International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 31, Pages 1623-1632
doi:10.1155/S0161171204308045
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 Nezam Iraniparast. 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.
Abstract
Consider the system Autt+Cuxx=f(x,t), (x,t)∈T for
u(x,t) in ℝ2, where A and C are real constant
2×2 matrices, and f is a continuous function in
ℝ2. We assume that detC≠0 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)≤t≤1,(−1+r+t)/r≤x≤(1+r−t)/r}, we will show that the system has a unique C2
solution in T.