Fast/slow diffusion and collapsing sandpiles,
The authors study the behaviour of solutions \(u_p\) of the initial value problem \[ u_{p,t}- \Delta_pu_p= 0\quad\text{in }\mathbb{R}^n\times (0,\infty),\quad u_p=g\quad\text{in }\mathbb{R}^n\times\{t= 0\}, \] \(1\leq p<\infty\), in the infinitely fast/infinitely slow diffusion limit \(p\to\infty\). Here \(\Delta_p\) is the \(p\)-Laplacian and \(g:\mathbb{R}^n\to\mathbb{R}\) denotes a compactly supported Lipschitz function with Lipschitz constant \(>1\). It is shown that \(u_p(\cdot,t)\) converges uniformly as \(p\to\infty\) to a time-independent function \(u\) such that \(|Du|_{L^\infty}\leq 1\). This can be interpreted as a model for the collapse of a sandpile from an initially unstable configuration. The function \(g\) determines the height of the sandpile at \(t=0\), \(u\) describes the stable profile after the motion ceases. It should be noted that in certain cases explicit solutions can be calculated.
scientific article; zbMATH DE number 1040628