Turning to general case,
we consider mth-order PDE's of the form,

We say that the system (para.8) is *weakly parabolic* of order
if

For problems with __ this leads to the
Gårding-Petrovski characterization of parabolicity of order ,
requiring
__

__
: Generically we have the order of
dissipation which is necessarily even.
__

__
__

__
The extension to problems with
(with Lipschitz
continuous coefficients) may proceed in one of two ways.
Either, we freeze the
coefficients and Fourier analyze the corresponding constant
coefficients problems;
or we may use the energy method, e.g., integration by parts shows that for
__

with , the corresponding systems (para.8) is parabolic of order 2.

__
__

__
: is weakly parabolic of
order two, yet it does not satisfy Petrovski parabolicity.
__

__
__

Thu Jan 22 19:07:34 PST 1998