Partial Differential Equations/The wave equation

Initial value problem

Definition 9.?:

Let $f,g\in {\mathcal {C}}(\mathbb {R} ^{d})$. The initial value problem for the wave equation is defined to be the problem to find a function $u:\mathbb {R} ^{d}\to \mathbb {R}$ such that

${\begin{cases}\end{cases}}$.

We will first solve the initial value problem in the case