On weak solutions of semilinear hyperbolic-parabolic equations

In this paper we prove the existence and uniqueness of weak solutions of the mixedproblem for the nonlinear hyperbolic-parabolic equation(K1(x,t)u′)′+K2(x,t)u′+A(t)u+F(u)=fwith null Dirichlet boundary conditions and zero initial data, where F(s) is a continuous function suchthat sF(s)≥0, ∀s∈R and {A(t);t≥0} is a family of operators of L(H01(Ω);H−1(Ω)). For theexistence we apply the Faedo-Galerkin method with an unusual a priori estimate and a result ofW. A. Strauss. Uniqueness is proved only for some particular classes of functions F