The Cauchy problem for nonstrictly second order hyperbolic equations with nonregular coefficients

Abstract

In this paper we consider the Cauchy problem for the equation\(\partial ^2 u\left( {t,x} \right)/\partial t^2 = - \sum\nolimits_{j,k = 1}^n {D_j } \left( {a_{jk} \left( {t,x} \right)D_k u\left( {t,x} \right)} \right) + f\left( {t,x} \right)\), where the matrix {ajk(x)} is non-negative, and the first derivatives of the coefficients have a singularity of orderq≥3 att=T>0; under these assumptions, the Cauchy problem is well-posed in all Gevrey classes of indexs<q/(q−1).