The Inhomogeneous Wave Equation with L^p Data

2019/11/05 by Foster, Benjamin
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1911.01584

Abstract

We prove existence and uniqueness of L² solutions to the inhomogeneous wave equation on ℝ^n-1×ℝ under the assumption that the inhomogeneous data lies in L^p(ℝⁿ) 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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