Differential and Integral Equations

On some Dirichlet and Cauchy problems for a singular diffusion equation

Kin Ming Hui

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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

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

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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