Singular perturbations of first-order hyperbolic systems with stiff sourceterms

This work develops a singular perturbation theory for initial-value problem s of nonlinear first-order hyperbolic systems with stiff source terms in se veral space variables. It is observed that under reasonable assumptions, ma ny equations of classical physics of that type admit a structural stability condition. This condition is equivalent to the well-known subcharacteristi c condition for one-dimensional 2 x 2-systems and the well-known time-like condition for one-dimensional scalar second-order hyperbolic equations with a small positive parameter multiplying the highest derivatives. Under this : stability condition, we construct formal asymptotic approximations of the initial-layer solution to the nonlinear problem. Furthermore, assuming som e regularity of the solutions to the limiting inner problem and the reduced problem, we prove the existence of classical solutions in the uniform time interval where the reduced problem has a smooth solution and justify the v alidity of the formal approximations in any fixed compact subset of the uni form time interval. The stability condition seems to be a key to problems o f this hind and can be easily verified. Moreover, this presentation unifies and improves earlier works for some specific equations. (C) 1999 Academic Press.