Sustained Oscillations in Hyperbolic–Parabolic Systems

We construct examples of oscillating solutions with persistent oscillations for various hyperbolic-parabolic systems with singular diffusion matrices that appear in mechanics. These include an example for the equations of nonlinear viscoelasticity of Kelvin–Voigt type with stored energy that violates rank-one convexity, which amounts to a time-dependent variant of twinning solutions. We also present an example pertaining to the system of gas dynamics with thermal effects for a viscous, adiabatic gas. Finally, we show an example for the compressible Navier–Stokes system in one-space dimension with nonmonotone pressure function. We also study the existence of oscillating solutions for linear hyperbolic-parabolic systems with singular diffusion matrices.