Lp-Lq decay estimates for dissipative linear hyperbolic systems in 1D

Given A, B in R^{n×n}, we consider the Cauchy problem for partially dissipative hyperbolic systems having the formwith the aim of providing a detailed description of the large-time behavior. Sharp Lp–Lqestimates, for 1 ≤q≤p≤∞, 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. They show that the solution to the system decays to the diffusion waves 1/2faster than the diffusive decay rate (1/q−1/p)/2while the exponential decaying waves are negligible for large time. In particular, under a symmetry property, it decays 1faster than. The proof is based on a complex interpolation argument once Lrestimates for the fundamental solution to the system in the frequency space and Fourier multiplier estimates are accomplished.