A characteristic initial value problem for a strictly hyperbolic system (Q1774782)

Summary: Consider the system \(Au_{tt}+Cu_{xx}=f(x,t)\), \((x, t) \in T\) for \(u(x,t)\) in \(\mathbb{R}^2\), where \(A\) and \(C\) are real constant \(2 \times 2\) matrices, and \(f\) is a continuous function in \(\mathbb{R}^2\). We assume that \(\det C\neq 0\) and that the system is strictly hyperbolic in the sense that there are four distinct characteristic curves \(\Gamma_i\), \(i=1,\dotsc,4\), in \(xt\)-plane whose gradients \((\xi_{1i},\xi_{2i})\) satisfy \(\det[A\xi_{1i}^2+C\xi_{2i}^2]=0\). We allow the characteristics of the system to be given by \({dt}/{dx}=\pm 1\) and \({dt}/{dx}=\pm r\), \(r \in(0,1)\). Under special conditions on the boundaries of the region \(T = \{(x, t):0\leq t\leq 1,({-1+r+t})/{r}\leq x \leq({1+r-t})/{r}\}\), we will show that the system has a unique \(C^2\) solution in \(T\).