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• W2902390410 abstract "Geodesics escape is widely used to study the scattering of hyperbolic equations. However, there are few progresses except in a simply connected complete Riemannian manifold with nonpositive curvature. We propose a kind of complete Riemannian metrics in $mathbb{R}^n$, which is called as escape metrics. We expose the relationship between escape metrics and geodesics escape in $mathbb{R}^n$. Under the escape metric $g$, we prove that each geodesic of $(mathbb{R}^n,g)$ escapes, that is, $lim_{trightarrow +infty} |gamma (t)|=+infty$ for any $xin mathbb{R}^n$ and any unit-speed geodesic $gamma (t)$ starting at $x$. We also obtain the geodesics escape velocity and give the counterexample that if escape metrics are not satisfied, then there exists an unit-speed geodesic $gamma (t)$ such that $overline{lim}_{trightarrow +infty} |gamma (t)|<+infty$. In addition, we establish Morawetz multipliers in Riemannian geometry to derive dispersive estimates for the wave equation on an exterior domain of $mathbb{R}^n$ with an escape metric. More concretely, for radial solutions, the uniform decay rate of the local energy is independent of the parity of the dimension $n$. For general solutions, we prove the space-time estimation of the energy and uniform decay rate $t^{-1}$ of the local energy. It is worth pointing out that different from the assumption of an Euclidean metric at infinity in the existing studies, escape metrics are more general Riemannian metrics." @default.
• W2902390410 created "2018-12-11" @default.
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• W2902390410 date "2018-11-30" @default.
• W2902390410 modified "2023-09-27" @default.
• W2902390410 title "Escape Metrics and its Applications" @default.
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