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:

- by the
*Fourier method*- one characterize the hyperbolicity of (hyper.25) in terms of the algebraic properties of the pseudodifferential symbol, ; - alternatively, we can also work
directly in physical space with the
*energy method*. For example, if we assume that*P*(*x*,*t*,*D*) is*semi-bounded*, i.e., if

then we have hyperbolicity (1.1.18).

__: 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.
__

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Thu Jan 22 19:07:34 PST 1998