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 {\sl structural stability condition}. This condition is equivalent to the well-known subcharacteristic condition for one-dimensional $2\times 2$-systems and the well-known time-like condition for scalar second-order hyperbolic equations with a small positive parameter multiplying the highest derivatives. Under this stability condition, we construct a formal asymptotic expansion for the solution to the nonlinear problems. Furthermore, assuming some regularity of the solutions to the limiting inner problems and the reduced problems, 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 expansion.
The stability condition seems to be a key to the problems of this kind and can be easily verified. Moreover, this presentation unifies and improves earlier works for some specific equations.
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