The Inhomogeneous Wave Equation with $L^p$ Data

Benjamin Foster

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arXiv · 1911.01584

The Inhomogeneous Wave Equation with $L^p$ Data

Benjamin Foster

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Abstract

We prove existence and uniqueness of $L^2$ solutions to the inhomogeneous wave equation on $\mathbb{R}^{n-1}\times\mathbb{R}$ under the assumption that the inhomogeneous data lies in $L^p(\mathbb{R}^n)$ for $p=2n/(n+4)$. 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 $p$. This extends work of Michael Goldberg, in which similar Fourier-analytic techniques were used to study the inhomogeneous Helmholtz equation.

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Benjamin Foster. 2019-11-05. The Inhomogeneous Wave Equation with $L^p$ Data. https://arxiv.org/abs/1911.01584

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