Nonsymmetrisable quasilinear hyperbolic systems (Q1324970)

The object of this paper is the Cauchy problem for a weakly hyperbolic, quasilinear \(3\times 3\) system, having constant rank and multiplicity of the characteristics. Making use of the Nash-Moser theorem and the techniques introduced by \textit{N. Iwasaki} for quasilinear effectively hyperbolic systems [J. Math. Kyoto Univ. 25, 727-743 (1985; Zbl 0613.35046)] together with the linear theory developed by \textit{J. Vaillant} [J. Math. Pures Appl., IX. Sér. 47, 1-40 (1968; Zbl 0159.386); C. R. Acad. Sci., Paris, Sér. I 310, No. 13, 865-868 (1990; Zbl 0708.35008)], the author proves the local well-posedness in \(C^ \infty\) for a special class of systems of the above type. An example of such a system is given by \[ u_ t= [\lambda(t,x)I+ \mu(t,x,u)A] u_ x+ f(t,x,u), \qquad t\geq 0, \quad x\in\mathbb{R}, \] where \(u= (u_ 1,u_ 2,u_ 3)\), \(f=(f_ 1, f_ 2,f_ 3)\), \(I\)= identity matrix, \(A= [a_{ij}]\) with \(a_{i,j}= 0\) if \((i,j)\neq (3,1)\) and \(a_{3,1}=1\), while \(\lambda\), \(\mu\), \(f_ i\) are real \(C^ \infty\) functions such that \(\mu(0,0,0) \neq 0\), \(\partial f_ 1/ \partial u_ 2 (0,0,0)\neq 0\), \(\partial f_ 1/\partial u_ 3= \lambda_ x/2\), \(\partial f_ 2/\partial u_ 3=0\). Further results on the same topics can be found in two recent joint notes of \textit{D. Gourdin} with \textit{E. Munoz} [C. R. Acad. Sci., Paris, Sér. I 316, No. 11, 1183-1186 (1993; Zbl 0781.35038); ibid., No. 12, 1283-1288 (1993; Zbl 0781.35035)] and in the `Thèse de Doctorat' of \textit{E. Munoz} [Univ. de Paris VI, Oct. 1993].