Well... that's just great
This system is not trivial to solve -- at least in general. Start by scaling and then define , to obtain , where . By saying this is hyperbolic, I presume you mean the (constant!) matrix has no real eigenvalues. Then, prescribing differentiable Cauchy data might not mean there is a solution at all! In fact, has to be at least analytic, and I have no idea on obtaining an actual solution.
For the sake of argument, suppose...
- The matrix of the system has real and distinct eigenvalues. Then the Cauchy problem (P)&(C) has always a solution of the form
where we have to pick the constants involved so that the integrals converge and is truly a solution, the latter part being algebraic manipulations.
- The matrix of the system is self adjoint. Then, for the Cauchy data (C) being a tempered distribution, there is a unique solution for (P)&(C). In fact, a stronger result on regularity is true: This solution must necessarily be
Hope this helps.