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 & Continuous Dynamical Systems - A, 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. Google Scholar

[2]

P. Daskalopoulos and R. Hamilton, Geometric estimates for the logarithmic fast diffusion equation,, Comm. Anal. Geom., 12 (2004), 143. Google Scholar

[3]

P. Daskalopoulos and M. A. del Pino, On a singular diffusion equation,, Comm. Anal. Geom., 3 (1995), 523. Google Scholar

[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. Google Scholar

[5]

P. Daskalopoulos and N. Sesum, Eternal solutions to the Ricci flow on $\mathbbR^2$,, Int. Math. Res. Not., 2006 (8361). Google Scholar

[6]

P. Daskalopoulos and N. Sesum, Type II extinction profile of maximal solutions to the Ricci flow equation,, J. Geom. Anal., 20 (2010), 565. doi: 10.1007/s12220-010-9128-1. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

P. G. de Gennes, Wetting: Statics and dynamics,, Rev. Modern Phys., 57 (1985), 827. doi: 10.1103/RevModPhys.57.827. Google Scholar

[10]

R. Hamilton and S. T. Yau, The Harnack estimate for the Ricci flow on a surface-revisited,, Asian J. Math., 1 (1997), 418. Google Scholar

[11]

S. Y. Hsu, Large time behaviour of solutions of the Ricci flow equation on $R^2$,, Pacific J. Math., 197 (2001), 25. doi: 10.2140/pjm.2001.197.25. Google Scholar

[12]

S. Y. Hsu, Asymptotic profile of a singular diffusion equation as $t\to\infty$,, Nonlinear Analysis, 48 (2002), 781. doi: 10.1016/S0362-546X(00)00214-5. Google Scholar

[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. Google Scholar

[14]

S. Y. Hsu, Behaviour of solutions of a singular diffusion equation near the extinction time,, Nonlinear Analysis, 56 (2004), 63. doi: 10.1016/j.na.2003.07.018. Google Scholar

[15]

K. M. Hui, Existence of solutions of the equation $u_t=\Delta\log u$,, Nonlinear Analysis, 37 (1999), 875. doi: 10.1016/S0362-546X(98)00081-9. Google Scholar

[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. doi: 10.2140/pjm.1999.187.297. Google Scholar

[17]

J. R. King, Self-similar behaviour for the equation of fast nonlinear diffusion,, Phil. Trans. Royal Soc. London Series A, 343 (1993), 337. doi: 10.1098/rsta.1993.0052. Google Scholar

[18]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva, "Linear and Quasilinear Equations of Parabolic Type,", Transl. Math. Mono., (1968). Google Scholar

[19]

J. L. Vazquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type,, J. Math. Pures Appl. (9), 71 (1992), 503. Google Scholar

[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. Google Scholar

[21]

L. F. Wu, The Ricci flow on complete $R^2$,, Comm. Anal. Geom., 1 (1993), 439. Google Scholar

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. Google Scholar

[2]

P. Daskalopoulos and R. Hamilton, Geometric estimates for the logarithmic fast diffusion equation,, Comm. Anal. Geom., 12 (2004), 143. Google Scholar

[3]

P. Daskalopoulos and M. A. del Pino, On a singular diffusion equation,, Comm. Anal. Geom., 3 (1995), 523. Google Scholar

[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. Google Scholar

[5]

P. Daskalopoulos and N. Sesum, Eternal solutions to the Ricci flow on $\mathbbR^2$,, Int. Math. Res. Not., 2006 (8361). Google Scholar

[6]

P. Daskalopoulos and N. Sesum, Type II extinction profile of maximal solutions to the Ricci flow equation,, J. Geom. Anal., 20 (2010), 565. doi: 10.1007/s12220-010-9128-1. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

P. G. de Gennes, Wetting: Statics and dynamics,, Rev. Modern Phys., 57 (1985), 827. doi: 10.1103/RevModPhys.57.827. Google Scholar

[10]

R. Hamilton and S. T. Yau, The Harnack estimate for the Ricci flow on a surface-revisited,, Asian J. Math., 1 (1997), 418. Google Scholar

[11]

S. Y. Hsu, Large time behaviour of solutions of the Ricci flow equation on $R^2$,, Pacific J. Math., 197 (2001), 25. doi: 10.2140/pjm.2001.197.25. Google Scholar

[12]

S. Y. Hsu, Asymptotic profile of a singular diffusion equation as $t\to\infty$,, Nonlinear Analysis, 48 (2002), 781. doi: 10.1016/S0362-546X(00)00214-5. Google Scholar

[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. Google Scholar

[14]

S. Y. Hsu, Behaviour of solutions of a singular diffusion equation near the extinction time,, Nonlinear Analysis, 56 (2004), 63. doi: 10.1016/j.na.2003.07.018. Google Scholar

[15]

K. M. Hui, Existence of solutions of the equation $u_t=\Delta\log u$,, Nonlinear Analysis, 37 (1999), 875. doi: 10.1016/S0362-546X(98)00081-9. Google Scholar

[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. doi: 10.2140/pjm.1999.187.297. Google Scholar

[17]

J. R. King, Self-similar behaviour for the equation of fast nonlinear diffusion,, Phil. Trans. Royal Soc. London Series A, 343 (1993), 337. doi: 10.1098/rsta.1993.0052. Google Scholar

[18]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva, "Linear and Quasilinear Equations of Parabolic Type,", Transl. Math. Mono., (1968). Google Scholar

[19]

J. L. Vazquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type,, J. Math. Pures Appl. (9), 71 (1992), 503. Google Scholar

[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. Google Scholar

[21]

L. F. Wu, The Ricci flow on complete $R^2$,, Comm. Anal. Geom., 1 (1993), 439. Google Scholar

[1]

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

[2]

Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991

[3]

Iryna Pankratova, Andrey Piatnitski. On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 935-970. doi: 10.3934/dcdsb.2009.11.935

[4]

Liviu I. Ignat, Ademir F. Pazoto. Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3575-3589. doi: 10.3934/dcds.2014.34.3575

[5]

Abdelaziz Rhandi, Roland Schnaubelt. Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 663-683. doi: 10.3934/dcds.1999.5.663

[6]

Kin Ming Hui, Sunghoon Kim. Existence of Neumann and singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4859-4887. doi: 10.3934/dcds.2015.35.4859

[7]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013

[8]

Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613

[9]

Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335

[10]

Yuanzhen Shao. Continuous maximal regularity on singular manifolds and its applications. Evolution Equations & Control Theory, 2016, 5 (2) : 303-335. doi: 10.3934/eect.2016006

[11]

Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019

[12]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[13]

Tomás Caraballo, Antonio M. Márquez-Durán, Rivero Felipe. Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1817-1833. doi: 10.3934/dcdsb.2017108

[14]

Pavel Krejčí, Jürgen Sprekels. Long time behaviour of a singular phase transition model. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1119-1135. doi: 10.3934/dcds.2006.15.1119

[15]

S. Dumont, Noureddine Igbida. On the collapsing sandpile problem. Communications on Pure & Applied Analysis, 2011, 10 (2) : 625-638. doi: 10.3934/cpaa.2011.10.625

[16]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[17]

Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223

[18]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

[19]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

[20]

Toru Sasaki, Takashi Suzuki. Asymptotic behaviour of the solutions to a virus dynamics model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 525-541. doi: 10.3934/dcdsb.2017206

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]