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
continuous coefficients) may proceed in one of two ways.
Either, we freeze the
coefficients and Fourier analyze the corresponding constant
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.