On a class of second order Fuchsian hyperbolic equations (Q1100628)

Let P be of the form: \[ (1)\quad P=-tD^ 2_ t+A(t,x,D_ y)+a(t,x)D_ t+b(t,x)D_ y+c(t,x), \] where: (i) the principal part of P; \(-tD\) \(2_ t+A(t,x,D_ y)\) is strictly hyperbolic for \(t>0.\) (ii) A(t,x,\(\cdot)\), b(t,x) and \(c(t,x)\in C^{\infty}(R_ t\times M)\), where M is a \(C^{\infty}\)-manifold without boundary. The aim of this paper is to solve the Cauchy problem: \[ (2)\quad Pu(t,x)=f(t,x),\quad t>0;\quad u(0,x)=g(x),\quad x\in M, \] together with obtaining the propagation of singularities of solution. The case of the homogeneous equation: f(t,x)\(\equiv 0\) the author mentions the works of \textit{A. Bove}, \textit{J. E. Lewis} and \textit{C. Parenti} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12, 1-42 (1985; Zbl 0593.35050), Hokkaido Math. J. 14, 175-248 (1985; Zbl 0582.35074) and Math. Ann. 273, 553-571 (1986; Zbl 0567.35057)]. The main theorem is stated in Theorem 1 in which the Fuchs condition: \[ (F)\quad -ia(0,x)\not\in \{0,-1,-2,-3,...\}\text{ for every } x\in M, \] is essential for \(WF(u|_{t>0})\subseteq WF(f|_{t>0})\) being valid.