Asymptotic estimates of solutions of \(u_ t - \frac12\Delta u =-|\nabla u|\) in \(\mathbb{R}_ + \times \mathbb{R}^ d,\quad d\geq 2\) (Q676192)
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scientific article; zbMATH DE number 992046
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Asymptotic estimates of solutions of \(u_ t - \frac12\Delta u =-|\nabla u|\) in \(\mathbb{R}_ + \times \mathbb{R}^ d,\quad d\geq 2\) |
scientific article; zbMATH DE number 992046 |
Statements
Asymptotic estimates of solutions of \(u_ t - \frac12\Delta u =-|\nabla u|\) in \(\mathbb{R}_ + \times \mathbb{R}^ d,\quad d\geq 2\) (English)
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24 June 1997
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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.
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\(d\)-dimensional Bessel process
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probabilistic representation
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