Article

# A decay estimate for a wave equation with trapping and a complex potential

International Mathematics Research Notices (Impact Factor: 1.12). 07/2011; DOI: 10.1093/imrn/rnr237

Source: arXiv

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**ABSTRACT:**We study solutions of the decoupled Maxwell equations in the exterior region of a Schwarzschild black hole. In stationary regions, where the Schwarzschild coordinate $r$ ranges over $2M < r_1 < r < r_2$, we obtain a decay rate of $t^{-1}$ for all components of the Maxwell field. We use vector field methods and do not require a spherical harmonic decomposition. In outgoing regions, where the Regge-Wheeler tortoise coordinate is large, $r_*>\epsilon t$, we obtain decay for the null components with rates of $|\phi_+| \sim |\alpha| < C r^{-5/2}$, $|\phi_0| \sim |\rho| + |\sigma| < C r^{-2} |t-r_*|^{-1/2}$, and $|\phi_{-1}| \sim |\underline{\alpha}| < C r^{-1} |t-r_*|^{-1}$. Along the event horizon and in ingoing regions, where $r_*<0$, and when $t+r_*1$, all components (normalized with respect to an ingoing null basis) decay at a rate of $C \uout^{-1}$ with $\uout=t+r_*$ in the exterior region.Journal of Hyperbolic Differential Equations 11/2007; · 0.85 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The semilinear wave equation on the (outer) Schwarzschild manifold is studied. We prove local decay estimates for general (non-radial) data, deriving a-priori Morawetz type estimates.Advances in Differential Equations 11/2003; · 0.63 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The Schwarzschild and Reissner–Nordstrøm solutions to Einstein's equations describe space–times which contain spherically symmetric black holes. We consider solutions to the linear wave equation in the exterior of a fixed black hole space–time of this type. We show that for solutions with initial data which decay at infinity and at the bifurcation sphere, a weighted L6 norm in space decays like . This weight vanishes at the event horizon, but not at infinity. To obtain this control, we require only an ϵ loss of angular derivatives.Journal of Functional Analysis 01/2009; · 1.25 Impact Factor

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