Decay structure of two hyperbolic relaxation models with regularity-loss

The paper aims at investigating two types of decay structure for linear symmetric hyperbolic systems with non-symmetric relaxation. Precisely, the system is of the type $(p,q)$ if the real part of all eigenvalues admits an upper bound $-c|\xi|^{2p}/(1+|\xi|^2)^{q}$, where $c$ is a generic positive constant and $\xi$ is the frequency variable, and the system enjoys the regularity-loss property if $p<q$. It is well known that the standard type $(1,1)$ can be assured by the classical Kawashima-Shizuta condition. A new structural condition was introduced in \cite{UDK} to analyze the regularity-loss type $(1,2)$ system with non-symmetric relaxation. In the paper, we construct two more complex models of the regularity-loss type corresponding to $p=m-3$, $q=m-2$ and $p=(3m-10)/2$, $q=2(m-3)$, respectively, where $m$ denotes phase dimensions. The proof is based on the delicate Fourier energy method as well as the suitable linear combination of series of energy inequalities. Due to arbitrary higher dimensions, it is not obvious to capture the energy dissipation rate with respect to the degenerate components. Thus, for each model, the analysis always starts from the case of low phase dimensions in order to understand the basic dissipative structure in the general case, and in the mean time, we also give the explicit construction of the compensating symmetric matrix $K$ and skew-symmetric matrix $S$.