# American Institute of Mathematical Sciences

March  2007, 19(1): 1-35. doi: 10.3934/dcds.2007.19.1

## Finite-time blow-down in the evolution of point masses by planar logarithmic diffusion

Received  August 2006 Revised  February 2007 Published  June 2007

We are interested in a remarkable property of certain nonlinear diffusion equations, which we call blow-down or delayed regularization. The following happens: a solution of one of these equations is shown to exist in some generalized sense, and it is also shown to be non-smooth for some time $0 < t < t_1$, after which it becomes smooth and still nontrivial. We use the logarithmic diffusion equation to examine an example of occurrence of this phenomenon starting from data that contain Dirac deltas, which persist for a finite time. The interpretation of the results in terms of diffusion is also unusual: if the process starts with one or several point masses surrounded by a continuous distribution, then the masses decay into the medium over a finite period of time. The study of the phenomenon implies consideration of a new concept of measure solution which seems natural for these diffusion processes.
Citation: Juan Luis Vázquez. Finite-time blow-down in the evolution of point masses by planar logarithmic diffusion. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 1-35. doi: 10.3934/dcds.2007.19.1
 [1] Vakhtang Putkaradze, Stuart Rogers. Numerical simulations of a rolling ball robot actuated by internal point masses. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 143-207. doi: 10.3934/naco.2020021 [2] Nejib Mahmoudi. Single-point blow-up for a multi-component reaction-diffusion system. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 209-230. doi: 10.3934/dcds.2018010 [3] Michael Herty, Axel Klar, Sébastien Motsch, Ferdinand Olawsky. A smooth model for fiber lay-down processes and its diffusion approximations. Kinetic and Related Models, 2009, 2 (3) : 489-502. doi: 10.3934/krm.2009.2.489 [4] Ugo Bessi. Another point of view on Kusuoka's measure. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3241-3271. doi: 10.3934/dcds.2020404 [5] Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881 [6] Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042 [7] Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051 [8] Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28 (1) : 263-289. doi: 10.3934/era.2020016 [9] Wenjun Liu, Jiangyong Yu, Gang Li. Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4337-4366. doi: 10.3934/dcdss.2021121 [10] Nadjat Doudi, Salah Boulaaras, Nadia Mezouar, Rashid Jan. Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022106 [11] Jaeyoung Byeon, Sungwon Cho, Junsang Park. On the location of a peak point of a least energy solution for Hénon equation. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1055-1081. doi: 10.3934/dcds.2011.30.1055 [12] Simona Fornaro, Stefano Lisini, Giuseppe Savaré, Giuseppe Toscani. Measure valued solutions of sub-linear diffusion equations with a drift term. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1675-1707. doi: 10.3934/dcds.2012.32.1675 [13] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure and Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 [14] Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2607-2623. doi: 10.3934/dcdss.2021032 [15] Fang Li, Kimie Nakashima, Wei-Ming Ni. Stability from the point of view of diffusion, relaxation and spatial inhomogeneity. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 259-274. doi: 10.3934/dcds.2008.20.259 [16] Monica Marras, Stella Vernier Piro. Blow-up phenomena in reaction-diffusion systems. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 4001-4014. doi: 10.3934/dcds.2012.32.4001 [17] Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182 [18] Jiao Chen, Weike Wang. The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Communications on Pure and Applied Analysis, 2014, 13 (1) : 307-330. doi: 10.3934/cpaa.2014.13.307 [19] Mohammad Eslamian, Ahmad Kamandi. A novel algorithm for approximating common solution of a system of monotone inclusion problems and common fixed point problem. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021210 [20] Abd-semii Oluwatosin-Enitan Owolabi, Timilehin Opeyemi Alakoya, Adeolu Taiwo, Oluwatosin Temitope Mewomo. A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 255-278. doi: 10.3934/naco.2021004

2021 Impact Factor: 1.588