Large-Time Behavior of Periodic Entropy Solutions to Anisotropic Degenerate Parabolic-Hyperbolic Equations

We are interested in the large-time behavior of periodic entropy solutions in $L^\infty$ to anisotropic degenerate parabolic-hyperbolic equations of second-order. Unlike the pure hyperbolic case, the nonlinear equation is no longer self-similar invariant and the diffusion term in the equation significantly affects the large-time behavior of solutions; thus the approach developed earlier based on the self-similar scaling does not directly apply. In this paper, we develop another approach for establishing the decay of periodic solutions for anisotropic degenerate parabolic-hyperbolic equations. The proof is based on the kinetic formulation of entropy solutions. It involves time translations and a monotonicity-in-time property of entropy solutions, and employs the advantages of the precise kinetic equation for the solutions in order to recognize the role of nonlinearity-diffusivity of the equation.