Differential and Integral Equations

Uniqueness of solutions of a singular diffusion equation

Shu-Yu Hsu

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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

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060683

Mathematical Reviews number (MathSciNet)
MR1947091

Zentralblatt MATH identifier
1036.35103

Subjects
Primary: 35K57: Reaction-diffusion equations
Secondary: 35B40: Asymptotic behavior of solutions 35K65: Degenerate parabolic equations

Citation

Hsu, Shu-Yu. Uniqueness of solutions of a singular diffusion equation. Differential Integral Equations 16 (2003), no. 2, 181--200. https://projecteuclid.org/euclid.die/1356060683


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