In this paper we prove the existence and uniqueness of weak solutions of the mixed problem for the nonlinear hyperbolic-parabolic equation ( K 1 ( x , t ) u ′ ) ′ + K 2 ( x , t ) u ′ + A ( t ) u + F ( u ) = f with null Dirichlet boundary conditions and zero initial data, where F ( s ) is a continuous function such that s F ( s ) ≥ 0 , ∀ s ∈ R and { A ( t ) ; t ≥ 0 } is a family of operators of L ( H 0 1 ( Ω ) ; H − 1 ( Ω ) ) . For the existence we apply the Faedo-Galerkin method with an unusual a priori estimate and a result of W. A. Strauss. Uniqueness is proved only for some particular classes of functions F .