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In this paper we prove the C∞-well posedness of the Cauchy problem for quasilinear hyperbolic equations of second order with coefficients non-Lipschitz in t ∈ [0, T ] and smooth in x ∈ R. 0. Introduction. In this paper we consider Cauchy problems for quasi-linear hyperbolic equations with coefficients non-Lipschitz in the time variable and smooth in spatial variables. Our goal is to prove C∞-well posedness for the Cauchy problem (CP1)    Pu[u] = utt − n ∑ j,k=1 ajk(t, x;u, ut,∇u)uxjxk + ρ(t, x;u, ut,∇u) = 0 for (t, x) ∈ (0, T )× R, u = φ(x), ut = ψ(x) at t = 0. (0.1) The paper [2] is devoted to the study of Cauchy problems for second order hyperbolic equations with coefficients depending on the time variable of the form utt − n ∑ j,k=1 ajk(t)uxjxk + n ∑ j=1 bj(t)uxj + c(t)u = 0, (0.2) Research is supported by DFG No. 446 JAP 17/1/01. 2000 Mathematics Subject Classification: Primary 35L80; Secondary 35L15. The paper is in final form and no version of it will be published elsewhere.

C∞-well posedness of the Cauchy problem for quasi-linear hyperbolic equations with coefficients non-Lipschitz in time and smooth in space

C^∞-well posedness of the Cauchy problem for quasi-linear hyperbolic equations with coefficients non-Lipschitz in time and smooth in space