# American Institute of Mathematical Sciences

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.

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##### 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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