A coupled non-linear hyperbolic-sobolev system

Abstract

A boundary-initial value problem for a quasilinear hyperbolic system in one space variable is coupled to a boundary-initial value problem for a quasilinear equation of Sobolev type in two space variables of the form Mut(x, t)+L(t) u (x, t)=f(x, t, u(x, t)) where M and L(t) are second order elliptic spacial operators. The coupling occurs through one of the boundary conditions for the hyperbolic system and the source term in the equation of Sobolev type. Such a coupling can arise in the consideration of oil flowing in a fissured medium and out of that medium via a pipe. Barenblatt, Zheltov, and Kochina[2] have modeled flow in a fissured medium via a special case of the above equation. A local existence and uniqueness theorem is demonstrated. The proof involves the method of characteristics, some applications of results of R. Showalter and the contraction mapping theorem.