Asymptotic and Viscous Stability of Large-amplitude Solutions of a Hyperbolic System Arising from Biology

In this paper, we study the qualitative behavior of the Cauchy problem of a hyperbolic model { pt −∇· (pq) = ∆p, x ∈ R, t > 0, qt −∇(ε|q| + p) = ε∆q, which is transformed from a singular chemotaxis system describing the effect of reinforced random walk in [17, 27]. When d = 1 and the initial data are prescribed around a constant ground state (p̄, 0) with p̄ ≥ 0, we prove the global asymptotic stability of constant ground states, and identify the explicit decay rate of solutions under very mild conditions on initial data. Moreover, we study the diffusion (viscous) limit of solutions as ε → 0 with convergence rates toward solutions of the non-diffusible (inviscid) problem. While the existence of global large solutions of the system in multi-dimensions remains an outstanding open question, we show that the model exhibits a strong parabolic smoothing effect, namely, solutions are spatially analytic for short time provided that the initial data belong to L(R) for any q > d ≥ 1. In fact, when d = 1, we obtain that the solution remains real analytic for all time.