Fourier transform (hyper.4b) to get the ODE
whose solution is
where 
is the Fourier transform of the initial data. Now, for
we find
put differently, we have
and hence (since the diagonal matrix inside the brackets on
the right is clearly unitary), the 
-norm
of 
is conserved in time,i.e.,
Summing over all modes and using Parseval's equality
we end up with energy conservation
as asserted.
We note that the only tool used in the energy method was the existence of
a positive symmetrizer for A, while the only tool used in the Fourier method
was the
real diagonalization of A; in fact the two are related,
for if 
,
then with 
we have
Energy conservation implies (in view of linearity) uniqueness, and serves as
a basic tool to prove existence. It will be taken as the definition of
hyperbolicity. It implies and is implied by the qualitative
properties (1)--(4) which opened our discussion on page 
.
We now turn
to consider general PDE's of the form
with 
-periodic boundary conditions and subject to prescribed initial
conditions,
u(x,0) = f(x) . Motivated by the example of the wave equation, we
make the definition of
:
We say that the system (1.1.17) is hyperbolic if the
following a priori energy
estimate holds:
As we shall see later on, this notion of hyperbolicity is equivalent with energy conservation ( -- measured with respect to an appropriate renormed weighted 'energy'), in analogy with what we have seen in the special case of the wave equation. Here are the basic facts concerning such systems.