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.