The degenerate drift-diffusion system with the Sobolev critical exponent
T. Ogawa
Discrete & Continuous Dynamical Systems - S 2011, 4(4): 875-886 doi: 10.3934/dcdss.2011.4.875
We consider the drift-diffusion system of degenerated type. For $n\ge 3$,

$\partial_t \rho -\Delta \rho^\alpha + \kappa\nabla\cdot (\rho \nabla \psi ) =0, t>0, x \in R^n,$

$-\Delta \psi = \rho, t>0, x \in R^n,$

$\rho(0,x) = \rho_0(x)\ge 0, x \in R^n,$

where $\alpha>1$ and $\kappa=1$. There exists a critical exponent that classifies the global behavior of the weak solution. In particular, we consider the critical case $\alpha_*=\frac{2 n}{n+2}=(2^*)'$, where the Talenti function $U(x)$ solving $-2^*\Delta U^{\frac{n-2}{n+2}}=U$ in $R^n$ classifies the global existence of the weak solution and finite blow-up of the solution.

keywords: Degenerate drift-diffusion global weak solution. blow-up Sobolev critical
T. Ogawa Tohru Ozawa
Communications on Pure & Applied Analysis 2015, 14(4): i-iii doi: 10.3934/cpaa.2015.14.4i
This special issue of Discrete and Continuous Dynamical Systems is dedicated to Professor Gustavo Ponce on the occasion of his sixtieth birthday.

Gustavo Ponce was born on April 20, 1952, in Venezuela. He received his B.A. in 1976 from Universidad Central de Venezuela and his Ph. D. in 1982 with the dissertation entitled ``Long time stability of solutions of nonlinear evolution equations" under the supervision of Sergiu Klainerman and Louis Nirenberg at Courant Institute, New York University. After professional experiences at University of California at Berkely (1982-1984), Universidad Central de Venezuela (1984-1986), University of Chicago (1986-1989), and Pennsylvania State University (1989-1991), he was appointed to a full professorship at Department of Mathematics, University of California at Santa Barbara in 1991, where he has remained up until now.
Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature
Y. Goto K. Ishii T. Ogawa
Communications on Pure & Applied Analysis 2005, 4(2): 311-339 doi: 10.3934/cpaa.2005.4.311
A new proof of the convergence of the Bence-Merriman-Osher algorithm for the motion of mean curvature is given. The idea is making use of the approximate distance function to the interface and analogous argument in the singular limiting problem for the Allen-Cahn equation via an auxiliary function given by the primitive function of the heat kernel.
keywords: signed distance function Motionby mean curvature viscosity solutions. Bence-Merrina-Osher algorithm
The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system
Masaki Kurokiba Toshitaka Nagai T. Ogawa
Communications on Pure & Applied Analysis 2006, 5(1): 97-106 doi: 10.3934/cpaa.2006.5.97
In this paper, we discuss the global existence and uniform boundedness of the radial solutions to the drift-diffusion system in two space dimension, which is derived from the simulation of semiconductor device design and self-interacting particles. It is shown that the time global existence and the uniform boundedness of the solution to the problem below the sharp threshold condition.
keywords: radial solution Drift-diffusion system threshold condition. uniform boundedness

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