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DCDS-S

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.

CPAA

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.

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.

keywords:

CPAA

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.

CPAA

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.

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