986), 119.  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.  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.  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.  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.   Google Scholar

[5]

M. S. Floater, Blow-up at the boundary for degenerate semilinear parabolic equations,, Arch. Rational Mech. Anal., 114 (1991), 57.  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.  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.  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.  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.  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.  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.  doi: 10.1006/jmaa.1997.5452.  Google Scholar

[12]

I. Stakgold, Reaction-diffusion problems in chemical engineering,, in, 1224 (1986), 119.  doi: 10.1007/BFb0072689.  Google Scholar

[1]

Xinfu Chen, Jong-Shenq Guo, Bei Hu. Dead-core rates for the porous medium equation with a strong absorption. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1761-1774. doi: 10.3934/dcdsb.2012.17.1761

[2]

Chunlai Mu, Jun Zhou, Yuhuan Li. Fast rate of dead core for fast diffusion equation with strong absorption. Communications on Pure & Applied Analysis, 2010, 9 (2) : 397-411. doi: 10.3934/cpaa.2010.9.397

[3]

Tai Nguyen Phuoc, Laurent Véron. Initial trace of positive solutions of a class of degenerate heat equation with absorption. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2033-2063. doi: 10.3934/dcds.2013.33.2033

[4]

Shin-Yi Lee, Shin-Hwa Wang, Chiou-Ping Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem. Conference Publications, 2005, 2005 (Special) : 587-596. doi: 10.3934/proc.2005.2005.587

[5]

Martin Fraas, David Krejčiřík, Yehuda Pinchover. On some strong ratio limit theorems for heat kernels. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 495-509. doi: 10.3934/dcds.2010.28.495

[6]

Joana Terra, Noemi Wolanski. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581

[7]

Youcef Amirat, Kamel Hamdache. Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3289-3320. doi: 10.3934/dcds.2013.33.3289

[8]

C. Brändle, E. Chasseigne, Raúl Ferreira. Unbounded solutions of the nonlocal heat equation. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1663-1686. doi: 10.3934/cpaa.2011.10.1663

[9]

Arthur Ramiandrisoa. Nonlinear heat equation: the radial case. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 849-870. doi: 10.3934/dcds.1999.5.849

[10]

Delio Mugnolo. Gaussian estimates for a heat equation on a network. Networks & Heterogeneous Media, 2007, 2 (1) : 55-79. doi: 10.3934/nhm.2007.2.55

[11]

Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002

[12]

Seda İǧret Araz, Murat Subașı. On the optimal coefficient control in a heat equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020124

[13]

Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767

[14]

Hiroshi Inoue, Kei Matsuura, Mitsuharu Ôtani. Strong solutions of magneto-micropolar fluid equation. Conference Publications, 2003, 2003 (Special) : 439-448. doi: 10.3934/proc.2003.2003.439

[15]

Chao Zhang, Xia Zhang, Shulin Zhou. Gradient estimates for the strong $p(x)$-Laplace equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4109-4129. doi: 10.3934/dcds.2017175

[16]

Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643

[17]

Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control & Related Fields, 2017, 7 (3) : 347-367. doi: 10.3934/mcrf.2017012

[18]

Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143

[19]

Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221

[20]

Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Universal solutions of the heat equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1105-1132. doi: 10.3934/dcds.2003.9.1105

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]