Perron's method for quasilinear hyperbolic systems, Part II

Abstract

In part I (P. Smith, Perron's method for quasilinear hyperbolic systems, part I, J. Math. Anal., in press) of this paper we defined a notion of viscosity solution (sub-(super-)solution) for these systems, proved a comparison principle for viscosity sub-and supersolutions. Here, in part II, we prove existence of viscosity solutions to the Cauchy problem, using a Perron-like method, for long time, and for all time.  2005 Elsevier Inc. All rights reserved.

Perron's method for quasilinear hyperbolic systems: Part I

“…This nice behavior at discontinuous points will allow us to simplify and generalize some of the proofs in the author's previous papers [1][2][3].…”

“…As in our previous papers [1][2][3], we define a notion of sub-and supersolution (and solution) for our P.D.E. system by using our technique of regularization.…”

“…In several previous papers [1][2][3] the author proved a comparison principle for a semilinear second order wave equation, defined a notion of viscosity solution for such equations, and showed that Perron's method extended to these hyperbolic equations to prove existence of viscosity solutions of the Cauchy problem for these equations.…”