Singular limit of solutions of the \(p\)-Laplacian equation (Q5929080)

The aim of the article under review is to study the asymptotic behavior of the solution \(u_p(x,t)\) as \(p\to\infty\) to the Cauchy problem for the \(p\)-Laplacian equation NEWLINE\[NEWLINE \begin{aligned} & \frac{\partial u}{\partial t} = \text{div}(|\nabla u|^{p-2}\nabla u),\quad (x,t)\in \mathbb R^N\times (0,T),\\ & u(x,0) = f(x)\geq 0,\quad x\in\mathbb R^N. \end{aligned} NEWLINE\]NEWLINE The main result reads as following: Let \(f\in L^{\infty}(\mathbb R^N)\), \(|\nabla f|\in L^{\infty}(\mathbb R^N)\). Then there exists a function \(u_{\infty}\in C(\mathbb R^N)\), \(|\nabla u_{\infty}|\leq 1\) such that for any compact set \(G\subset Q_T\), \(\lim_{p\to\infty}u_p(x,t) = u_{\infty}(x)\) uniformly on \(G\) and \(\int_{\mathbb R^N}u_{\infty}(x) dx = \int_{\mathbb R^N}f(x)dx\).