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

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 $2\lambda \geq m+1$.