The Evans function and stability criteria for degenerate viscous shock waves
The authors study stability of degenerate (or sonic) viscous shock waves for the system \[ u_t+f(u)_x=u_{xx}, \quad u,f\in {\mathbb R}^2. \] This problem reduces to the problem of spectrum stability for the linear operator found by linearizing the original system about the wave. To solve this problem the device of Evans functions is traditionally used. Results obtained through working with the Evans function have made critical use of some analiticity conditions. These conditions seem to fail in the case of degenerate waves. The authors surmount this obstacle by dividing the Evans function into two pieces: one analytic in a neighborhood of the origin, and one sufficiently small.
scientific article; zbMATH DE number 2113138