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Abstract
It is well known that the stability of certain distinguished waves arising
in evolutionary PDE can be determined by the spectrum of the linear operator
found by linearizing the PDE about the wave. Indeed, work over the last
fifteen years has shown that spectral stability implies nonlinear
stability in a broad range of cases, including asymptotically constant
traveling waves in both reaction--diffusion equations and viscous conservation
laws. A critical step toward analyzing the spectrum of such operators
was taken in the late eighties by Alexander, Gardner, and Jones, whose
Evans function (generalizing earlier work of John W. Evans) serves
as a characteristic function for the above-mentioned operators. Thus far,
results obtained through working with the Evans function have made
critical use of the function's analyticity at the origin (or its
analyticity over an appropriate Riemann surface). In the case of
degenerate (or sonic) viscous shock waves, however, the Evans function
is certainly not analytic in a neighborhood of the origin, and does not
appear to admit analytic extension to a Riemann manifold. We surmount
this obstacle by dividing the Evans function (plus related objects)
into two pieces: one analytic in a neighborhood of the origin, and one
sufficiently small.
Mathematics Subject Classification: Primary: 35B35, 34E10, 35K12, 35P05.
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