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

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.
Citation: Xinfu Chen, Jong-Shenq Guo, Bei Hu. Dead-core rates for the porous medium equation with a strong absorption. Discrete and 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), 1268-1278. 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]

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

[4]

J.-S. Guo and B. Hu, Quenching profile for a quasilinear parabolic equation, Quarterly Appl. Math., 58 (2000), 613-626.

[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), 657-673. doi: 10.1088/0951-7715/23/3/013.

[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), 835-855. doi: 10.1007/s00028-010-0072-0.

[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), 651-667. doi: 10.1007/s00208-004-0601-7.

[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), 37-70. doi: 10.2748/tmj/1206734406.

[9]

Y. Seki, On exact dead-core rates for a semilinear heat equation with strong absorption, Commun. Contemp. Math., 13 (2011), 1-52.

[10]

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.

[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), 17-22.

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]

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

[4]

J.-S. Guo and B. Hu, Quenching profile for a quasilinear parabolic equation, Quarterly Appl. Math., 58 (2000), 613-626.

[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), 657-673. doi: 10.1088/0951-7715/23/3/013.

[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), 835-855. doi: 10.1007/s00028-010-0072-0.

[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), 651-667. doi: 10.1007/s00208-004-0601-7.

[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), 37-70. doi: 10.2748/tmj/1206734406.

[9]

Y. Seki, On exact dead-core rates for a semilinear heat equation with strong absorption, Commun. Contemp. Math., 13 (2011), 1-52.

[10]

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.

[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), 17-22.

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