Differential and Integral Equations

Uniqueness and nonuniqueness for the porous medium equation with non linear boundary conditions

Carmen Cortazar, Manuel Elgueta, and Julio D. Rossi

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Abstract

We study the uniqueness problem for nonnegative solutions of $u_t=\Delta u^m$ in $\Omega \times [0,T)$, $-\frac{\partial u^m}{\partial \hat{n}}(x,t)=u^{\lambda}(x,t)$ on $\partial \Omega \times (0,T)$ and $u(x,0) \equiv 0$ on $\Omega$ where $m > 1$, $\lambda \ge 1$, and $\Omega$ is a bounded domain with smooth boundary in $\mathbf {R}^N$. We prove that the solution $u \equiv 0$ is unique if and only if $2\lambda \geq m+1$.

Article information

Source
Differential Integral Equations, Volume 16, Number 10 (2003), 1215-1222.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060545

Mathematical Reviews number (MathSciNet)
MR2014807

Zentralblatt MATH identifier
1073.35083

Subjects
Primary: 35K57: Reaction-diffusion equations
Secondary: 35K60: Nonlinear initial value problems for linear parabolic equations 76S05: Flows in porous media; filtration; seepage

Citation

Cortazar, Carmen; Elgueta, Manuel; Rossi, Julio D. Uniqueness and nonuniqueness for the porous medium equation with non linear boundary conditions. Differential Integral Equations 16 (2003), no. 10, 1215--1222. https://projecteuclid.org/euclid.die/1356060545


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