We want to extend our previous analysis to linear systems of the form
This is the motivation for the definition of hyperbolicity (1.1.18) in the context of constant coefficient problems: freeze the coefficients and assume the hyperbolicity of the constant coefficient problem(s), , uniformly for each ; then - in contrast to the notion of weak hyperbolicity, the variable coefficients problem is also hyperbolic. This result is based on the invariance of the notion of hyperbolicity under low-order perturbations.
As before the study of the variable coefficients problem can be carried out by one of two ways:
: The symmetric hyperbolic case :
we can rewrite such symmetric problems in the equivalent form
In this case the symmetry of the 's implies that is skew-adjoint, i.e., integration by parts gives
Therefore we have
and hence the semi-boundedness requirement (hyper.26) holds with . Consequently, if are symmetric (or at least symmetrizable) then the system (1.1.17) is hyperbolic.