The Inhomogeneous Wave Equation with $L^p$ Data

TL;DR

This paper establishes existence and uniqueness of solutions to the inhomogeneous wave equation with data in a specific $L^p$ space, extending previous Fourier-analytic methods to analyze singularities and sharpness of exponents.

Contribution

It introduces new $L^p$ conditions for the wave equation, including Fourier transform vanishing on a cone, and proves sharpness of the exponent $p$, extending Goldberg's work on related equations.

Findings

Existence and uniqueness of $L^2$ solutions under $L^p$ data conditions.

Identification of the critical exponent $p=2n/(n+4)$ for the problem.

Proof of the sharpness of the exponent $p$.

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.