# American Institute of Mathematical Sciences

June  2012, 32(6): 2165-2185. doi: 10.3934/dcds.2012.32.2165

## Collasping behaviour of a singular diffusion equation

 1 Institute of Mathematics, Academia sinica, Taiwan

Received  April 2011 Revised  August 2011 Published  February 2012

Let $0\le u_0(x)\in L^1(\mathbb{R}^2)\cap L^{\infty}(\mathbb{R}^2)$ be such that $u_0(x) =u_0(|x|)$ for all $|x|\ge r_1$ and is monotone decreasing for all $|x|\ge r_1$ for some constant $r_1>0$ and $\mbox{ess}\inf_{2{B}_{r_1}(0)}u_0\ge\mbox{ess} \sup_{R^2\setminus B_{r_2}(0)}u_0$ for some constant $r_2>r_1$. Then under some mild decay conditions at infinity on the initial value $u_0$ we will extend the result of P. Daskalopoulos, M.A. del Pino and N. Sesum [4], [6], and prove the collapsing behaviour of the maximal solution of the equation $u_t=\Delta\log u$ in $\mathbb{R}^2\times (0,T)$, $u(x,0)=u_0(x)$ in $\mathbb{R}^2$, near its extinction time $T=\int_{R^2}u_0dx/4\pi$ by a simplified method without using the Hamilton-Yau Harnack inequality.
Citation: Kin Ming Hui. Collasping behaviour of a singular diffusion equation. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2165-2185. doi: 10.3934/dcds.2012.32.2165
##### References:
 [1] D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Transactions A. M. S., 280 (1983), 351-366. [2] P. Daskalopoulos and R. Hamilton, Geometric estimates for the logarithmic fast diffusion equation, Comm. Anal. Geom., 12 (2004), 143-164. [3] P. Daskalopoulos and M. A. del Pino, On a singular diffusion equation, Comm. Anal. Geom., 3 (1995), 523-542. [4] P. Daskalopoulos and M. A. del Pino, Type II collapsing of maximal solutions to the Ricci flow in $\mathbbR^2$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 851-874. [5] P. Daskalopoulos and N. Sesum, Eternal solutions to the Ricci flow on $\mathbbR^2$, Int. Math. Res. Not., 2006, Art. ID 83610, 20 pp. [6] P. Daskalopoulos and N. Sesum, Type II extinction profile of maximal solutions to the Ricci flow equation, J. Geom. Anal., 20 (2010), 565-591. doi: 10.1007/s12220-010-9128-1. [7] J. R. Esteban, A. Rodríguez and J. L. Vazquez, The fast diffusion equation with logarithmic nonlinearity and the evolution of conformal metrics in the plane, Advances in Differential Equations, 1 (1996), 21-50. [8] J. R. Esteban, A. Rodriguez and J. L. Vazquez, The maximal solution of the logarithmic fast diffusion equation in two space dimensions, Advances in Differential Equations, 2 (1997), 867-894. [9] P. G. de Gennes, Wetting: Statics and dynamics, Rev. Modern Phys., 57 (1985), 827-863. doi: 10.1103/RevModPhys.57.827. [10] R. Hamilton and S. T. Yau, The Harnack estimate for the Ricci flow on a surface-revisited, Asian J. Math., 1 (1997), 418-421. [11] S. Y. Hsu, Large time behaviour of solutions of the Ricci flow equation on $R^2$, Pacific J. Math., 197 (2001), 25-41. doi: 10.2140/pjm.2001.197.25. [12] S. Y. Hsu, Asymptotic profile of a singular diffusion equation as $t\to\infty$, Nonlinear Analysis, 48 (2002), 781-790. doi: 10.1016/S0362-546X(00)00214-5. [13] S. Y. Hsu, Asymptotic behaviour of solutions of the equation $u_t=\Delta\log u$ near the extinction time, Advances in Differential Equations, 8 (2003), 161-187. [14] S. Y. Hsu, Behaviour of solutions of a singular diffusion equation near the extinction time, Nonlinear Analysis, 56 (2004), 63-104. doi: 10.1016/j.na.2003.07.018. [15] K. M. Hui, Existence of solutions of the equation $u_t=\Delta\log u$, Nonlinear Analysis, 37 (1999), 875-914. doi: 10.1016/S0362-546X(98)00081-9. [16] K. M. Hui, Singular limit of solutions of the equation $u_t=\Delta (u^m/m)$ as $m\to 0$, Pacific J. Math., 187 (1999), 297-316. doi: 10.2140/pjm.1999.187.297. [17] J. R. King, Self-similar behaviour for the equation of fast nonlinear diffusion, Phil. Trans. Royal Soc. London Series A, 343 (1993), 337-375. doi: 10.1098/rsta.1993.0052. [18] O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva, "Linear and Quasilinear Equations of Parabolic Type," Transl. Math. Mono., Vol. 23, Amer. Math. Soc., Providence, R.I., 1968. [19] J. L. Vazquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl. (9), 71 (1992), 503-526. [20] L. F. Wu, A new result for the porous medium equation derived from the Ricci flow, Bull. Amer. Math. Soc. (N.S.), 28 (1993), 90-94. [21] L. F. Wu, The Ricci flow on complete $R^2$, Comm. Anal. Geom., 1 (1993), 439-472.

