Solution of the Hadamard problem for a class of hyperbolic equations with variable coefficients (Q1842698)

In [ibid. 25, No. 5, 43-54 (1990); transl. from ibid. 25, No. 5m 462-473 (1990; Zbl 0771.35029)] the author studied the strictly hyperbolic equation \[ u_{tt}- u_{xx}- \sum_{i,j=1}^ n d^{ij} (t- x)u_{y_ i y_ j}+ c(t-x, y)u=0, \qquad (t,x,y)\in \Omega \subseteq \mathbb{R}^{n+2}, \tag{1} \] where \(n\) is even and the coefficients are sufficiently smooth functions. An algorithm was provided for the construction of functions \(c(t-x, y)\) for which (1) satisfies the Huygens principle in the ``narrow sense'' in Hadamard's terminology. The constructions were examined in detail on model equation \[ L_ n (u)\equiv u_{tt}- u_{xx}- \sum_{i=1}^ n a^ i (t-x) u_{y_ i y_ i}+ a^ l (t-x) b(y) u=0. \tag{2} \] The present article proves that the equations described in (loc. cit.) exhaust all equations of type (2) that satisfy the Huygens principle. This solves the Hadamard problem for equations of type (2).