On the decay of almost periodic solutions of anisotropic degenerate parabolic-hyperbolic equations

Abstract

We prove the well-posedness and decay of Besicovitch almost periodic solutions for nonlinear degenerate anisotropic hyperbolic-parabolic equations. The decay property is proven for the case where the diffusion term is given by a non-degenerate nonlinear $d’’\times d’’$ diffusion matrix and the complementary $d’$ components of flux-function form a non-degenerate flux in ${\mathbb{R}}^{d’}$, with $d’+d’’=d$. For this special case we also prove that the strong trace property at the initial time holds, which allows, in particular, to require the assumption of the initial data only in a weak sense, and gives the continuity in time of the solution with values in $L_{\mathrm{loc}}^1({\mathbb{R}}^d)$). So far, for the decay property, we need also to require that the bounded Besicovitch almost periodic initial function can be approximated in the Besicovitch norm by almost periodic functions whose $ϵ$-inclusion intervals $l_{ϵ}$ satisfy $l_{ϵ}|\log ϵ{|}^1/2\to 0$ as $ϵ\to 0$. This includes, in particular, generalized limit periodic functions, that is, limits in the Besicovitch norm of purely periodic functions.