On weak solutions of semilinear hyperbolic-parabolic equations

ABSTRACT. In this paper we prove the existence and uniqueness of weak solutions of the mixed problem for the nonlinear hyperbolic-parabolic equation (K (x, t)u’) ’ + K(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 sF(s)> _ O, Vs E R and {A(t);t> _ 0} is a family of operators of L(H(2);H-I(gt)) 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