A simple construction of a parametrix for a regular hyperbolic operator
Let \(P=D^ 2_ t-\phi (t,x)A(t,x,D_ x)+B(t,x,D_ t,D_ x)\) with \(\phi \in C^{\infty}({\mathbb{R}}\times {\mathbb{R}}^ N)\), \(\phi\geq 0\) for \(t\geq 0\), \(\partial \phi /\partial t\neq 0\) where \(\phi =0\). \(A(t,x,D_ x)=\sum^{N}_{i,j=1}a_{ij}(t,x)D_ iD_ j\) is an elliptic differential operator and B is a differential operator of order one. The author describes the construction of a local parametrix for the Cauchy problem for P on \(t\geq 0\) and x near to \(x_ 0\). Moreover he gives the tools to have a local parametrix for \(Pu=f.\) The techniques used here are introduced in two earlier papers of the author [Ann. Mat. Pura Appl., IV. Ser. 140, 285-299 (1985; Zbl 0592.35097) and 146, 311-336 (1987; Zbl 0637.35020)].
scientific article; zbMATH DE number 4175558