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Collasping behaviour of a singular diffusion equation
1. | Institute of Mathematics, Academia sinica, Taiwan |
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 $\mathbb{R}^2$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 851-874. |
[5] |
P. Daskalopoulos and N. Sesum, Eternal solutions to the Ricci flow on $\mathbb{R}^2$, 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. |
show all references
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 $\mathbb{R}^2$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 851-874. |
[5] |
P. Daskalopoulos and N. Sesum, Eternal solutions to the Ricci flow on $\mathbb{R}^2$, 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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