On the Cauchy Problem for Hyperbolic Operators with Non-Regular Coefficients

Abstract

In this work we collect some new results on the well posedness of the Cauchy problem for a class of strictly hyperbolic operators. Let T > 0. We are concerned with the equation

$$u{}_{tt} - \sum\limits_{i,j = 1}^n {{a_{ij}}} \left( t \right){u_{xixj}} + \sum\limits_{i = 1}^n {{b_i}} \left( t \right){u_{xi}} + c\left( t \right)u = 0{\kern 1pt} in\left[ {0,T} \right] \times {\mathbb{R}^n},{\kern 1pt}$$

(1.1)

with initial data

$$u\left( {0,x} \right) = {u_0}\left( x \right),{\kern 1pt} {u_t}\left( {0,x} \right) = {u_1}\left( x \right){\kern 1pt} in{\mathbb{R}^n},$$

(1.2)

where (a _ij ) is a real symmetric matrix such that

$$a\left( {t,\xi } \right) = \sum\limits_{i,j = 1}^n {{a_{ij}}} \left( t \right){\xi _i}{\xi _j}/{\left| \xi \right|^2} \geqslant {\lambda _0} > 0,$$

(1.3)

for all t and for all ξ =≠ 0, and the coefficients b _i and c are measurable and bounded.

To the Professor J. Leray