Differential and Integral Equations

On some Dirichlet and Cauchy problems for a singular diffusion equation

Kin Ming Hui

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We will prove the existence and uniqueness of solutions of the Dirichlet problem $u_t=\Delta$ log $u$, $u>0$, in $\Omega\times (0,\infty)$, $u=g$ on $\partial \Omega\times (0,\infty)$, $u(x,0)=u_0(x)\ge 0$ on $\Omega$ where $\Omega\subset R^n$ is a smooth bounded domain. We will also prove the local and global existence and uniqueness of maximal solutions of the above equation in $R^n\times (0,\infty )$ for $n\ge 3$ under very general condition on $u_0$ and we will prove finite time extinction of solution for $u_0\in L_{loc}^{\infty}(R^n)$ satisfying $0\le u_0(x)\le C/|x|^2$ for all $|x|\ge R_0$ for some constants $C>0$, $R_0>0$.

Article information

Source
Differential Integral Equations, Volume 15, Number 7 (2002), 769-804.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060798

Mathematical Reviews number (MathSciNet)
MR1895566

Zentralblatt MATH identifier
1020.35038

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions 35D05 35K65: Degenerate parabolic equations

Citation

Hui, Kin Ming. On some Dirichlet and Cauchy problems for a singular diffusion equation. Differential Integral Equations 15 (2002), no. 7, 769--804. https://projecteuclid.org/euclid.die/1356060798


Export citation