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International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 31, Pages 1623-1632

A characteristic initial value problem for a strictly hyperbolic system

Department of Mathematics, Western Kentucky University, Bowling Green 42101-3576, KY, USA

Received 6 August 2003

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Consider the system Autt+Cuxx=f(x,t), (x,t)T for u(x,t) in 2, where A and C are real constant 2×2 matrices, and f is a continuous function in 2. We assume that detC0 and that the system is strictly hyperbolic in the sense that there are four distinct characteristic curves Γi, i=1,,4, in xt-plane whose gradients (ξ1i,ξ2i) satisfy det[Aξ1i2+Cξ1i2]=0. We allow the characteristics of the system to be given by dt/dx=±1 and dt/dx=±r, r(0,1). Under special conditions on the boundaries of the region T={(x,t)t1,(1+r+t)/rx(1+rt)/r}, we will show that the system has a unique C2 solution in T.