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

Abstract 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.