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

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Given A,B∈Rn×n, we consider the Cauchy problem for partially dissipative hyperbolic systems having the form∂tu+A∂xu+Bu=0, with the aim of providing a detailed description of the large-time behavior. Sharp Lp–Lq estimates, 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 1/2 faster than the diffusive decay rate (1/q−1/p)/2 while the exponential decaying waves are negligible for large time. In particular, under a symmetry property, it decays 1 faster than. The proof is based on a complex interpolation argument once Lr estimates for the fundamental solution to the system in the frequency space and Fourier multiplier estimates are accomplished.