Partial Differential Equations/The wave equation

Definition 9.?:

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

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