Consider the quasilinear hyperbolic system \[(1) \{ {gathered} v_t - u_x = 0, \\ u_t + p(v,s)_x = - α u, α > 0, p_v < 0{ for }v > 0, \\ s_t = 0 \\ {gathered} .\] with initial data \[ (2) u(x,0) = u_0 (x), v(x,0) = v + v_o (x), s(x,0) = s + s_0 (x), \] where v > 0, v and s are constants, (u₀ (x),v₀ (x)) ∈ C¹ with a compact support, and s₀ (x) ∈ C² with a compact support. It is proved in this paper that there exists a globally defined classical solution for the Cauchy problem if the C¹-norm of (u₀ (x),v₀ (x)) and the C²-norm of s₀ (x) are small.
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Hsiao et al. (1996) studied this question.