Differential and Integral Equations

Uniqueness of solutions of a singular diffusion equation

Shu-Yu Hsu

Abstract

In this paper we will show that if $0\le u_0\in L^1(R^2)\cap L^p(R^2)$ for some constant $p>1$ and $f\in C([0,\infty))$ with $f\ge 2$ on $[0,\infty)$, then the solution of the equation $u_t=\Delta$ log $u$, $u>0$, in $R^2\times (0,T)$, $u(x,0)=u_0(x)$ on $R^2$, which satisfies the conditions $\int_{R^2}u(x,t)dx=\int_{R^2}u_0(x)dx-2\pi\int_0^tf(s)ds$ for all $0\le t\le T$ and log $u/$log $|x|\to -f$ uniformly as $|x|\to\infty$ on any compact subset of $(0,T)$ is unique where $T$ is given by $\int_{R^2}u_0(x)dx=2\pi\int_0^Tf(s)ds$. For $0\le u_0(x)\le C\min (1,(|x|\text{log }|x|)^{-2})$ or $u_0\in L^1(R^2)\cap L^p(R^2)$ for some $p>1$ with compact support and $f\in C([0,\infty))$, $f\ge 2$, we prove existence of solution satisfying the decay condition log $u/$log $|x|\to -f$ uniformly as $|x|\to\infty$ on any compact subset of $(0,T)$. We will also show that any solution obtained as the limit of solutions of certain Dirichlet problems in bounded domains will satisfy the same decay condition and is hence unique.

Article information

Source
Differential Integral Equations, Volume 16, Number 2 (2003), 181-200.

Dates
First available in Project Euclid: 21 December 2012