On Mixed Problems for Quasilinear Second-Order Systems

The paper is devoted to the study of initial-boundary value problems for quasilinear second-order systems. Existence and uniqueness of the solution in the space H-s ((Omega) over bar x vertical bar 0, T vertical bar), with s > d/2 + 3, is proved in the case where Omega is a half-space of R-d. The proof of the main theorem relies on two preliminary results: existence of the solution to mixed problems for linear second-order systems with smooth coefficients, and existence of the solution to initial-boundary value problems for linear second-order operators whose coefficients depend on the variables x and t through a function upsilon is an element of H-s (Rd-1). By means of the results proved for linear operators, the well posedness of the mixed problem for the quasi-linear system is established by studying the convergence of a suitable iteration scheme.