$L^p$-$L^q$ decay estimates for dissipative linear hyperbolic systems in 1D

Given $A,B\in M_n(\mathbb R)$, we consider the Cauchy problem for partially dissipative hyperbolic systems having the form \begin{equation*} \partial_{t}u+A\partial_{x}u+Bu=0, \end{equation*} with the aim of providing a detailed description of the large-time behavior. Sharp $L^p$-$L^q$ estimates are established for the distance between the solution to the system and a time-asymptotic profile, where the profile is the superposition of diffusion waves and exponentially decaying waves.