A strongly hyperbolic system with homogeneous initial and boundary conditions on an infinite cylinder is considered. For the case of a finite cylinder see the paper of one of the author Nguyen Manh Hung
[Sb. Math. 190, No. 7, 1035–1058 (1999); translation from Mat. Sb. 190, No. 7, 103–126 (1999; Zbl 0962.35036
)]. Under some specific conditions for the coefficients, one proves the existence and uniqueness of a generalized solution. The proof is based on Galerkin approximations and uses an extension of the Garding inequality.