American Institute of Mathematical Sciences

Advanced
September  2012, 17(6): 1761-1774. doi: 10.3934/dcdsb.2012.17.1761

Dead-core rates for the porous medium equation with a strong absorption

Xinfu Chen 1, , Jong-Shenq Guo 2, and  Bei Hu 3,

1. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260
2. 

Department of Mathematics, Tamkang University, Tamsui, Taipei County 25137
3. 

Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556

Received  March 2011 Revised  August 2011 Published  May 2012

We study the dead-core rate for the solution of the porous medium equation with a strong absorption. It is known that solutions with certain class of initial data develop a dead-core in finite time. We prove that, unlike the cases of semilinear heat equation and fast diffusion equation, there are solutions with the self-similar dead-core rate. This result is based on the construction of a Lyapunov functional, some a priori estimates, and a delicate analysis of the associated re-scaled ordinary differential equation.
Keywords: porous medium equation, strong absorption., Dead-core rate.
Mathematics Subject Classification: Primary: 35K20, 35K55; Secondary: 35B4.
Citation: 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
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]

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

[4]

J.-S. Guo and B. Hu, Quenching profile for a quasilinear parabolic equation,, Quarterly Appl. Math., 58 (2000), 613. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

Y. Seki, On exact dead-core rates for a semilinear heat equation with strong absorption,, Commun. Contemp. Math., 13 (2011), 1. Google Scholar

[10]

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

[11]

T. I. Zelenjak, Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable,, Differential Equations, 4 (1968), 17. 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]

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

[4]

J.-S. Guo and B. Hu, Quenching profile for a quasilinear parabolic equation,, Quarterly Appl. Math., 58 (2000), 613. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

Y. Seki, On exact dead-core rates for a semilinear heat equation with strong absorption,, Commun. Contemp. Math., 13 (2011), 1. Google Scholar

[10]

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

[11]

T. I. Zelenjak, Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable,, Differential Equations, 4 (1968), 17. Google Scholar

[1]

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
[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]

Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107
[4]

Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks & Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337
[5]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393
[6]

Ansgar Jüngel, Ingrid Violet. Mixed entropy estimates for the porous-medium equation with convection. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 783-796. doi: 10.3934/dcdsb.2009.12.783
[7]

Jing Li, Yifu Wang, Jingxue Yin. Non-sharp travelling waves for a dual porous medium equation. Communications on Pure & Applied Analysis, 2016, 15 (2) : 623-636. doi: 10.3934/cpaa.2016.15.623
[8]

Sofía Nieto, Guillermo Reyes. Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1123-1139. doi: 10.3934/cpaa.2013.12.1123
[9]

Gabriele Grillo, Matteo Muratori, Fabio Punzo. On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5927-5962. doi: 10.3934/dcds.2015.35.5927
[10]

Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure & Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183
[11]

Zhilei Liang. On the critical exponents for porous medium equation with a localized reaction in high dimensions. Communications on Pure & Applied Analysis, 2012, 11 (2) : 649-658. doi: 10.3934/cpaa.2012.11.649
[12]

Edoardo Mainini. On the signed porous medium flow. Networks & Heterogeneous Media, 2012, 7 (3) : 525-541. doi: 10.3934/nhm.2012.7.525
[13]

Liangwei Wang, Jingxue Yin, Chunhua Jin. $\omega$-limit sets for porous medium equation with initial data in some weighted spaces. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 223-236. doi: 10.3934/dcdsb.2013.18.223
[14]

Ansgar Jüngel, Stefan Schuchnigg. A discrete Bakry-Emery method and its application to the porous-medium equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5541-5560. doi: 10.3934/dcds.2017241
[15]

Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665
[16]

Guofu Lu. Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1567-1586. doi: 10.3934/dcdsb.2016011
[17]

Wen Wang, Dapeng Xie, Hui Zhou. Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1957-1974. doi: 10.3934/cpaa.2018093
[18]

Jean-Daniel Djida, Juan J. Nieto, Iván Area. Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4031-4053. doi: 10.3934/dcdsb.2019049
[19]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355
[20]

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

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences