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Parabolic systems

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 tex2html_wrap_inline11109 if
For problems with this leads to the Gårding-Petrovski characterization of parabolicity of order tex2html_wrap_inline11111, requiring

: Generically we have tex2html_wrap_inline11113 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 tex2html_wrap_inline11115, the corresponding systems (para.8) is parabolic of order 2.

: tex2html_wrap_inline11117 is weakly parabolic of order two, yet it does not satisfy Petrovski parabolicity.

Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998