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##### References:
 [1] D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Transactions A. M. S., 280 (1983), 351-366. [2] P. Daskalopoulos and R. Hamilton, Geometric estimates for the logarithmic fast diffusion equation, Comm. Anal. Geom., 12 (2004), 143-164. [3] P. Daskalopoulos and M. A. del Pino, On a singular diffusion equation, Comm. Anal. Geom., 3 (1995), 523-542. [4] P. Daskalopoulos and M. A. del Pino, Type II collapsing of maximal solutions to the Ricci flow in $\mathbbR^2$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 851-874. [5] P. Daskalopoulos and N. Sesum, Eternal solutions to the Ricci flow on $\mathbbR^2$, Int. Math. Res. Not., 2006, Art. ID 83610, 20 pp. [6] P. Daskalopoulos and N. Sesum, Type II extinction profile of maximal solutions to the Ricci flow equation, J. Geom. Anal., 20 (2010), 565-591. doi: 10.1007/s12220-010-9128-1. [7] J. R. Esteban, A. Rodríguez and J. L. Vazquez, The fast diffusion equation with logarithmic nonlinearity and the evolution of conformal metrics in the plane, Advances in Differential Equations, 1 (1996), 21-50. [8] J. R. Esteban, A. Rodriguez and J. L. Vazquez, The maximal solution of the logarithmic fast diffusion equation in two space dimensions, Advances in Differential Equations, 2 (1997), 867-894. [9] P. G. de Gennes, Wetting: Statics and dynamics, Rev. Modern Phys., 57 (1985), 827-863. doi: 10.1103/RevModPhys.57.827. [10] R. Hamilton and S. T. Yau, The Harnack estimate for the Ricci flow on a surface-revisited, Asian J. Math., 1 (1997), 418-421. [11] S. Y. Hsu, Large time behaviour of solutions of the Ricci flow equation on $R^2$, Pacific J. Math., 197 (2001), 25-41. doi: 10.2140/pjm.2001.197.25. [12] S. Y. Hsu, Asymptotic profile of a singular diffusion equation as $t\to\infty$, Nonlinear Analysis, 48 (2002), 781-790. doi: 10.1016/S0362-546X(00)00214-5. [13] S. Y. Hsu, Asymptotic behaviour of solutions of the equation $u_t=\Delta\log u$ near the extinction time, Advances in Differential Equations, 8 (2003), 161-187. [14] S. Y. Hsu, Behaviour of solutions of a singular diffusion equation near the extinction time, Nonlinear Analysis, 56 (2004), 63-104. doi: 10.1016/j.na.2003.07.018. [15] K. M. Hui, Existence of solutions of the equation $u_t=\Delta\log u$, Nonlinear Analysis, 37 (1999), 875-914. doi: 10.1016/S0362-546X(98)00081-9. [16] K. M. Hui, Singular limit of solutions of the equation $u_t=\Delta (u^m/m)$ as $m\to 0$, Pacific J. Math., 187 (1999), 297-316. doi: 10.2140/pjm.1999.187.297. [17] J. R. King, Self-similar behaviour for the equation of fast nonlinear diffusion, Phil. Trans. Royal Soc. London Series A, 343 (1993), 337-375. doi: 10.1098/rsta.1993.0052. [18] O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva, "Linear and Quasilinear Equations of Parabolic Type," Transl. Math. Mono., Vol. 23, Amer. Math. Soc., Providence, R.I., 1968. [19] J. L. Vazquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl. (9), 71 (1992), 503-526. [20] L. F. Wu, A new result for the porous medium equation derived from the Ricci flow, Bull. Amer. Math. Soc. (N.S.), 28 (1993), 90-94. [21] L. F. Wu, The Ricci flow on complete $R^2$, Comm. Anal. Geom., 1 (1993), 439-472.
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