Some results on the well-posedness for second order linear equations

Abstract

We investigate the Cauchy problem for second order hyperbolic equations of complete form, and we prove an extension of a classical result of Oleĭnik [10] concerning the well-posedness for equations in which are absent the terms with mixed time-space derivatives. Then, in space dimension $n=1$, we compare our results with those in [8] for equations with analytic coefficients, and those of [7] and [11] for homogeneous equations with coefficients depending only either on $t$ or on $x$. Moreover we exhibit, in space dimension $n\ge 2$, an equation of the form \begin{equation*} u_{tt} - \sum_{i,j=1}^{n} (a_{ij}(t,x)u_{x_{j}})_{x_{i}} = 0{,} \quad\text{with}\quad \sum a_{ij} \xi_{i}\xi_{j} \ge 0, \end{equation*} where the coefficients are analytic functions, for which the Cauchy problem is ill-posed. Finally, we present a sufficient condition for the well-posedness of $2 \times 2$ systems.