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.