The Evans function and stability criteria for degenerate viscous shock waves
Peter Howard K. Zumbrun
Discrete & Continuous Dynamical Systems - A 2004, 10(4): 837-855 doi: 10.3934/dcds.2004.10.837
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.
keywords: degeneracy Conservation laws stability Evans function.
Spectral analysis for transition front solutions in Cahn-Hilliard systems
Peter Howard Bongsuk Kwon
Discrete & Continuous Dynamical Systems - A 2012, 32(1): 125-166 doi: 10.3934/dcds.2012.32.125
We consider the spectrum associated with the linear operator obtained when a Cahn--Hilliard system on $\mathbb{R}$ is linearized about a transition wave solution. In many cases it's possible to show that the only non-negative eigenvalue is $\lambda = 0$, and so stability depends entirely on the nature of this neutral eigenvalue. In such cases, we identify a stability condition based on an appropriate Evans function, and we verify this condition under strong structural conditions on our equations. More generally, we discuss and implement a straightforward numerical check of our condition, valid under mild structural conditions.
keywords: Evans function. transition fronts stability Cahn-Hilliard systems

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