On Mixed Problems for Quasilinear Second-Order Systems

Abstract

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 $H^s\left(\overline{\Omega }\times \left[0,T\right]\right)$, with $s>d/2+3$, is proved in the case where $\Omega$ is a half-space of ${\Re }^d$. 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 $v\in H^s\left({\Re }^{d+1}\right)$. 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.