International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 29-32, Pages 1623-1632
doi:10.1155/S0161171204308045

A characteristic initial value problem for a strictly hyperbolic system

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.