International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 4, Pages 751-758
doi:10.1155/S0161171296001044

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 (K1(x,t)u′)′+K2(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)≥0, ∀s∈R and {A(t);t≥0} is a family of operators of L(H01(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.