Fourier Analysis of Wave Equations

Partial Differential Equation is one of the major influential and useful subjects in Mathematical Sciences. In this thesis, I mainly use Fourier analysis and PDE methods to study the solution to an inhomogeneous wave equation on Euclidean spaces. I obtained the existence, uniqueness, and regularity for the solution. In the classical case, the datum involved is required to have up to C2 smoothness; my results treat the low regularity case and only require Hs smoothness with fractional order s. The main tools include Fourier transform, Duhamel principle, and the method of energy estimates. The results have potential applications to the solutions of nonlinear wave equations with low regularity datum, which have background and implications in Physics and Engineering.