On the Mesa problem (Q1094585)

Let \(u_ m(x,t)\) denote the solution of \(u_ t=\Delta u^ m\) in \({\mathbb{R}}^ d\times (0,\infty)\) with \(u(x,0)=f(x)\), a prescribed function which is integrable, bounded, continuously differentiable, and star shaped in \({\mathbb{R}}^ d\). The authors study the asymptotic behavior of \(u_ m\) as \(m\to \infty\), proving weak star convergence of (rescaled) solutions in \(L^{\infty}_{loc}({\mathbb{R}}^ d\times (0,\infty))\) to a function V whose profile is a time-dependent mesa with plateau of height \(t^{-1/m}\) and remaining surface coinciding with the initial surface f.