Singular Perturbations of First-Order Hyperbolic Systems with Stiff Source Terms

This work develops a singular perturbation theory for initial-value problems of nonlinear first-order hyperbolic systems with stiff source terms in several space variables. It is observed that under reasonable assumptions, many equations of classical physics of that type admit a structural stability condition. This condition is equivalent to the well-known subcharacteristic condition for one-dimensional 2×2-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 some 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 validity of the formal approximations in any fixed compact subset of the uniform time interval. The stability condition seems to be a key to problems of this kind and can be easily verified. Moreover, this presentation unifies and improves earlier works for some specific equations.