A range problem for homogeneous, hyperbolic partial differential equations (Q913120)

Let \(Pu=f\) be an hyperbolic equation on \({\mathbb{R}}^ n\times {\mathbb{R}}\), where \(P=P(\partial /\partial x_ 1,...,\partial /\partial x_ n,\partial /\partial_ t)\) is linear and homogeneous with constant coefficients. The author shows that f can be recovered from u in the following sense: \(\Omega \subset {\mathbb{R}}^ n\times {\mathbb{R}}\) open, convex and bounded, \(f\in L^ 2_ 0(\Omega)\). Suppose that \(u=u(x,t)=0\) for \(t\ll 0\) and that \(Pu=f\). Then for \(t>\sup \{t |\) \(\Omega \cap ({\mathbb{R}}^ n\times \{t\})\neq \emptyset \}\) (a representation of) f can be calculated from the Cauchy data for u on \({\mathbb{R}}^ n\times \{t\}\).