## Differential and Integral Equations

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

### On some Dirichlet and Cauchy problems for a singular diffusion equation

#### 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