Uniqueness of solutions of a singular diffusion equation
Shu-Yu Hsu · Differential and Integral Equations · 2003
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.