Decay of Almost Periodic Solutions of Anisotropic Degenerate Parabolic-Hyperbolic Equations

摘要

We prove the well-posedness and decay of Besicovitch almost periodicsolutions for nonlinear degenerate anisotropic hyperbolic-parabolic equations.The decay property is proven for the case where the diffusion term is given bya 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 thestrong 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 givesthe continuity in time of the solution with values in$L_{\operatorname{loc}}^1(\mathbb{R}^d)$. So far, for the decay property, weneed also to impose that the bounded Besicovitch almost periodic initialfunction can be approximated in the Besicovitch norm by almost periodicfunctions whose $\varepsilon$-inclusion intervals $l_\varepsilon$ satisfy$l_\varepsilon/|\log \varepsilon|^{1/2}\to 0$ as $\varepsilon\to 0$. Thisincludes, in particular, generalized limit periodic functions, that is, limitsin the Besicovitch norm of purely periodic functions.