The inhomogeneous wave equation with 𝐿^{𝑝} data,Proceedings of the American Mathematical Society

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The inhomogeneous wave equation with 𝐿^{𝑝} data
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2020-07-20 , DOI: 10.1090/proc/15123
Benjamin Foster

Abstract:We prove existence and uniqueness of

solutions to the inhomogeneous wave equation on $\mathbb{R}^{n}\times \mathbb{R}$ under the assumption that the inhomogeneous data lies in $L^p(\mathbb{R}^{n+1})$ for

and $n+1\ge 5$ . We also require the Fourier transform of the inhomogeneous data to vanish on an infinite cone where the solution could become singular. Subsequently, we show sharpness of the exponent

. This extends work of Michael Goldberg, in which similar Fourier-analytic techniques were used to study the inhomogeneous Helmholtz equation.

中文翻译：

具有𝐿^ {𝑝}数据的非均匀波动方程

摘要：我们基于不均匀数据位于for和的假设，证明

了不均匀波动方程解的存在性和唯一性。我们还需要对不均匀数据进行傅立叶变换，以使其在无限锥上消失，在无限锥上，解可能变得很奇异。随后，我们展示了指数的清晰度。这扩展了迈克尔·戈德堡的工作，其中使用了类似的傅里叶分析技术来研究非均质的亥姆霍兹方程。 $\ mathbb {R} ^ {n} \ times \ mathbb {R}$ $L ^ p（\ mathbb {R} ^ {n + 1}）$

$n + 1 \ ge 5$

更新日期：2020-09-01

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