Asymptotic estimates of solutions of \(u_ t - \frac12\Delta u =-|\nabla u|\) in \(\mathbb{R}_ + \times \mathbb{R}^ d,\quad d\geq 2\)
The authors express the solution \(u_\mu\) of \(u_t -\frac{1}{2}\triangle u =-|\nabla u|\) in \(\mathbb{R}_+ \times \mathbb{R}^d, \;d\geq 2; \;u_\mu(0,.)=\mu\) by a functional of a \(d\)-dimensional Bessel process, where \(\mu\) is invariant by rotations and profiled. Using this result they estimate the asymptotic behaviour of \(|u_\mu (t,\cdot)|_1\) and \(|u_\mu (t,\cdot)|_\infty\) as \(t\) goes to infinity.
scientific article; zbMATH DE number 992046