## Abstract

Consider the Cauchy problem for the 3-D linear wave equation ∂^{2}_{t} U - c^{2}ΔU = 0 with radial initial data U(0, x) = Φ(x) = φ(|x|), U_{t}(0, x) = Ψ(x) = ψ (|x|). A standard result states that U belongs to C([0, T];H^{s}(ℝ^{3})) whenever (Φ, Ψ) ∈ H^{s} × H^{s-1}(ℝ^{3}). In this article we are interested in the question of how U can be realized as a limit of solutions to initial-boundary value problems on the exterior of vanishing balls B_{ε} about the origin. We note that, as the solutions we compare are defined on different domains, the answer is not an immediate consequence of H^{s} well-posedness for the wave equation. We show how explicit solution formulae yield convergence and optimal reg- ularity for the Cauchy solution via exterior solutions, when the latter are extended continuously as constants on B_{ε} at each time. We establish that for s = 2 the solution U can be realized as an H^{2}-limit (uniformly in time) of exterior solutions on ℝ^{3} \ B_{ε} satisfying vanishing Neumann conditions along |x| = ε, as ε ↓ 0. Similarly for s = 1: U is then an H^{1}-limit of exterior solutions satisfying vanishing Dirichlet conditions along |x| = ε.

Original language | English (US) |
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Article number | 266 |

Journal | Electronic Journal of Differential Equations |

Volume | 2016 |

State | Published - Sep 30 2016 |

## All Science Journal Classification (ASJC) codes

- Analysis

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